From 7cf62f6a8aece479f5126e8bb95af168a4b2c372 Mon Sep 17 00:00:00 2001
From: autodeploy <autodeploy>
Date: Fri, 13 Dec 2024 19:27:52 +0000
Subject: [PATCH] Auto-deploy: rendered output

---
 .nojekyll                                     |    0
 docs                                          |    1 -
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 docs/book.js                                  |  697 +++++
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 docs/css/chrome.css                           |  587 ++++
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 86 files changed, 21271 insertions(+), 1 deletion(-)
 create mode 100644 .nojekyll
 delete mode 120000 docs
 create mode 100644 docs/.nojekyll
 create mode 100644 docs/404.html
 create mode 100644 docs/FontAwesome/css/font-awesome.css
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diff --git a/.nojekyll b/.nojekyll
new file mode 100644
index 0000000..e69de29
diff --git a/docs b/docs
deleted file mode 120000
index b2dbf19..0000000
--- a/docs
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diff --git a/docs/.nojekyll b/docs/.nojekyll
new file mode 100644
index 0000000..f173110
--- /dev/null
+++ b/docs/.nojekyll
@@ -0,0 +1 @@
+This file makes sure that Github Pages doesn't process mdBook's output.
diff --git a/docs/404.html b/docs/404.html
new file mode 100644
index 0000000..59af5df
--- /dev/null
+++ b/docs/404.html
@@ -0,0 +1,217 @@
+<!DOCTYPE HTML>
+<html lang="en" class="light" dir="ltr">
+    <head>
+        <!-- Book generated using mdBook -->
+        <meta charset="UTF-8">
+        <title>Page not found - Zcash Trailing Finality Layer - v0.1.0</title>
+        <base href="/">
+
+
+        <!-- Custom HTML head -->
+        
+        <meta name="description" content="">
+        <meta name="viewport" content="width=device-width, initial-scale=1">
+        <meta name="theme-color" content="#ffffff">
+
+        <link rel="icon" href="favicon.svg">
+        <link rel="shortcut icon" href="favicon.png">
+        <link rel="stylesheet" href="css/variables.css">
+        <link rel="stylesheet" href="css/general.css">
+        <link rel="stylesheet" href="css/chrome.css">
+        <link rel="stylesheet" href="css/print.css" media="print">
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+        <!-- Fonts -->
+        <link rel="stylesheet" href="FontAwesome/css/font-awesome.css">
+        <link rel="stylesheet" href="fonts/fonts.css">
+
+        <!-- Highlight.js Stylesheets -->
+        <link rel="stylesheet" href="highlight.css">
+        <link rel="stylesheet" href="tomorrow-night.css">
+        <link rel="stylesheet" href="ayu-highlight.css">
+
+        <!-- Custom theme stylesheets -->
+        <link rel="stylesheet" href="./css/mdbook-admonish.css">
+
+    </head>
+    <body class="sidebar-visible no-js">
+    <div id="body-container">
+        <!-- Provide site root to javascript -->
+        <script>
+            var path_to_root = "";
+            var default_theme = window.matchMedia("(prefers-color-scheme: dark)").matches ? "navy" : "light";
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+
+        <!-- Work around some values being stored in localStorage wrapped in quotes -->
+        <script>
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+                var theme = localStorage.getItem('mdbook-theme');
+                var sidebar = localStorage.getItem('mdbook-sidebar');
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+
+                if (sidebar.startsWith('"') && sidebar.endsWith('"')) {
+                    localStorage.setItem('mdbook-sidebar', sidebar.slice(1, sidebar.length - 1));
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+            } catch (e) { }
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+
+        <!-- Set the theme before any content is loaded, prevents flash -->
+        <script>
+            var theme;
+            try { theme = localStorage.getItem('mdbook-theme'); } catch(e) { }
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+            var html = document.querySelector('html');
+            html.classList.remove('light')
+            html.classList.add(theme);
+            var body = document.querySelector('body');
+            body.classList.remove('no-js')
+            body.classList.add('js');
+        </script>
+
+        <input type="checkbox" id="sidebar-toggle-anchor" class="hidden">
+
+        <!-- Hide / unhide sidebar before it is displayed -->
+        <script>
+            var body = document.querySelector('body');
+            var sidebar = null;
+            var sidebar_toggle = document.getElementById("sidebar-toggle-anchor");
+            if (document.body.clientWidth >= 1080) {
+                try { sidebar = localStorage.getItem('mdbook-sidebar'); } catch(e) { }
+                sidebar = sidebar || 'visible';
+            } else {
+                sidebar = 'hidden';
+            }
+            sidebar_toggle.checked = sidebar === 'visible';
+            body.classList.remove('sidebar-visible');
+            body.classList.add("sidebar-" + sidebar);
+        </script>
+
+        <nav id="sidebar" class="sidebar" aria-label="Table of contents">
+            <div class="sidebar-scrollbox">
+                <ol class="chapter"><li class="chapter-item expanded "><a href="introduction.html"><strong aria-hidden="true">1.</strong> Introduction</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="introduction/a-path-to-pos-zcash.html"><strong aria-hidden="true">1.1.</strong> A Path to Proof-of-Stake Zcash</a></li><li class="chapter-item expanded "><a href="introduction/trailing-finality-layer-in-a-nutshell.html"><strong aria-hidden="true">1.2.</strong> Trailing Finality Layer in a Nutshell</a></li><li class="chapter-item expanded "><a href="introduction/status-and-next-steps.html"><strong aria-hidden="true">1.3.</strong> Status and Next Steps</a></li><li class="chapter-item expanded "><a href="introduction/motivating-finality.html"><strong aria-hidden="true">1.4.</strong> Motivating Finality</a></li><li class="chapter-item expanded "><a href="introduction/get-involved.html"><strong aria-hidden="true">1.5.</strong> Get Involved</a></li></ol></li><li class="chapter-item expanded "><a href="terminology.html"><strong aria-hidden="true">2.</strong> Terminology</a></li><li class="chapter-item expanded "><a href="design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/design-at-a-glance.html"><strong aria-hidden="true">3.1.</strong> Design at a Glance</a></li><li class="chapter-item expanded "><a href="design/goals.html"><strong aria-hidden="true">3.2.</strong> Design Goals</a></li><li class="chapter-item expanded "><a href="design/crosslink.html"><strong aria-hidden="true">3.3.</strong> Crosslink</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/crosslink/the-arguments-for-bounded-availability-and-finality-overrides.html"><strong aria-hidden="true">3.3.1.</strong> The Arguments for Bounded Availability and Finality Overrides</a></li><li class="chapter-item expanded "><a href="design/crosslink/notes-on-snap-and-chat.html"><strong aria-hidden="true">3.3.2.</strong> Notes On Snap‑and‑Chat</a></li><li class="chapter-item expanded "><a href="design/crosslink/construction.html"><strong aria-hidden="true">3.3.3.</strong> The Crosslink 2 Construction</a></li><li class="chapter-item expanded "><a href="design/crosslink/security-analysis.html"><strong aria-hidden="true">3.3.4.</strong> Security Analysis of Crosslink 2</a></li><li class="chapter-item expanded "><a href="design/crosslink/questions.html"><strong aria-hidden="true">3.3.5.</strong> Questions About Crosslink</a></li><li class="chapter-item expanded "><a href="design/crosslink/potential-changes.html"><strong aria-hidden="true">3.3.6.</strong> Potential Changes to Crosslink</a></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">3.4.</strong> Model Code Architecture</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">3.5.</strong> Network Architecture</div></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">4.</strong> Protocol Specification</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">4.1.</strong> Abstract Protocol</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">4.1.1.</strong> Token Dynamics</div></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">4.2.</strong> Concrete Protocol</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">4.2.1.</strong> Token Dynamics</div></li></ol></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.</strong> Security</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.</strong> Abstract Analysis</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.1.</strong> Ebb-and-Flow analysis</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.2.</strong> Subprotocol Compromise Analysis</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.3.</strong> Model Code Architecture Analysis</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.4.</strong> Network Architecture Analysis</div></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.2.</strong> Concrete Analysis</div></li></ol></li><li class="chapter-item expanded "><a href="references.html"><strong aria-hidden="true">6.</strong> References</a></li><li class="chapter-item expanded "><a href="version-history.html"><strong aria-hidden="true">7.</strong> Version History</a></li></ol>
+            </div>
+            <div id="sidebar-resize-handle" class="sidebar-resize-handle"></div>
+        </nav>
+
+        <!-- Track and set sidebar scroll position -->
+        <script>
+            var sidebarScrollbox = document.querySelector('#sidebar .sidebar-scrollbox');
+            sidebarScrollbox.addEventListener('click', function(e) {
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+        <div id="page-wrapper" class="page-wrapper">
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+                        <label id="sidebar-toggle" class="icon-button" for="sidebar-toggle-anchor" title="Toggle Table of Contents" aria-label="Toggle Table of Contents" aria-controls="sidebar">
+                            <i class="fa fa-bars"></i>
+                        </label>
+                        <button id="theme-toggle" class="icon-button" type="button" title="Change theme" aria-label="Change theme" aria-haspopup="true" aria-expanded="false" aria-controls="theme-list">
+                            <i class="fa fa-paint-brush"></i>
+                        </button>
+                        <ul id="theme-list" class="theme-popup" aria-label="Themes" role="menu">
+                            <li role="none"><button role="menuitem" class="theme" id="light">Light</button></li>
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+                        <button id="search-toggle" class="icon-button" type="button" title="Search. (Shortkey: s)" aria-label="Toggle Searchbar" aria-expanded="false" aria-keyshortcuts="S" aria-controls="searchbar">
+                            <i class="fa fa-search"></i>
+                        </button>
+                    </div>
+
+                    <h1 class="menu-title">Zcash Trailing Finality Layer - v0.1.0</h1>
+
+                    <div class="right-buttons">
+                        <a href="print.html" title="Print this book" aria-label="Print this book">
+                            <i id="print-button" class="fa fa-print"></i>
+                        </a>
+
+                    </div>
+                </div>
+
+                <div id="search-wrapper" class="hidden">
+                    <form id="searchbar-outer" class="searchbar-outer">
+                        <input type="search" id="searchbar" name="searchbar" placeholder="Search this book ..." aria-controls="searchresults-outer" aria-describedby="searchresults-header">
+                    </form>
+                    <div id="searchresults-outer" class="searchresults-outer hidden">
+                        <div id="searchresults-header" class="searchresults-header"></div>
+                        <ul id="searchresults">
+                        </ul>
+                    </div>
+                </div>
+
+                <!-- Apply ARIA attributes after the sidebar and the sidebar toggle button are added to the DOM -->
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+                    Array.from(document.querySelectorAll('#sidebar a')).forEach(function(link) {
+                        link.setAttribute('tabIndex', sidebar === 'visible' ? 0 : -1);
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+                </script>
+
+                <div id="content" class="content">
+                    <main>
+                        <h1 id="document-not-found-404"><a class="header" href="#document-not-found-404">Document not found (404)</a></h1>
+<p>This URL is invalid, sorry. Please use the navigation bar or search to continue.</p>
+
+                    </main>
+
+                    <nav class="nav-wrapper" aria-label="Page navigation">
+                        <!-- Mobile navigation buttons -->
+
+
+                        <div style="clear: both"></div>
+                    </nav>
+                </div>
+            </div>
+
+            <nav class="nav-wide-wrapper" aria-label="Page navigation">
+
+            </nav>
+
+        </div>
+
+
+
+
+        <script>
+            window.playground_copyable = true;
+        </script>
+
+
+        <script src="elasticlunr.min.js"></script>
+        <script src="mark.min.js"></script>
+        <script src="searcher.js"></script>
+
+        <script src="clipboard.min.js"></script>
+        <script src="highlight.js"></script>
+        <script src="book.js"></script>
+
+        <!-- Custom JS scripts -->
+
+
+    </div>
+    </body>
+</html>
diff --git a/docs/FontAwesome/css/font-awesome.css b/docs/FontAwesome/css/font-awesome.css
new file mode 100644
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--- /dev/null
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+/*!
+ *  Font Awesome 4.7.0 by @davegandy - http://fontawesome.io - @fontawesome
+ *  License - http://fontawesome.io/license (Font: SIL OFL 1.1, CSS: MIT License)
+ */@font-face{font-family:'FontAwesome';src:url('../fonts/fontawesome-webfont.eot?v=4.7.0');src:url('../fonts/fontawesome-webfont.eot?#iefix&v=4.7.0') format('embedded-opentype'),url('../fonts/fontawesome-webfont.woff2?v=4.7.0') format('woff2'),url('../fonts/fontawesome-webfont.woff?v=4.7.0') format('woff'),url('../fonts/fontawesome-webfont.ttf?v=4.7.0') format('truetype'),url('../fonts/fontawesome-webfont.svg?v=4.7.0#fontawesomeregular') format('svg');font-weight:normal;font-style:normal}.fa{display:inline-block;font:normal normal normal 14px/1 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new file mode 100644
index 0000000..855c845
--- /dev/null
+++ b/docs/FontAwesome/fonts/fontawesome-webfont.svg
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diff --git a/docs/FontAwesome/fonts/fontawesome-webfont.woff2 b/docs/FontAwesome/fonts/fontawesome-webfont.woff2
new file mode 100644
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diff --git a/docs/ayu-highlight.css b/docs/ayu-highlight.css
new file mode 100644
index 0000000..32c9432
--- /dev/null
+++ b/docs/ayu-highlight.css
@@ -0,0 +1,78 @@
+/*
+Based off of the Ayu theme
+Original by Dempfi (https://github.com/dempfi/ayu)
+*/
+
+.hljs {
+  display: block;
+  overflow-x: auto;
+  background: #191f26;
+  color: #e6e1cf;
+}
+
+.hljs-comment,
+.hljs-quote {
+  color: #5c6773;
+  font-style: italic;
+}
+
+.hljs-variable,
+.hljs-template-variable,
+.hljs-attribute,
+.hljs-attr,
+.hljs-regexp,
+.hljs-link,
+.hljs-selector-id,
+.hljs-selector-class {
+  color: #ff7733;
+}
+
+.hljs-number,
+.hljs-meta,
+.hljs-builtin-name,
+.hljs-literal,
+.hljs-type,
+.hljs-params {
+  color: #ffee99;
+}
+
+.hljs-string,
+.hljs-bullet {
+  color: #b8cc52;
+}
+
+.hljs-title,
+.hljs-built_in,
+.hljs-section {
+  color: #ffb454;
+}
+
+.hljs-keyword,
+.hljs-selector-tag,
+.hljs-symbol {
+  color: #ff7733;
+}
+
+.hljs-name {
+    color: #36a3d9;
+}
+
+.hljs-tag {
+    color: #00568d;
+}
+
+.hljs-emphasis {
+  font-style: italic;
+}
+
+.hljs-strong {
+  font-weight: bold;
+}
+
+.hljs-addition {
+  color: #91b362;
+}
+
+.hljs-deletion {
+  color: #d96c75;
+}
diff --git a/docs/book.js b/docs/book.js
new file mode 100644
index 0000000..aa12e7e
--- /dev/null
+++ b/docs/book.js
@@ -0,0 +1,697 @@
+"use strict";
+
+// Fix back button cache problem
+window.onunload = function () { };
+
+// Global variable, shared between modules
+function playground_text(playground, hidden = true) {
+    let code_block = playground.querySelector("code");
+
+    if (window.ace && code_block.classList.contains("editable")) {
+        let editor = window.ace.edit(code_block);
+        return editor.getValue();
+    } else if (hidden) {
+        return code_block.textContent;
+    } else {
+        return code_block.innerText;
+    }
+}
+
+(function codeSnippets() {
+    function fetch_with_timeout(url, options, timeout = 6000) {
+        return Promise.race([
+            fetch(url, options),
+            new Promise((_, reject) => setTimeout(() => reject(new Error('timeout')), timeout))
+        ]);
+    }
+
+    var playgrounds = Array.from(document.querySelectorAll(".playground"));
+    if (playgrounds.length > 0) {
+        fetch_with_timeout("https://play.rust-lang.org/meta/crates", {
+            headers: {
+                'Content-Type': "application/json",
+            },
+            method: 'POST',
+            mode: 'cors',
+        })
+        .then(response => response.json())
+        .then(response => {
+            // get list of crates available in the rust playground
+            let playground_crates = response.crates.map(item => item["id"]);
+            playgrounds.forEach(block => handle_crate_list_update(block, playground_crates));
+        });
+    }
+
+    function handle_crate_list_update(playground_block, playground_crates) {
+        // update the play buttons after receiving the response
+        update_play_button(playground_block, playground_crates);
+
+        // and install on change listener to dynamically update ACE editors
+        if (window.ace) {
+            let code_block = playground_block.querySelector("code");
+            if (code_block.classList.contains("editable")) {
+                let editor = window.ace.edit(code_block);
+                editor.addEventListener("change", function (e) {
+                    update_play_button(playground_block, playground_crates);
+                });
+                // add Ctrl-Enter command to execute rust code
+                editor.commands.addCommand({
+                    name: "run",
+                    bindKey: {
+                        win: "Ctrl-Enter",
+                        mac: "Ctrl-Enter"
+                    },
+                    exec: _editor => run_rust_code(playground_block)
+                });
+            }
+        }
+    }
+
+    // updates the visibility of play button based on `no_run` class and
+    // used crates vs ones available on https://play.rust-lang.org
+    function update_play_button(pre_block, playground_crates) {
+        var play_button = pre_block.querySelector(".play-button");
+
+        // skip if code is `no_run`
+        if (pre_block.querySelector('code').classList.contains("no_run")) {
+            play_button.classList.add("hidden");
+            return;
+        }
+
+        // get list of `extern crate`'s from snippet
+        var txt = playground_text(pre_block);
+        var re = /extern\s+crate\s+([a-zA-Z_0-9]+)\s*;/g;
+        var snippet_crates = [];
+        var item;
+        while (item = re.exec(txt)) {
+            snippet_crates.push(item[1]);
+        }
+
+        // check if all used crates are available on play.rust-lang.org
+        var all_available = snippet_crates.every(function (elem) {
+            return playground_crates.indexOf(elem) > -1;
+        });
+
+        if (all_available) {
+            play_button.classList.remove("hidden");
+        } else {
+            play_button.classList.add("hidden");
+        }
+    }
+
+    function run_rust_code(code_block) {
+        var result_block = code_block.querySelector(".result");
+        if (!result_block) {
+            result_block = document.createElement('code');
+            result_block.className = 'result hljs language-bash';
+
+            code_block.append(result_block);
+        }
+
+        let text = playground_text(code_block);
+        let classes = code_block.querySelector('code').classList;
+        let edition = "2015";
+        if(classes.contains("edition2018")) {
+            edition = "2018";
+        } else if(classes.contains("edition2021")) {
+            edition = "2021";
+        }
+        var params = {
+            version: "stable",
+            optimize: "0",
+            code: text,
+            edition: edition
+        };
+
+        if (text.indexOf("#![feature") !== -1) {
+            params.version = "nightly";
+        }
+
+        result_block.innerText = "Running...";
+
+        fetch_with_timeout("https://play.rust-lang.org/evaluate.json", {
+            headers: {
+                'Content-Type': "application/json",
+            },
+            method: 'POST',
+            mode: 'cors',
+            body: JSON.stringify(params)
+        })
+        .then(response => response.json())
+        .then(response => {
+            if (response.result.trim() === '') {
+                result_block.innerText = "No output";
+                result_block.classList.add("result-no-output");
+            } else {
+                result_block.innerText = response.result;
+                result_block.classList.remove("result-no-output");
+            }
+        })
+        .catch(error => result_block.innerText = "Playground Communication: " + error.message);
+    }
+
+    // Syntax highlighting Configuration
+    hljs.configure({
+        tabReplace: '    ', // 4 spaces
+        languages: [],      // Languages used for auto-detection
+    });
+
+    let code_nodes = Array
+        .from(document.querySelectorAll('code'))
+        // Don't highlight `inline code` blocks in headers.
+        .filter(function (node) {return !node.parentElement.classList.contains("header"); });
+
+    if (window.ace) {
+        // language-rust class needs to be removed for editable
+        // blocks or highlightjs will capture events
+        code_nodes
+            .filter(function (node) {return node.classList.contains("editable"); })
+            .forEach(function (block) { block.classList.remove('language-rust'); });
+
+        code_nodes
+            .filter(function (node) {return !node.classList.contains("editable"); })
+            .forEach(function (block) { hljs.highlightBlock(block); });
+    } else {
+        code_nodes.forEach(function (block) { hljs.highlightBlock(block); });
+    }
+
+    // Adding the hljs class gives code blocks the color css
+    // even if highlighting doesn't apply
+    code_nodes.forEach(function (block) { block.classList.add('hljs'); });
+
+    Array.from(document.querySelectorAll("code.hljs")).forEach(function (block) {
+
+        var lines = Array.from(block.querySelectorAll('.boring'));
+        // If no lines were hidden, return
+        if (!lines.length) { return; }
+        block.classList.add("hide-boring");
+
+        var buttons = document.createElement('div');
+        buttons.className = 'buttons';
+        buttons.innerHTML = "<button class=\"fa fa-eye\" title=\"Show hidden lines\" aria-label=\"Show hidden lines\"></button>";
+
+        // add expand button
+        var pre_block = block.parentNode;
+        pre_block.insertBefore(buttons, pre_block.firstChild);
+
+        pre_block.querySelector('.buttons').addEventListener('click', function (e) {
+            if (e.target.classList.contains('fa-eye')) {
+                e.target.classList.remove('fa-eye');
+                e.target.classList.add('fa-eye-slash');
+                e.target.title = 'Hide lines';
+                e.target.setAttribute('aria-label', e.target.title);
+
+                block.classList.remove('hide-boring');
+            } else if (e.target.classList.contains('fa-eye-slash')) {
+                e.target.classList.remove('fa-eye-slash');
+                e.target.classList.add('fa-eye');
+                e.target.title = 'Show hidden lines';
+                e.target.setAttribute('aria-label', e.target.title);
+
+                block.classList.add('hide-boring');
+            }
+        });
+    });
+
+    if (window.playground_copyable) {
+        Array.from(document.querySelectorAll('pre code')).forEach(function (block) {
+            var pre_block = block.parentNode;
+            if (!pre_block.classList.contains('playground')) {
+                var buttons = pre_block.querySelector(".buttons");
+                if (!buttons) {
+                    buttons = document.createElement('div');
+                    buttons.className = 'buttons';
+                    pre_block.insertBefore(buttons, pre_block.firstChild);
+                }
+
+                var clipButton = document.createElement('button');
+                clipButton.className = 'fa fa-copy clip-button';
+                clipButton.title = 'Copy to clipboard';
+                clipButton.setAttribute('aria-label', clipButton.title);
+                clipButton.innerHTML = '<i class=\"tooltiptext\"></i>';
+
+                buttons.insertBefore(clipButton, buttons.firstChild);
+            }
+        });
+    }
+
+    // Process playground code blocks
+    Array.from(document.querySelectorAll(".playground")).forEach(function (pre_block) {
+        // Add play button
+        var buttons = pre_block.querySelector(".buttons");
+        if (!buttons) {
+            buttons = document.createElement('div');
+            buttons.className = 'buttons';
+            pre_block.insertBefore(buttons, pre_block.firstChild);
+        }
+
+        var runCodeButton = document.createElement('button');
+        runCodeButton.className = 'fa fa-play play-button';
+        runCodeButton.hidden = true;
+        runCodeButton.title = 'Run this code';
+        runCodeButton.setAttribute('aria-label', runCodeButton.title);
+
+        buttons.insertBefore(runCodeButton, buttons.firstChild);
+        runCodeButton.addEventListener('click', function (e) {
+            run_rust_code(pre_block);
+        });
+
+        if (window.playground_copyable) {
+            var copyCodeClipboardButton = document.createElement('button');
+            copyCodeClipboardButton.className = 'fa fa-copy clip-button';
+            copyCodeClipboardButton.innerHTML = '<i class="tooltiptext"></i>';
+            copyCodeClipboardButton.title = 'Copy to clipboard';
+            copyCodeClipboardButton.setAttribute('aria-label', copyCodeClipboardButton.title);
+
+            buttons.insertBefore(copyCodeClipboardButton, buttons.firstChild);
+        }
+
+        let code_block = pre_block.querySelector("code");
+        if (window.ace && code_block.classList.contains("editable")) {
+            var undoChangesButton = document.createElement('button');
+            undoChangesButton.className = 'fa fa-history reset-button';
+            undoChangesButton.title = 'Undo changes';
+            undoChangesButton.setAttribute('aria-label', undoChangesButton.title);
+
+            buttons.insertBefore(undoChangesButton, buttons.firstChild);
+
+            undoChangesButton.addEventListener('click', function () {
+                let editor = window.ace.edit(code_block);
+                editor.setValue(editor.originalCode);
+                editor.clearSelection();
+            });
+        }
+    });
+})();
+
+(function themes() {
+    var html = document.querySelector('html');
+    var themeToggleButton = document.getElementById('theme-toggle');
+    var themePopup = document.getElementById('theme-list');
+    var themeColorMetaTag = document.querySelector('meta[name="theme-color"]');
+    var stylesheets = {
+        ayuHighlight: document.querySelector("[href$='ayu-highlight.css']"),
+        tomorrowNight: document.querySelector("[href$='tomorrow-night.css']"),
+        highlight: document.querySelector("[href$='highlight.css']"),
+    };
+
+    function showThemes() {
+        themePopup.style.display = 'block';
+        themeToggleButton.setAttribute('aria-expanded', true);
+        themePopup.querySelector("button#" + get_theme()).focus();
+    }
+
+    function updateThemeSelected() {
+        themePopup.querySelectorAll('.theme-selected').forEach(function (el) {
+            el.classList.remove('theme-selected');
+        });
+        themePopup.querySelector("button#" + get_theme()).classList.add('theme-selected');
+    }
+
+    function hideThemes() {
+        themePopup.style.display = 'none';
+        themeToggleButton.setAttribute('aria-expanded', false);
+        themeToggleButton.focus();
+    }
+
+    function get_theme() {
+        var theme;
+        try { theme = localStorage.getItem('mdbook-theme'); } catch (e) { }
+        if (theme === null || theme === undefined) {
+            return default_theme;
+        } else {
+            return theme;
+        }
+    }
+
+    function set_theme(theme, store = true) {
+        let ace_theme;
+
+        if (theme == 'coal' || theme == 'navy') {
+            stylesheets.ayuHighlight.disabled = true;
+            stylesheets.tomorrowNight.disabled = false;
+            stylesheets.highlight.disabled = true;
+
+            ace_theme = "ace/theme/tomorrow_night";
+        } else if (theme == 'ayu') {
+            stylesheets.ayuHighlight.disabled = false;
+            stylesheets.tomorrowNight.disabled = true;
+            stylesheets.highlight.disabled = true;
+            ace_theme = "ace/theme/tomorrow_night";
+        } else {
+            stylesheets.ayuHighlight.disabled = true;
+            stylesheets.tomorrowNight.disabled = true;
+            stylesheets.highlight.disabled = false;
+            ace_theme = "ace/theme/dawn";
+        }
+
+        setTimeout(function () {
+            themeColorMetaTag.content = getComputedStyle(document.documentElement).backgroundColor;
+        }, 1);
+
+        if (window.ace && window.editors) {
+            window.editors.forEach(function (editor) {
+                editor.setTheme(ace_theme);
+            });
+        }
+
+        var previousTheme = get_theme();
+
+        if (store) {
+            try { localStorage.setItem('mdbook-theme', theme); } catch (e) { }
+        }
+
+        html.classList.remove(previousTheme);
+        html.classList.add(theme);
+        updateThemeSelected();
+    }
+
+    // Set theme
+    var theme = get_theme();
+
+    set_theme(theme, false);
+
+    themeToggleButton.addEventListener('click', function () {
+        if (themePopup.style.display === 'block') {
+            hideThemes();
+        } else {
+            showThemes();
+        }
+    });
+
+    themePopup.addEventListener('click', function (e) {
+        var theme;
+        if (e.target.className === "theme") {
+            theme = e.target.id;
+        } else if (e.target.parentElement.className === "theme") {
+            theme = e.target.parentElement.id;
+        } else {
+            return;
+        }
+        set_theme(theme);
+    });
+
+    themePopup.addEventListener('focusout', function(e) {
+        // e.relatedTarget is null in Safari and Firefox on macOS (see workaround below)
+        if (!!e.relatedTarget && !themeToggleButton.contains(e.relatedTarget) && !themePopup.contains(e.relatedTarget)) {
+            hideThemes();
+        }
+    });
+
+    // Should not be needed, but it works around an issue on macOS & iOS: https://github.com/rust-lang/mdBook/issues/628
+    document.addEventListener('click', function(e) {
+        if (themePopup.style.display === 'block' && !themeToggleButton.contains(e.target) && !themePopup.contains(e.target)) {
+            hideThemes();
+        }
+    });
+
+    document.addEventListener('keydown', function (e) {
+        if (e.altKey || e.ctrlKey || e.metaKey || e.shiftKey) { return; }
+        if (!themePopup.contains(e.target)) { return; }
+
+        switch (e.key) {
+            case 'Escape':
+                e.preventDefault();
+                hideThemes();
+                break;
+            case 'ArrowUp':
+                e.preventDefault();
+                var li = document.activeElement.parentElement;
+                if (li && li.previousElementSibling) {
+                    li.previousElementSibling.querySelector('button').focus();
+                }
+                break;
+            case 'ArrowDown':
+                e.preventDefault();
+                var li = document.activeElement.parentElement;
+                if (li && li.nextElementSibling) {
+                    li.nextElementSibling.querySelector('button').focus();
+                }
+                break;
+            case 'Home':
+                e.preventDefault();
+                themePopup.querySelector('li:first-child button').focus();
+                break;
+            case 'End':
+                e.preventDefault();
+                themePopup.querySelector('li:last-child button').focus();
+                break;
+        }
+    });
+})();
+
+(function sidebar() {
+    var body = document.querySelector("body");
+    var sidebar = document.getElementById("sidebar");
+    var sidebarLinks = document.querySelectorAll('#sidebar a');
+    var sidebarToggleButton = document.getElementById("sidebar-toggle");
+    var sidebarResizeHandle = document.getElementById("sidebar-resize-handle");
+    var firstContact = null;
+
+    function showSidebar() {
+        body.classList.remove('sidebar-hidden')
+        body.classList.add('sidebar-visible');
+        Array.from(sidebarLinks).forEach(function (link) {
+            link.setAttribute('tabIndex', 0);
+        });
+        sidebarToggleButton.setAttribute('aria-expanded', true);
+        sidebar.setAttribute('aria-hidden', false);
+        try { localStorage.setItem('mdbook-sidebar', 'visible'); } catch (e) { }
+    }
+
+
+    var sidebarAnchorToggles = document.querySelectorAll('#sidebar a.toggle');
+
+    function toggleSection(ev) {
+        ev.currentTarget.parentElement.classList.toggle('expanded');
+    }
+
+    Array.from(sidebarAnchorToggles).forEach(function (el) {
+        el.addEventListener('click', toggleSection);
+    });
+
+    function hideSidebar() {
+        body.classList.remove('sidebar-visible')
+        body.classList.add('sidebar-hidden');
+        Array.from(sidebarLinks).forEach(function (link) {
+            link.setAttribute('tabIndex', -1);
+        });
+        sidebarToggleButton.setAttribute('aria-expanded', false);
+        sidebar.setAttribute('aria-hidden', true);
+        try { localStorage.setItem('mdbook-sidebar', 'hidden'); } catch (e) { }
+    }
+
+    // Toggle sidebar
+    sidebarToggleButton.addEventListener('click', function sidebarToggle() {
+        if (body.classList.contains("sidebar-hidden")) {
+            var current_width = parseInt(
+                document.documentElement.style.getPropertyValue('--sidebar-width'), 10);
+            if (current_width < 150) {
+                document.documentElement.style.setProperty('--sidebar-width', '150px');
+            }
+            showSidebar();
+        } else if (body.classList.contains("sidebar-visible")) {
+            hideSidebar();
+        } else {
+            if (getComputedStyle(sidebar)['transform'] === 'none') {
+                hideSidebar();
+            } else {
+                showSidebar();
+            }
+        }
+    });
+
+    sidebarResizeHandle.addEventListener('mousedown', initResize, false);
+
+    function initResize(e) {
+        window.addEventListener('mousemove', resize, false);
+        window.addEventListener('mouseup', stopResize, false);
+        body.classList.add('sidebar-resizing');
+    }
+    function resize(e) {
+        var pos = (e.clientX - sidebar.offsetLeft);
+        if (pos < 20) {
+            hideSidebar();
+        } else {
+            if (body.classList.contains("sidebar-hidden")) {
+                showSidebar();
+            }
+            pos = Math.min(pos, window.innerWidth - 100);
+            document.documentElement.style.setProperty('--sidebar-width', pos + 'px');
+        }
+    }
+    //on mouseup remove windows functions mousemove & mouseup
+    function stopResize(e) {
+        body.classList.remove('sidebar-resizing');
+        window.removeEventListener('mousemove', resize, false);
+        window.removeEventListener('mouseup', stopResize, false);
+    }
+
+    document.addEventListener('touchstart', function (e) {
+        firstContact = {
+            x: e.touches[0].clientX,
+            time: Date.now()
+        };
+    }, { passive: true });
+
+    document.addEventListener('touchmove', function (e) {
+        if (!firstContact)
+            return;
+
+        var curX = e.touches[0].clientX;
+        var xDiff = curX - firstContact.x,
+            tDiff = Date.now() - firstContact.time;
+
+        if (tDiff < 250 && Math.abs(xDiff) >= 150) {
+            if (xDiff >= 0 && firstContact.x < Math.min(document.body.clientWidth * 0.25, 300))
+                showSidebar();
+            else if (xDiff < 0 && curX < 300)
+                hideSidebar();
+
+            firstContact = null;
+        }
+    }, { passive: true });
+})();
+
+(function chapterNavigation() {
+    document.addEventListener('keydown', function (e) {
+        if (e.altKey || e.ctrlKey || e.metaKey || e.shiftKey) { return; }
+        if (window.search && window.search.hasFocus()) { return; }
+        var html = document.querySelector('html');
+
+        function next() {
+            var nextButton = document.querySelector('.nav-chapters.next');
+            if (nextButton) {
+                window.location.href = nextButton.href;
+            }
+        }
+        function prev() {
+            var previousButton = document.querySelector('.nav-chapters.previous');
+            if (previousButton) {
+                window.location.href = previousButton.href;
+            }
+        }
+        switch (e.key) {
+            case 'ArrowRight':
+                e.preventDefault();
+                if (html.dir == 'rtl') {
+                    prev();
+                } else {
+                    next();
+                }
+                break;
+            case 'ArrowLeft':
+                e.preventDefault();
+                if (html.dir == 'rtl') {
+                    next();
+                } else {
+                    prev();
+                }
+                break;
+        }
+    });
+})();
+
+(function clipboard() {
+    var clipButtons = document.querySelectorAll('.clip-button');
+
+    function hideTooltip(elem) {
+        elem.firstChild.innerText = "";
+        elem.className = 'fa fa-copy clip-button';
+    }
+
+    function showTooltip(elem, msg) {
+        elem.firstChild.innerText = msg;
+        elem.className = 'fa fa-copy tooltipped';
+    }
+
+    var clipboardSnippets = new ClipboardJS('.clip-button', {
+        text: function (trigger) {
+            hideTooltip(trigger);
+            let playground = trigger.closest("pre");
+            return playground_text(playground, false);
+        }
+    });
+
+    Array.from(clipButtons).forEach(function (clipButton) {
+        clipButton.addEventListener('mouseout', function (e) {
+            hideTooltip(e.currentTarget);
+        });
+    });
+
+    clipboardSnippets.on('success', function (e) {
+        e.clearSelection();
+        showTooltip(e.trigger, "Copied!");
+    });
+
+    clipboardSnippets.on('error', function (e) {
+        showTooltip(e.trigger, "Clipboard error!");
+    });
+})();
+
+(function scrollToTop () {
+    var menuTitle = document.querySelector('.menu-title');
+
+    menuTitle.addEventListener('click', function () {
+        document.scrollingElement.scrollTo({ top: 0, behavior: 'smooth' });
+    });
+})();
+
+(function controllMenu() {
+    var menu = document.getElementById('menu-bar');
+
+    (function controllPosition() {
+        var scrollTop = document.scrollingElement.scrollTop;
+        var prevScrollTop = scrollTop;
+        var minMenuY = -menu.clientHeight - 50;
+        // When the script loads, the page can be at any scroll (e.g. if you reforesh it).
+        menu.style.top = scrollTop + 'px';
+        // Same as parseInt(menu.style.top.slice(0, -2), but faster
+        var topCache = menu.style.top.slice(0, -2);
+        menu.classList.remove('sticky');
+        var stickyCache = false; // Same as menu.classList.contains('sticky'), but faster
+        document.addEventListener('scroll', function () {
+            scrollTop = Math.max(document.scrollingElement.scrollTop, 0);
+            // `null` means that it doesn't need to be updated
+            var nextSticky = null;
+            var nextTop = null;
+            var scrollDown = scrollTop > prevScrollTop;
+            var menuPosAbsoluteY = topCache - scrollTop;
+            if (scrollDown) {
+                nextSticky = false;
+                if (menuPosAbsoluteY > 0) {
+                    nextTop = prevScrollTop;
+                }
+            } else {
+                if (menuPosAbsoluteY > 0) {
+                    nextSticky = true;
+                } else if (menuPosAbsoluteY < minMenuY) {
+                    nextTop = prevScrollTop + minMenuY;
+                }
+            }
+            if (nextSticky === true && stickyCache === false) {
+                menu.classList.add('sticky');
+                stickyCache = true;
+            } else if (nextSticky === false && stickyCache === true) {
+                menu.classList.remove('sticky');
+                stickyCache = false;
+            }
+            if (nextTop !== null) {
+                menu.style.top = nextTop + 'px';
+                topCache = nextTop;
+            }
+            prevScrollTop = scrollTop;
+        }, { passive: true });
+    })();
+    (function controllBorder() {
+        function updateBorder() {
+            if (menu.offsetTop === 0) {
+                menu.classList.remove('bordered');
+            } else {
+                menu.classList.add('bordered');
+            }
+        }
+        updateBorder();
+        document.addEventListener('scroll', updateBorder, { passive: true });
+    })();
+})();
diff --git a/docs/clipboard.min.js b/docs/clipboard.min.js
new file mode 100644
index 0000000..02c549e
--- /dev/null
+++ b/docs/clipboard.min.js
@@ -0,0 +1,7 @@
+/*!
+ * clipboard.js v2.0.4
+ * https://zenorocha.github.io/clipboard.js
+ * 
+ * Licensed MIT © Zeno Rocha
+ */
+!function(t,e){"object"==typeof exports&&"object"==typeof module?module.exports=e():"function"==typeof define&&define.amd?define([],e):"object"==typeof exports?exports.ClipboardJS=e():t.ClipboardJS=e()}(this,function(){return function(n){var o={};function r(t){if(o[t])return o[t].exports;var e=o[t]={i:t,l:!1,exports:{}};return n[t].call(e.exports,e,e.exports,r),e.l=!0,e.exports}return r.m=n,r.c=o,r.d=function(t,e,n){r.o(t,e)||Object.defineProperty(t,e,{enumerable:!0,get:n})},r.r=function(t){"undefined"!=typeof Symbol&&Symbol.toStringTag&&Object.defineProperty(t,Symbol.toStringTag,{value:"Module"}),Object.defineProperty(t,"__esModule",{value:!0})},r.t=function(e,t){if(1&t&&(e=r(e)),8&t)return e;if(4&t&&"object"==typeof e&&e&&e.__esModule)return e;var n=Object.create(null);if(r.r(n),Object.defineProperty(n,"default",{enumerable:!0,value:e}),2&t&&"string"!=typeof e)for(var o in e)r.d(n,o,function(t){return e[t]}.bind(null,o));return n},r.n=function(t){var e=t&&t.__esModule?function(){return t.default}:function(){return t};return r.d(e,"a",e),e},r.o=function(t,e){return Object.prototype.hasOwnProperty.call(t,e)},r.p="",r(r.s=0)}([function(t,e,n){"use strict";var r="function"==typeof Symbol&&"symbol"==typeof Symbol.iterator?function(t){return typeof t}:function(t){return t&&"function"==typeof Symbol&&t.constructor===Symbol&&t!==Symbol.prototype?"symbol":typeof t},i=function(){function o(t,e){for(var n=0;n<e.length;n++){var o=e[n];o.enumerable=o.enumerable||!1,o.configurable=!0,"value"in o&&(o.writable=!0),Object.defineProperty(t,o.key,o)}}return function(t,e,n){return e&&o(t.prototype,e),n&&o(t,n),t}}(),a=o(n(1)),c=o(n(3)),u=o(n(4));function o(t){return t&&t.__esModule?t:{default:t}}var l=function(t){function o(t,e){!function(t,e){if(!(t instanceof e))throw new TypeError("Cannot call a class as a function")}(this,o);var n=function(t,e){if(!t)throw new ReferenceError("this hasn't been initialised - super() hasn't been 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e.onClick(t)})}},{key:"onClick",value:function(t){var e=t.delegateTarget||t.currentTarget;this.clipboardAction&&(this.clipboardAction=null),this.clipboardAction=new a.default({action:this.action(e),target:this.target(e),text:this.text(e),container:this.container,trigger:e,emitter:this})}},{key:"defaultAction",value:function(t){return s("action",t)}},{key:"defaultTarget",value:function(t){var e=s("target",t);if(e)return document.querySelector(e)}},{key:"defaultText",value:function(t){return s("text",t)}},{key:"destroy",value:function(){this.listener.destroy(),this.clipboardAction&&(this.clipboardAction.destroy(),this.clipboardAction=null)}}],[{key:"isSupported",value:function(){var t=0<arguments.length&&void 0!==arguments[0]?arguments[0]:["copy","cut"],e="string"==typeof t?[t]:t,n=!!document.queryCommandSupported;return e.forEach(function(t){n=n&&!!document.queryCommandSupported(t)}),n}}]),o}();function s(t,e){var n="data-clipboard-"+t;if(e.hasAttribute(n))return e.getAttribute(n)}t.exports=l},function(t,e,n){"use strict";var o,r="function"==typeof Symbol&&"symbol"==typeof Symbol.iterator?function(t){return typeof t}:function(t){return t&&"function"==typeof Symbol&&t.constructor===Symbol&&t!==Symbol.prototype?"symbol":typeof t},i=function(){function o(t,e){for(var n=0;n<e.length;n++){var o=e[n];o.enumerable=o.enumerable||!1,o.configurable=!0,"value"in o&&(o.writable=!0),Object.defineProperty(t,o.key,o)}}return function(t,e,n){return e&&o(t.prototype,e),n&&o(t,n),t}}(),a=n(2),c=(o=a)&&o.__esModule?o:{default:o};var u=function(){function e(t){!function(t,e){if(!(t instanceof e))throw new TypeError("Cannot call a class as a function")}(this,e),this.resolveOptions(t),this.initSelection()}return i(e,[{key:"resolveOptions",value:function(){var t=0<arguments.length&&void 0!==arguments[0]?arguments[0]:{};this.action=t.action,this.container=t.container,this.emitter=t.emitter,this.target=t.target,this.text=t.text,this.trigger=t.trigger,this.selectedText=""}},{key:"initSelection",value:function(){this.text?this.selectFake():this.target&&this.selectTarget()}},{key:"selectFake",value:function(){var t=this,e="rtl"==document.documentElement.getAttribute("dir");this.removeFake(),this.fakeHandlerCallback=function(){return t.removeFake()},this.fakeHandler=this.container.addEventListener("click",this.fakeHandlerCallback)||!0,this.fakeElem=document.createElement("textarea"),this.fakeElem.style.fontSize="12pt",this.fakeElem.style.border="0",this.fakeElem.style.padding="0",this.fakeElem.style.margin="0",this.fakeElem.style.position="absolute",this.fakeElem.style[e?"right":"left"]="-9999px";var n=window.pageYOffset||document.documentElement.scrollTop;this.fakeElem.style.top=n+"px",this.fakeElem.setAttribute("readonly",""),this.fakeElem.value=this.text,this.container.appendChild(this.fakeElem),this.selectedText=(0,c.default)(this.fakeElem),this.copyText()}},{key:"removeFake",value:function(){this.fakeHandler&&(this.container.removeEventListener("click",this.fakeHandlerCallback),this.fakeHandler=null,this.fakeHandlerCallback=null),this.fakeElem&&(this.container.removeChild(this.fakeElem),this.fakeElem=null)}},{key:"selectTarget",value:function(){this.selectedText=(0,c.default)(this.target),this.copyText()}},{key:"copyText",value:function(){var e=void 0;try{e=document.execCommand(this.action)}catch(t){e=!1}this.handleResult(e)}},{key:"handleResult",value:function(t){this.emitter.emit(t?"success":"error",{action:this.action,text:this.selectedText,trigger:this.trigger,clearSelection:this.clearSelection.bind(this)})}},{key:"clearSelection",value:function(){this.trigger&&this.trigger.focus(),window.getSelection().removeAllRanges()}},{key:"destroy",value:function(){this.removeFake()}},{key:"action",set:function(){var t=0<arguments.length&&void 0!==arguments[0]?arguments[0]:"copy";if(this._action=t,"copy"!==this._action&&"cut"!==this._action)throw new Error('Invalid "action" value, use either "copy" or "cut"')},get:function(){return this._action}},{key:"target",set:function(t){if(void 0!==t){if(!t||"object"!==(void 0===t?"undefined":r(t))||1!==t.nodeType)throw new Error('Invalid "target" value, use a valid Element');if("copy"===this.action&&t.hasAttribute("disabled"))throw new Error('Invalid "target" attribute. Please use "readonly" instead of "disabled" attribute');if("cut"===this.action&&(t.hasAttribute("readonly")||t.hasAttribute("disabled")))throw new Error('Invalid "target" attribute. You can\'t cut text from elements with "readonly" or "disabled" attributes');this._target=t}},get:function(){return this._target}}]),e}();t.exports=u},function(t,e){t.exports=function(t){var e;if("SELECT"===t.nodeName)t.focus(),e=t.value;else if("INPUT"===t.nodeName||"TEXTAREA"===t.nodeName){var n=t.hasAttribute("readonly");n||t.setAttribute("readonly",""),t.select(),t.setSelectionRange(0,t.value.length),n||t.removeAttribute("readonly"),e=t.value}else{t.hasAttribute("contenteditable")&&t.focus();var o=window.getSelection(),r=document.createRange();r.selectNodeContents(t),o.removeAllRanges(),o.addRange(r),e=o.toString()}return e}},function(t,e){function n(){}n.prototype={on:function(t,e,n){var o=this.e||(this.e={});return(o[t]||(o[t]=[])).push({fn:e,ctx:n}),this},once:function(t,e,n){var o=this;function r(){o.off(t,r),e.apply(n,arguments)}return r._=e,this.on(t,r,n)},emit:function(t){for(var e=[].slice.call(arguments,1),n=((this.e||(this.e={}))[t]||[]).slice(),o=0,r=n.length;o<r;o++)n[o].fn.apply(n[o].ctx,e);return this},off:function(t,e){var n=this.e||(this.e={}),o=n[t],r=[];if(o&&e)for(var i=0,a=o.length;i<a;i++)o[i].fn!==e&&o[i].fn._!==e&&r.push(o[i]);return r.length?n[t]=r:delete n[t],this}},t.exports=n},function(t,e,n){var d=n(5),h=n(6);t.exports=function(t,e,n){if(!t&&!e&&!n)throw new Error("Missing required arguments");if(!d.string(e))throw new TypeError("Second argument must be a String");if(!d.fn(n))throw new TypeError("Third argument must be a Function");if(d.node(t))return s=e,f=n,(l=t).addEventListener(s,f),{destroy:function(){l.removeEventListener(s,f)}};if(d.nodeList(t))return a=t,c=e,u=n,Array.prototype.forEach.call(a,function(t){t.addEventListener(c,u)}),{destroy:function(){Array.prototype.forEach.call(a,function(t){t.removeEventListener(c,u)})}};if(d.string(t))return o=t,r=e,i=n,h(document.body,o,r,i);throw new TypeError("First argument must be a String, HTMLElement, HTMLCollection, or NodeList");var o,r,i,a,c,u,l,s,f}},function(t,n){n.node=function(t){return void 0!==t&&t instanceof HTMLElement&&1===t.nodeType},n.nodeList=function(t){var e=Object.prototype.toString.call(t);return void 0!==t&&("[object NodeList]"===e||"[object HTMLCollection]"===e)&&"length"in t&&(0===t.length||n.node(t[0]))},n.string=function(t){return"string"==typeof t||t instanceof String},n.fn=function(t){return"[object Function]"===Object.prototype.toString.call(t)}},function(t,e,n){var a=n(7);function i(t,e,n,o,r){var i=function(e,n,t,o){return function(t){t.delegateTarget=a(t.target,n),t.delegateTarget&&o.call(e,t)}}.apply(this,arguments);return t.addEventListener(n,i,r),{destroy:function(){t.removeEventListener(n,i,r)}}}t.exports=function(t,e,n,o,r){return"function"==typeof t.addEventListener?i.apply(null,arguments):"function"==typeof n?i.bind(null,document).apply(null,arguments):("string"==typeof t&&(t=document.querySelectorAll(t)),Array.prototype.map.call(t,function(t){return i(t,e,n,o,r)}))}},function(t,e){if("undefined"!=typeof Element&&!Element.prototype.matches){var n=Element.prototype;n.matches=n.matchesSelector||n.mozMatchesSelector||n.msMatchesSelector||n.oMatchesSelector||n.webkitMatchesSelector}t.exports=function(t,e){for(;t&&9!==t.nodeType;){if("function"==typeof t.matches&&t.matches(e))return t;t=t.parentNode}}}])});
\ No newline at end of file
diff --git a/docs/css/chrome.css b/docs/css/chrome.css
new file mode 100644
index 0000000..2314f7a
--- /dev/null
+++ b/docs/css/chrome.css
@@ -0,0 +1,587 @@
+/* CSS for UI elements (a.k.a. chrome) */
+
+@import 'variables.css';
+
+html {
+    scrollbar-color: var(--scrollbar) var(--bg);
+}
+#searchresults a,
+.content a:link,
+a:visited,
+a > .hljs {
+    color: var(--links);
+}
+
+/*
+    body-container is necessary because mobile browsers don't seem to like
+    overflow-x on the body tag when there is a <meta name="viewport"> tag.
+*/
+#body-container {
+    /*
+        This is used when the sidebar pushes the body content off the side of
+        the screen on small screens. Without it, dragging on mobile Safari
+        will want to reposition the viewport in a weird way.
+    */
+    overflow-x: clip;
+}
+
+/* Menu Bar */
+
+#menu-bar,
+#menu-bar-hover-placeholder {
+    z-index: 101;
+    margin: auto calc(0px - var(--page-padding));
+}
+#menu-bar {
+    position: relative;
+    display: flex;
+    flex-wrap: wrap;
+    background-color: var(--bg);
+    border-block-end-color: var(--bg);
+    border-block-end-width: 1px;
+    border-block-end-style: solid;
+}
+#menu-bar.sticky,
+.js #menu-bar-hover-placeholder:hover + #menu-bar,
+.js #menu-bar:hover,
+.js.sidebar-visible #menu-bar {
+    position: -webkit-sticky;
+    position: sticky;
+    top: 0 !important;
+}
+#menu-bar-hover-placeholder {
+    position: sticky;
+    position: -webkit-sticky;
+    top: 0;
+    height: var(--menu-bar-height);
+}
+#menu-bar.bordered {
+    border-block-end-color: var(--table-border-color);
+}
+#menu-bar i, #menu-bar .icon-button {
+    position: relative;
+    padding: 0 8px;
+    z-index: 10;
+    line-height: var(--menu-bar-height);
+    cursor: pointer;
+    transition: color 0.5s;
+}
+@media only screen and (max-width: 420px) {
+    #menu-bar i, #menu-bar .icon-button {
+        padding: 0 5px;
+    }
+}
+
+.icon-button {
+    border: none;
+    background: none;
+    padding: 0;
+    color: inherit;
+}
+.icon-button i {
+    margin: 0;
+}
+
+.right-buttons {
+    margin: 0 15px;
+}
+.right-buttons a {
+    text-decoration: none;
+}
+
+.left-buttons {
+    display: flex;
+    margin: 0 5px;
+}
+.no-js .left-buttons button {
+    display: none;
+}
+
+.menu-title {
+    display: inline-block;
+    font-weight: 200;
+    font-size: 2.4rem;
+    line-height: var(--menu-bar-height);
+    text-align: center;
+    margin: 0;
+    flex: 1;
+    white-space: nowrap;
+    overflow: hidden;
+    text-overflow: ellipsis;
+}
+.js .menu-title {
+    cursor: pointer;
+}
+
+.menu-bar,
+.menu-bar:visited,
+.nav-chapters,
+.nav-chapters:visited,
+.mobile-nav-chapters,
+.mobile-nav-chapters:visited,
+.menu-bar .icon-button,
+.menu-bar a i {
+    color: var(--icons);
+}
+
+.menu-bar i:hover,
+.menu-bar .icon-button:hover,
+.nav-chapters:hover,
+.mobile-nav-chapters i:hover {
+    color: var(--icons-hover);
+}
+
+/* Nav Icons */
+
+.nav-chapters {
+    font-size: 2.5em;
+    text-align: center;
+    text-decoration: none;
+
+    position: fixed;
+    top: 0;
+    bottom: 0;
+    margin: 0;
+    max-width: 150px;
+    min-width: 90px;
+
+    display: flex;
+    justify-content: center;
+    align-content: center;
+    flex-direction: column;
+
+    transition: color 0.5s, background-color 0.5s;
+}
+
+.nav-chapters:hover {
+    text-decoration: none;
+    background-color: var(--theme-hover);
+    transition: background-color 0.15s, color 0.15s;
+}
+
+.nav-wrapper {
+    margin-block-start: 50px;
+    display: none;
+}
+
+.mobile-nav-chapters {
+    font-size: 2.5em;
+    text-align: center;
+    text-decoration: none;
+    width: 90px;
+    border-radius: 5px;
+    background-color: var(--sidebar-bg);
+}
+
+/* Only Firefox supports flow-relative values */
+.previous { float: left; }
+[dir=rtl] .previous { float: right; }
+
+/* Only Firefox supports flow-relative values */
+.next {
+    float: right;
+    right: var(--page-padding);
+}
+[dir=rtl] .next {
+    float: left;
+    right: unset;
+    left: var(--page-padding);
+}
+
+/* Use the correct buttons for RTL layouts*/
+[dir=rtl] .previous i.fa-angle-left:before {content:"\f105";}
+[dir=rtl] .next i.fa-angle-right:before { content:"\f104"; }
+
+@media only screen and (max-width: 1080px) {
+    .nav-wide-wrapper { display: none; }
+    .nav-wrapper { display: block; }
+}
+
+/* sidebar-visible */
+@media only screen and (max-width: 1380px) {
+    #sidebar-toggle-anchor:checked ~ .page-wrapper .nav-wide-wrapper { display: none; }
+    #sidebar-toggle-anchor:checked ~ .page-wrapper .nav-wrapper { display: block; }
+}
+
+/* Inline code */
+
+:not(pre) > .hljs {
+    display: inline;
+    padding: 0.1em 0.3em;
+    border-radius: 3px;
+}
+
+:not(pre):not(a) > .hljs {
+    color: var(--inline-code-color);
+    overflow-x: initial;
+}
+
+a:hover > .hljs {
+    text-decoration: underline;
+}
+
+pre {
+    position: relative;
+}
+pre > .buttons {
+    position: absolute;
+    z-index: 100;
+    right: 0px;
+    top: 2px;
+    margin: 0px;
+    padding: 2px 0px;
+
+    color: var(--sidebar-fg);
+    cursor: pointer;
+    visibility: hidden;
+    opacity: 0;
+    transition: visibility 0.1s linear, opacity 0.1s linear;
+}
+pre:hover > .buttons {
+    visibility: visible;
+    opacity: 1
+}
+pre > .buttons :hover {
+    color: var(--sidebar-active);
+    border-color: var(--icons-hover);
+    background-color: var(--theme-hover);
+}
+pre > .buttons i {
+    margin-inline-start: 8px;
+}
+pre > .buttons button {
+    cursor: inherit;
+    margin: 0px 5px;
+    padding: 3px 5px;
+    font-size: 14px;
+
+    border-style: solid;
+    border-width: 1px;
+    border-radius: 4px;
+    border-color: var(--icons);
+    background-color: var(--theme-popup-bg);
+    transition: 100ms;
+    transition-property: color,border-color,background-color;
+    color: var(--icons);
+}
+@media (pointer: coarse) {
+    pre > .buttons button {
+        /* On mobile, make it easier to tap buttons. */
+        padding: 0.3rem 1rem;
+    }
+}
+pre > code {
+    padding: 1rem;
+}
+
+/* FIXME: ACE editors overlap their buttons because ACE does absolute
+   positioning within the code block which breaks padding. The only solution I
+   can think of is to move the padding to the outer pre tag (or insert a div
+   wrapper), but that would require fixing a whole bunch of CSS rules.
+*/
+.hljs.ace_editor {
+  padding: 0rem 0rem;
+}
+
+pre > .result {
+    margin-block-start: 10px;
+}
+
+/* Search */
+
+#searchresults a {
+    text-decoration: none;
+}
+
+mark {
+    border-radius: 2px;
+    padding-block-start: 0;
+    padding-block-end: 1px;
+    padding-inline-start: 3px;
+    padding-inline-end: 3px;
+    margin-block-start: 0;
+    margin-block-end: -1px;
+    margin-inline-start: -3px;
+    margin-inline-end: -3px;
+    background-color: var(--search-mark-bg);
+    transition: background-color 300ms linear;
+    cursor: pointer;
+}
+
+mark.fade-out {
+    background-color: rgba(0,0,0,0) !important;
+    cursor: auto;
+}
+
+.searchbar-outer {
+    margin-inline-start: auto;
+    margin-inline-end: auto;
+    max-width: var(--content-max-width);
+}
+
+#searchbar {
+    width: 100%;
+    margin-block-start: 5px;
+    margin-block-end: 0;
+    margin-inline-start: auto;
+    margin-inline-end: auto;
+    padding: 10px 16px;
+    transition: box-shadow 300ms ease-in-out;
+    border: 1px solid var(--searchbar-border-color);
+    border-radius: 3px;
+    background-color: var(--searchbar-bg);
+    color: var(--searchbar-fg);
+}
+#searchbar:focus,
+#searchbar.active {
+    box-shadow: 0 0 3px var(--searchbar-shadow-color);
+}
+
+.searchresults-header {
+    font-weight: bold;
+    font-size: 1em;
+    padding-block-start: 18px;
+    padding-block-end: 0;
+    padding-inline-start: 5px;
+    padding-inline-end: 0;
+    color: var(--searchresults-header-fg);
+}
+
+.searchresults-outer {
+    margin-inline-start: auto;
+    margin-inline-end: auto;
+    max-width: var(--content-max-width);
+    border-block-end: 1px dashed var(--searchresults-border-color);
+}
+
+ul#searchresults {
+    list-style: none;
+    padding-inline-start: 20px;
+}
+ul#searchresults li {
+    margin: 10px 0px;
+    padding: 2px;
+    border-radius: 2px;
+}
+ul#searchresults li.focus {
+    background-color: var(--searchresults-li-bg);
+}
+ul#searchresults span.teaser {
+    display: block;
+    clear: both;
+    margin-block-start: 5px;
+    margin-block-end: 0;
+    margin-inline-start: 20px;
+    margin-inline-end: 0;
+    font-size: 0.8em;
+}
+ul#searchresults span.teaser em {
+    font-weight: bold;
+    font-style: normal;
+}
+
+/* Sidebar */
+
+.sidebar {
+    position: fixed;
+    left: 0;
+    top: 0;
+    bottom: 0;
+    width: var(--sidebar-width);
+    font-size: 0.875em;
+    box-sizing: border-box;
+    -webkit-overflow-scrolling: touch;
+    overscroll-behavior-y: contain;
+    background-color: var(--sidebar-bg);
+    color: var(--sidebar-fg);
+}
+[dir=rtl] .sidebar { left: unset; right: 0; }
+.sidebar-resizing {
+    -moz-user-select: none;
+    -webkit-user-select: none;
+    -ms-user-select: none;
+    user-select: none;
+}
+.no-js .sidebar,
+.js:not(.sidebar-resizing) .sidebar {
+    transition: transform 0.3s; /* Animation: slide away */
+}
+.sidebar code {
+    line-height: 2em;
+}
+.sidebar .sidebar-scrollbox {
+    overflow-y: auto;
+    position: absolute;
+    top: 0;
+    bottom: 0;
+    left: 0;
+    right: 0;
+    padding: 10px 10px;
+}
+.sidebar .sidebar-resize-handle {
+    position: absolute;
+    cursor: col-resize;
+    width: 0;
+    right: 0;
+    top: 0;
+    bottom: 0;
+}
+[dir=rtl] .sidebar .sidebar-resize-handle { right:  unset; left: 0; }
+.js .sidebar .sidebar-resize-handle {
+    cursor: col-resize;
+    width: 5px;
+}
+/* sidebar-hidden */
+#sidebar-toggle-anchor:not(:checked) ~ .sidebar {
+    transform: translateX(calc(0px - var(--sidebar-width)));
+    z-index: -1;
+}
+[dir=rtl] #sidebar-toggle-anchor:not(:checked) ~ .sidebar {
+    transform: translateX(var(--sidebar-width));
+}
+.sidebar::-webkit-scrollbar {
+    background: var(--sidebar-bg);
+}
+.sidebar::-webkit-scrollbar-thumb {
+    background: var(--scrollbar);
+}
+
+/* sidebar-visible */
+#sidebar-toggle-anchor:checked ~ .page-wrapper {
+    transform: translateX(var(--sidebar-width));
+}
+[dir=rtl] #sidebar-toggle-anchor:checked ~ .page-wrapper {
+    transform: translateX(calc(0px - var(--sidebar-width)));
+}
+@media only screen and (min-width: 620px) {
+    #sidebar-toggle-anchor:checked ~ .page-wrapper {
+        transform: none;
+        margin-inline-start: var(--sidebar-width);
+    }
+    [dir=rtl] #sidebar-toggle-anchor:checked ~ .page-wrapper {
+        transform: none;
+    }
+}
+
+.chapter {
+    list-style: none outside none;
+    padding-inline-start: 0;
+    line-height: 2.2em;
+}
+
+.chapter ol {
+    width: 100%;
+}
+
+.chapter li {
+    display: flex;
+    color: var(--sidebar-non-existant);
+}
+.chapter li a {
+    display: block;
+    padding: 0;
+    text-decoration: none;
+    color: var(--sidebar-fg);
+}
+
+.chapter li a:hover {
+    color: var(--sidebar-active);
+}
+
+.chapter li a.active {
+    color: var(--sidebar-active);
+}
+
+.chapter li > a.toggle {
+    cursor: pointer;
+    display: block;
+    margin-inline-start: auto;
+    padding: 0 10px;
+    user-select: none;
+    opacity: 0.68;
+}
+
+.chapter li > a.toggle div {
+    transition: transform 0.5s;
+}
+
+/* collapse the section */
+.chapter li:not(.expanded) + li > ol {
+    display: none;
+}
+
+.chapter li.chapter-item {
+    line-height: 1.5em;
+    margin-block-start: 0.6em;
+}
+
+.chapter li.expanded > a.toggle div {
+    transform: rotate(90deg);
+}
+
+.spacer {
+    width: 100%;
+    height: 3px;
+    margin: 5px 0px;
+}
+.chapter .spacer {
+    background-color: var(--sidebar-spacer);
+}
+
+@media (-moz-touch-enabled: 1), (pointer: coarse) {
+    .chapter li a { padding: 5px 0; }
+    .spacer { margin: 10px 0; }
+}
+
+.section {
+    list-style: none outside none;
+    padding-inline-start: 20px;
+    line-height: 1.9em;
+}
+
+/* Theme Menu Popup */
+
+.theme-popup {
+    position: absolute;
+    left: 10px;
+    top: var(--menu-bar-height);
+    z-index: 1000;
+    border-radius: 4px;
+    font-size: 0.7em;
+    color: var(--fg);
+    background: var(--theme-popup-bg);
+    border: 1px solid var(--theme-popup-border);
+    margin: 0;
+    padding: 0;
+    list-style: none;
+    display: none;
+    /* Don't let the children's background extend past the rounded corners. */
+    overflow: hidden;
+}
+[dir=rtl] .theme-popup { left: unset;  right: 10px; }
+.theme-popup .default {
+    color: var(--icons);
+}
+.theme-popup .theme {
+    width: 100%;
+    border: 0;
+    margin: 0;
+    padding: 2px 20px;
+    line-height: 25px;
+    white-space: nowrap;
+    text-align: start;
+    cursor: pointer;
+    color: inherit;
+    background: inherit;
+    font-size: inherit;
+}
+.theme-popup .theme:hover {
+    background-color: var(--theme-hover);
+}
+
+.theme-selected::before {
+    display: inline-block;
+    content: "✓";
+    margin-inline-start: -14px;
+    width: 14px;
+}
diff --git a/docs/css/general.css b/docs/css/general.css
new file mode 100644
index 0000000..e7d20da
--- /dev/null
+++ b/docs/css/general.css
@@ -0,0 +1,234 @@
+/* Base styles and content styles */
+
+@import 'variables.css';
+
+:root {
+    /* Browser default font-size is 16px, this way 1 rem = 10px */
+    font-size: 62.5%;
+    color-scheme: var(--color-scheme);
+}
+
+html {
+    font-family: "Open Sans", sans-serif;
+    color: var(--fg);
+    background-color: var(--bg);
+    text-size-adjust: none;
+    -webkit-text-size-adjust: none;
+}
+
+body {
+    margin: 0;
+    font-size: 1.6rem;
+    overflow-x: hidden;
+}
+
+code {
+    font-family: var(--mono-font) !important;
+    font-size: var(--code-font-size);
+    direction: ltr !important;
+}
+
+/* make long words/inline code not x overflow */
+main {
+    overflow-wrap: break-word;
+}
+
+/* make wide tables scroll if they overflow */
+.table-wrapper {
+    overflow-x: auto;
+}
+
+/* Don't change font size in headers. */
+h1 code, h2 code, h3 code, h4 code, h5 code, h6 code {
+    font-size: unset;
+}
+
+.left { float: left; }
+.right { float: right; }
+.boring { opacity: 0.6; }
+.hide-boring .boring { display: none; }
+.hidden { display: none !important; }
+
+h2, h3 { margin-block-start: 2.5em; }
+h4, h5 { margin-block-start: 2em; }
+
+.header + .header h3,
+.header + .header h4,
+.header + .header h5 {
+    margin-block-start: 1em;
+}
+
+h1:target::before,
+h2:target::before,
+h3:target::before,
+h4:target::before,
+h5:target::before,
+h6:target::before {
+    display: inline-block;
+    content: "»";
+    margin-inline-start: -30px;
+    width: 30px;
+}
+
+/* This is broken on Safari as of version 14, but is fixed
+   in Safari Technology Preview 117 which I think will be Safari 14.2.
+   https://bugs.webkit.org/show_bug.cgi?id=218076
+*/
+:target {
+    /* Safari does not support logical properties */
+    scroll-margin-top: calc(var(--menu-bar-height) + 0.5em);
+}
+
+.page {
+    outline: 0;
+    padding: 0 var(--page-padding);
+    margin-block-start: calc(0px - var(--menu-bar-height)); /* Compensate for the #menu-bar-hover-placeholder */
+}
+.page-wrapper {
+    box-sizing: border-box;
+    background-color: var(--bg);
+}
+.no-js .page-wrapper,
+.js:not(.sidebar-resizing) .page-wrapper {
+    transition: margin-left 0.3s ease, transform 0.3s ease; /* Animation: slide away */
+}
+[dir=rtl] .js:not(.sidebar-resizing) .page-wrapper {
+    transition: margin-right 0.3s ease, transform 0.3s ease; /* Animation: slide away */
+}
+
+.content {
+    overflow-y: auto;
+    padding: 0 5px 50px 5px;
+}
+.content main {
+    margin-inline-start: auto;
+    margin-inline-end: auto;
+    max-width: var(--content-max-width);
+}
+.content p { line-height: 1.45em; }
+.content ol { line-height: 1.45em; }
+.content ul { line-height: 1.45em; }
+.content a { text-decoration: none; }
+.content a:hover { text-decoration: underline; }
+.content img, .content video { max-width: 100%; }
+.content .header:link,
+.content .header:visited {
+    color: var(--fg);
+}
+.content .header:link,
+.content .header:visited:hover {
+    text-decoration: none;
+}
+
+table {
+    margin: 0 auto;
+    border-collapse: collapse;
+}
+table td {
+    padding: 3px 20px;
+    border: 1px var(--table-border-color) solid;
+}
+table thead {
+    background: var(--table-header-bg);
+}
+table thead td {
+    font-weight: 700;
+    border: none;
+}
+table thead th {
+    padding: 3px 20px;
+}
+table thead tr {
+    border: 1px var(--table-header-bg) solid;
+}
+/* Alternate background colors for rows */
+table tbody tr:nth-child(2n) {
+    background: var(--table-alternate-bg);
+}
+
+
+blockquote {
+    margin: 20px 0;
+    padding: 0 20px;
+    color: var(--fg);
+    background-color: var(--quote-bg);
+    border-block-start: .1em solid var(--quote-border);
+    border-block-end: .1em solid var(--quote-border);
+}
+
+.warning {
+    margin: 20px;
+    padding: 0 20px;
+    border-inline-start: 2px solid var(--warning-border);
+}
+
+.warning:before {
+    position: absolute;
+    width: 3rem;
+    height: 3rem;
+    margin-inline-start: calc(-1.5rem - 21px);
+    content: "ⓘ";
+    text-align: center;
+    background-color: var(--bg);
+    color: var(--warning-border);
+    font-weight: bold;
+    font-size: 2rem;
+}
+
+blockquote .warning:before {
+    background-color: var(--quote-bg);
+}
+
+kbd {
+    background-color: var(--table-border-color);
+    border-radius: 4px;
+    border: solid 1px var(--theme-popup-border);
+    box-shadow: inset 0 -1px 0 var(--theme-hover);
+    display: inline-block;
+    font-size: var(--code-font-size);
+    font-family: var(--mono-font);
+    line-height: 10px;
+    padding: 4px 5px;
+    vertical-align: middle;
+}
+
+:not(.footnote-definition) + .footnote-definition,
+.footnote-definition + :not(.footnote-definition) {
+    margin-block-start: 2em;
+}
+.footnote-definition {
+    font-size: 0.9em;
+    margin: 0.5em 0;
+}
+.footnote-definition p {
+    display: inline;
+}
+
+.tooltiptext {
+    position: absolute;
+    visibility: hidden;
+    color: #fff;
+    background-color: #333;
+    transform: translateX(-50%); /* Center by moving tooltip 50% of its width left */
+    left: -8px; /* Half of the width of the icon */
+    top: -35px;
+    font-size: 0.8em;
+    text-align: center;
+    border-radius: 6px;
+    padding: 5px 8px;
+    margin: 5px;
+    z-index: 1000;
+}
+.tooltipped .tooltiptext {
+    visibility: visible;
+}
+
+.chapter li.part-title {
+    color: var(--sidebar-fg);
+    margin: 5px 0px;
+    font-weight: bold;
+}
+
+.result-no-output {
+    font-style: italic;
+}
diff --git a/docs/css/mdbook-admonish.css b/docs/css/mdbook-admonish.css
new file mode 100644
index 0000000..e0a3365
--- /dev/null
+++ b/docs/css/mdbook-admonish.css
@@ -0,0 +1,341 @@
+@charset "UTF-8";
+:root {
+  --md-admonition-icon--admonish-note: url("data:image/svg+xml;charset=utf-8,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 24 24'><path d='M20.71 7.04c.39-.39.39-1.04 0-1.41l-2.34-2.34c-.37-.39-1.02-.39-1.41 0l-1.84 1.83 3.75 3.75M3 17.25V21h3.75L17.81 9.93l-3.75-3.75L3 17.25z'/></svg>");
+  --md-admonition-icon--admonish-abstract: url("data:image/svg+xml;charset=utf-8,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 24 24'><path d='M17 9H7V7h10m0 6H7v-2h10m-3 6H7v-2h7M12 3a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1m7 0h-4.18C14.4 1.84 13.3 1 12 1c-1.3 0-2.4.84-2.82 2H5a2 2 0 0 0-2 2v14a2 2 0 0 0 2 2h14a2 2 0 0 0 2-2V5a2 2 0 0 0-2-2z'/></svg>");
+  --md-admonition-icon--admonish-info: url("data:image/svg+xml;charset=utf-8,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 24 24'><path d='M13 9h-2V7h2m0 10h-2v-6h2m-1-9A10 10 0 0 0 2 12a10 10 0 0 0 10 10 10 10 0 0 0 10-10A10 10 0 0 0 12 2z'/></svg>");
+  --md-admonition-icon--admonish-tip: url("data:image/svg+xml;charset=utf-8,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 24 24'><path d='M17.66 11.2c-.23-.3-.51-.56-.77-.82-.67-.6-1.43-1.03-2.07-1.66C13.33 7.26 13 4.85 13.95 3c-.95.23-1.78.75-2.49 1.32-2.59 2.08-3.61 5.75-2.39 8.9.04.1.08.2.08.33 0 .22-.15.42-.35.5-.23.1-.47.04-.66-.12a.58.58 0 0 1-.14-.17c-1.13-1.43-1.31-3.48-.55-5.12C5.78 10 4.87 12.3 5 14.47c.06.5.12 1 .29 1.5.14.6.41 1.2.71 1.73 1.08 1.73 2.95 2.97 4.96 3.22 2.14.27 4.43-.12 6.07-1.6 1.83-1.66 2.47-4.32 1.53-6.6l-.13-.26c-.21-.46-.77-1.26-.77-1.26m-3.16 6.3c-.28.24-.74.5-1.1.6-1.12.4-2.24-.16-2.9-.82 1.19-.28 1.9-1.16 2.11-2.05.17-.8-.15-1.46-.28-2.23-.12-.74-.1-1.37.17-2.06.19.38.39.76.63 1.06.77 1 1.98 1.44 2.24 2.8.04.14.06.28.06.43.03.82-.33 1.72-.93 2.27z'/></svg>");
+  --md-admonition-icon--admonish-success: url("data:image/svg+xml;charset=utf-8,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 24 24'><path d='m9 20.42-6.21-6.21 2.83-2.83L9 14.77l9.88-9.89 2.83 2.83L9 20.42z'/></svg>");
+  --md-admonition-icon--admonish-question: url("data:image/svg+xml;charset=utf-8,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 24 24'><path d='m15.07 11.25-.9.92C13.45 12.89 13 13.5 13 15h-2v-.5c0-1.11.45-2.11 1.17-2.83l1.24-1.26c.37-.36.59-.86.59-1.41a2 2 0 0 0-2-2 2 2 0 0 0-2 2H8a4 4 0 0 1 4-4 4 4 0 0 1 4 4 3.2 3.2 0 0 1-.93 2.25M13 19h-2v-2h2M12 2A10 10 0 0 0 2 12a10 10 0 0 0 10 10 10 10 0 0 0 10-10c0-5.53-4.5-10-10-10z'/></svg>");
+  --md-admonition-icon--admonish-warning: url("data:image/svg+xml;charset=utf-8,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 24 24'><path d='M13 14h-2V9h2m0 9h-2v-2h2M1 21h22L12 2 1 21z'/></svg>");
+  --md-admonition-icon--admonish-failure: url("data:image/svg+xml;charset=utf-8,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 24 24'><path d='M20 6.91 17.09 4 12 9.09 6.91 4 4 6.91 9.09 12 4 17.09 6.91 20 12 14.91 17.09 20 20 17.09 14.91 12 20 6.91z'/></svg>");
+  --md-admonition-icon--admonish-danger: url("data:image/svg+xml;charset=utf-8,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 24 24'><path d='M11 15H6l7-14v8h5l-7 14v-8z'/></svg>");
+  --md-admonition-icon--admonish-bug: url("data:image/svg+xml;charset=utf-8,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 24 24'><path d='M14 12h-4v-2h4m0 6h-4v-2h4m6-6h-2.81a5.985 5.985 0 0 0-1.82-1.96L17 4.41 15.59 3l-2.17 2.17a6.002 6.002 0 0 0-2.83 0L8.41 3 7 4.41l1.62 1.63C7.88 6.55 7.26 7.22 6.81 8H4v2h2.09c-.05.33-.09.66-.09 1v1H4v2h2v1c0 .34.04.67.09 1H4v2h2.81c1.04 1.79 2.97 3 5.19 3s4.15-1.21 5.19-3H20v-2h-2.09c.05-.33.09-.66.09-1v-1h2v-2h-2v-1c0-.34-.04-.67-.09-1H20V8z'/></svg>");
+  --md-admonition-icon--admonish-example: url("data:image/svg+xml;charset=utf-8,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 24 24'><path d='M7 13v-2h14v2H7m0 6v-2h14v2H7M7 7V5h14v2H7M3 8V5H2V4h2v4H3m-1 9v-1h3v4H2v-1h2v-.5H3v-1h1V17H2m2.25-7a.75.75 0 0 1 .75.75c0 .2-.08.39-.21.52L3.12 13H5v1H2v-.92L4 11H2v-1h2.25z'/></svg>");
+  --md-admonition-icon--admonish-quote: url("data:image/svg+xml;charset=utf-8,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 24 24'><path d='M14 17h3l2-4V7h-6v6h3M6 17h3l2-4V7H5v6h3l-2 4z'/></svg>");
+  --md-details-icon: url("data:image/svg+xml;charset=utf-8,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 24 24'><path d='M8.59 16.58 13.17 12 8.59 7.41 10 6l6 6-6 6-1.41-1.42Z'/></svg>");
+}
+
+:is(.admonition) {
+  display: flow-root;
+  margin: 1.5625em 0;
+  padding: 0 1.2rem;
+  color: var(--fg);
+  page-break-inside: avoid;
+  background-color: var(--bg);
+  border: 0 solid black;
+  border-inline-start-width: 0.4rem;
+  border-radius: 0.2rem;
+  box-shadow: 0 0.2rem 1rem rgba(0, 0, 0, 0.05), 0 0 0.1rem rgba(0, 0, 0, 0.1);
+}
+@media print {
+  :is(.admonition) {
+    box-shadow: none;
+  }
+}
+:is(.admonition) > * {
+  box-sizing: border-box;
+}
+:is(.admonition) :is(.admonition) {
+  margin-top: 1em;
+  margin-bottom: 1em;
+}
+:is(.admonition) > .tabbed-set:only-child {
+  margin-top: 0;
+}
+html :is(.admonition) > :last-child {
+  margin-bottom: 1.2rem;
+}
+
+a.admonition-anchor-link {
+  display: none;
+  position: absolute;
+  left: -1.2rem;
+  padding-right: 1rem;
+}
+a.admonition-anchor-link:link, a.admonition-anchor-link:visited {
+  color: var(--fg);
+}
+a.admonition-anchor-link:link:hover, a.admonition-anchor-link:visited:hover {
+  text-decoration: none;
+}
+a.admonition-anchor-link::before {
+  content: "§";
+}
+
+:is(.admonition-title, summary.admonition-title) {
+  position: relative;
+  min-height: 4rem;
+  margin-block: 0;
+  margin-inline: -1.6rem -1.2rem;
+  padding-block: 0.8rem;
+  padding-inline: 4.4rem 1.2rem;
+  font-weight: 700;
+  background-color: rgba(68, 138, 255, 0.1);
+  display: flex;
+}
+:is(.admonition-title, summary.admonition-title) p {
+  margin: 0;
+}
+html :is(.admonition-title, summary.admonition-title):last-child {
+  margin-bottom: 0;
+}
+:is(.admonition-title, summary.admonition-title)::before {
+  position: absolute;
+  top: 0.625em;
+  inset-inline-start: 1.6rem;
+  width: 2rem;
+  height: 2rem;
+  background-color: #448aff;
+  mask-image: url('data:image/svg+xml;charset=utf-8,<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"></svg>');
+  -webkit-mask-image: url('data:image/svg+xml;charset=utf-8,<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"></svg>');
+  mask-repeat: no-repeat;
+  -webkit-mask-repeat: no-repeat;
+  mask-size: contain;
+  -webkit-mask-size: contain;
+  content: "";
+}
+:is(.admonition-title, summary.admonition-title):hover a.admonition-anchor-link {
+  display: initial;
+}
+
+details.admonition > summary.admonition-title::after {
+  position: absolute;
+  top: 0.625em;
+  inset-inline-end: 1.6rem;
+  height: 2rem;
+  width: 2rem;
+  background-color: currentcolor;
+  mask-image: var(--md-details-icon);
+  -webkit-mask-image: var(--md-details-icon);
+  mask-repeat: no-repeat;
+  -webkit-mask-repeat: no-repeat;
+  mask-size: contain;
+  -webkit-mask-size: contain;
+  content: "";
+  transform: rotate(0deg);
+  transition: transform 0.25s;
+}
+details[open].admonition > summary.admonition-title::after {
+  transform: rotate(90deg);
+}
+
+:is(.admonition):is(.admonish-note) {
+  border-color: #448aff;
+}
+
+:is(.admonish-note) > :is(.admonition-title, summary.admonition-title) {
+  background-color: rgba(68, 138, 255, 0.1);
+}
+:is(.admonish-note) > :is(.admonition-title, summary.admonition-title)::before {
+  background-color: #448aff;
+  mask-image: var(--md-admonition-icon--admonish-note);
+  -webkit-mask-image: var(--md-admonition-icon--admonish-note);
+  mask-repeat: no-repeat;
+  -webkit-mask-repeat: no-repeat;
+  mask-size: contain;
+  -webkit-mask-repeat: no-repeat;
+}
+
+:is(.admonition):is(.admonish-abstract, .admonish-summary, .admonish-tldr) {
+  border-color: #00b0ff;
+}
+
+:is(.admonish-abstract, .admonish-summary, .admonish-tldr) > :is(.admonition-title, summary.admonition-title) {
+  background-color: rgba(0, 176, 255, 0.1);
+}
+:is(.admonish-abstract, .admonish-summary, .admonish-tldr) > :is(.admonition-title, summary.admonition-title)::before {
+  background-color: #00b0ff;
+  mask-image: var(--md-admonition-icon--admonish-abstract);
+  -webkit-mask-image: var(--md-admonition-icon--admonish-abstract);
+  mask-repeat: no-repeat;
+  -webkit-mask-repeat: no-repeat;
+  mask-size: contain;
+  -webkit-mask-repeat: no-repeat;
+}
+
+:is(.admonition):is(.admonish-info, .admonish-todo) {
+  border-color: #00b8d4;
+}
+
+:is(.admonish-info, .admonish-todo) > :is(.admonition-title, summary.admonition-title) {
+  background-color: rgba(0, 184, 212, 0.1);
+}
+:is(.admonish-info, .admonish-todo) > :is(.admonition-title, summary.admonition-title)::before {
+  background-color: #00b8d4;
+  mask-image: var(--md-admonition-icon--admonish-info);
+  -webkit-mask-image: var(--md-admonition-icon--admonish-info);
+  mask-repeat: no-repeat;
+  -webkit-mask-repeat: no-repeat;
+  mask-size: contain;
+  -webkit-mask-repeat: no-repeat;
+}
+
+:is(.admonition):is(.admonish-tip, .admonish-hint, .admonish-important) {
+  border-color: #00bfa5;
+}
+
+:is(.admonish-tip, .admonish-hint, .admonish-important) > :is(.admonition-title, summary.admonition-title) {
+  background-color: rgba(0, 191, 165, 0.1);
+}
+:is(.admonish-tip, .admonish-hint, .admonish-important) > :is(.admonition-title, summary.admonition-title)::before {
+  background-color: #00bfa5;
+  mask-image: var(--md-admonition-icon--admonish-tip);
+  -webkit-mask-image: var(--md-admonition-icon--admonish-tip);
+  mask-repeat: no-repeat;
+  -webkit-mask-repeat: no-repeat;
+  mask-size: contain;
+  -webkit-mask-repeat: no-repeat;
+}
+
+:is(.admonition):is(.admonish-success, .admonish-check, .admonish-done) {
+  border-color: #00c853;
+}
+
+:is(.admonish-success, .admonish-check, .admonish-done) > :is(.admonition-title, summary.admonition-title) {
+  background-color: rgba(0, 200, 83, 0.1);
+}
+:is(.admonish-success, .admonish-check, .admonish-done) > :is(.admonition-title, summary.admonition-title)::before {
+  background-color: #00c853;
+  mask-image: var(--md-admonition-icon--admonish-success);
+  -webkit-mask-image: var(--md-admonition-icon--admonish-success);
+  mask-repeat: no-repeat;
+  -webkit-mask-repeat: no-repeat;
+  mask-size: contain;
+  -webkit-mask-repeat: no-repeat;
+}
+
+:is(.admonition):is(.admonish-question, .admonish-help, .admonish-faq) {
+  border-color: #64dd17;
+}
+
+:is(.admonish-question, .admonish-help, .admonish-faq) > :is(.admonition-title, summary.admonition-title) {
+  background-color: rgba(100, 221, 23, 0.1);
+}
+:is(.admonish-question, .admonish-help, .admonish-faq) > :is(.admonition-title, summary.admonition-title)::before {
+  background-color: #64dd17;
+  mask-image: var(--md-admonition-icon--admonish-question);
+  -webkit-mask-image: var(--md-admonition-icon--admonish-question);
+  mask-repeat: no-repeat;
+  -webkit-mask-repeat: no-repeat;
+  mask-size: contain;
+  -webkit-mask-repeat: no-repeat;
+}
+
+:is(.admonition):is(.admonish-warning, .admonish-caution, .admonish-attention) {
+  border-color: #ff9100;
+}
+
+:is(.admonish-warning, .admonish-caution, .admonish-attention) > :is(.admonition-title, summary.admonition-title) {
+  background-color: rgba(255, 145, 0, 0.1);
+}
+:is(.admonish-warning, .admonish-caution, .admonish-attention) > :is(.admonition-title, summary.admonition-title)::before {
+  background-color: #ff9100;
+  mask-image: var(--md-admonition-icon--admonish-warning);
+  -webkit-mask-image: var(--md-admonition-icon--admonish-warning);
+  mask-repeat: no-repeat;
+  -webkit-mask-repeat: no-repeat;
+  mask-size: contain;
+  -webkit-mask-repeat: no-repeat;
+}
+
+:is(.admonition):is(.admonish-failure, .admonish-fail, .admonish-missing) {
+  border-color: #ff5252;
+}
+
+:is(.admonish-failure, .admonish-fail, .admonish-missing) > :is(.admonition-title, summary.admonition-title) {
+  background-color: rgba(255, 82, 82, 0.1);
+}
+:is(.admonish-failure, .admonish-fail, .admonish-missing) > :is(.admonition-title, summary.admonition-title)::before {
+  background-color: #ff5252;
+  mask-image: var(--md-admonition-icon--admonish-failure);
+  -webkit-mask-image: var(--md-admonition-icon--admonish-failure);
+  mask-repeat: no-repeat;
+  -webkit-mask-repeat: no-repeat;
+  mask-size: contain;
+  -webkit-mask-repeat: no-repeat;
+}
+
+:is(.admonition):is(.admonish-danger, .admonish-error) {
+  border-color: #ff1744;
+}
+
+:is(.admonish-danger, .admonish-error) > :is(.admonition-title, summary.admonition-title) {
+  background-color: rgba(255, 23, 68, 0.1);
+}
+:is(.admonish-danger, .admonish-error) > :is(.admonition-title, summary.admonition-title)::before {
+  background-color: #ff1744;
+  mask-image: var(--md-admonition-icon--admonish-danger);
+  -webkit-mask-image: var(--md-admonition-icon--admonish-danger);
+  mask-repeat: no-repeat;
+  -webkit-mask-repeat: no-repeat;
+  mask-size: contain;
+  -webkit-mask-repeat: no-repeat;
+}
+
+:is(.admonition):is(.admonish-bug) {
+  border-color: #f50057;
+}
+
+:is(.admonish-bug) > :is(.admonition-title, summary.admonition-title) {
+  background-color: rgba(245, 0, 87, 0.1);
+}
+:is(.admonish-bug) > :is(.admonition-title, summary.admonition-title)::before {
+  background-color: #f50057;
+  mask-image: var(--md-admonition-icon--admonish-bug);
+  -webkit-mask-image: var(--md-admonition-icon--admonish-bug);
+  mask-repeat: no-repeat;
+  -webkit-mask-repeat: no-repeat;
+  mask-size: contain;
+  -webkit-mask-repeat: no-repeat;
+}
+
+:is(.admonition):is(.admonish-example) {
+  border-color: #7c4dff;
+}
+
+:is(.admonish-example) > :is(.admonition-title, summary.admonition-title) {
+  background-color: rgba(124, 77, 255, 0.1);
+}
+:is(.admonish-example) > :is(.admonition-title, summary.admonition-title)::before {
+  background-color: #7c4dff;
+  mask-image: var(--md-admonition-icon--admonish-example);
+  -webkit-mask-image: var(--md-admonition-icon--admonish-example);
+  mask-repeat: no-repeat;
+  -webkit-mask-repeat: no-repeat;
+  mask-size: contain;
+  -webkit-mask-repeat: no-repeat;
+}
+
+:is(.admonition):is(.admonish-quote, .admonish-cite) {
+  border-color: #9e9e9e;
+}
+
+:is(.admonish-quote, .admonish-cite) > :is(.admonition-title, summary.admonition-title) {
+  background-color: rgba(158, 158, 158, 0.1);
+}
+:is(.admonish-quote, .admonish-cite) > :is(.admonition-title, summary.admonition-title)::before {
+  background-color: #9e9e9e;
+  mask-image: var(--md-admonition-icon--admonish-quote);
+  -webkit-mask-image: var(--md-admonition-icon--admonish-quote);
+  mask-repeat: no-repeat;
+  -webkit-mask-repeat: no-repeat;
+  mask-size: contain;
+  -webkit-mask-repeat: no-repeat;
+}
+
+.navy :is(.admonition) {
+  background-color: var(--sidebar-bg);
+}
+
+.ayu :is(.admonition),
+.coal :is(.admonition) {
+  background-color: var(--theme-hover);
+}
+
+.rust :is(.admonition) {
+  background-color: var(--sidebar-bg);
+  color: var(--sidebar-fg);
+}
+.rust .admonition-anchor-link:link, .rust .admonition-anchor-link:visited {
+  color: var(--sidebar-fg);
+}
diff --git a/docs/css/print.css b/docs/css/print.css
new file mode 100644
index 0000000..dcf0ba6
--- /dev/null
+++ b/docs/css/print.css
@@ -0,0 +1,50 @@
+
+#sidebar,
+#menu-bar,
+.nav-chapters,
+.mobile-nav-chapters {
+    display: none;
+}
+
+#page-wrapper.page-wrapper {
+    transform: none;
+    margin-inline-start: 0px;
+    overflow-y: initial;
+}
+
+#content {
+    max-width: none;
+    margin: 0;
+    padding: 0;
+}
+
+.page {
+    overflow-y: initial;
+}
+
+code {
+    direction: ltr !important;
+}
+
+pre > .buttons {
+    z-index: 2;
+}
+
+a, a:visited, a:active, a:hover {
+    color: #4183c4;
+    text-decoration: none;
+}
+
+h1, h2, h3, h4, h5, h6 {
+    page-break-inside: avoid;
+    page-break-after: avoid;
+}
+
+pre, code {
+    page-break-inside: avoid;
+    white-space: pre-wrap;
+}
+
+.fa {
+    display: none !important;
+}
diff --git a/docs/css/variables.css b/docs/css/variables.css
new file mode 100644
index 0000000..10a7590
--- /dev/null
+++ b/docs/css/variables.css
@@ -0,0 +1,277 @@
+
+/* Globals */
+
+:root {
+    --sidebar-width: 300px;
+    --page-padding: 15px;
+    --content-max-width: 750px;
+    --menu-bar-height: 50px;
+    --mono-font: "Source Code Pro", Consolas, "Ubuntu Mono", Menlo, "DejaVu Sans Mono", monospace, monospace;
+    --code-font-size: 0.875em /* please adjust the ace font size accordingly in editor.js */
+}
+
+/* Themes */
+
+.ayu {
+    --bg: hsl(210, 25%, 8%);
+    --fg: #c5c5c5;
+
+    --sidebar-bg: #14191f;
+    --sidebar-fg: #c8c9db;
+    --sidebar-non-existant: #5c6773;
+    --sidebar-active: #ffb454;
+    --sidebar-spacer: #2d334f;
+
+    --scrollbar: var(--sidebar-fg);
+
+    --icons: #737480;
+    --icons-hover: #b7b9cc;
+
+    --links: #0096cf;
+
+    --inline-code-color: #ffb454;
+
+    --theme-popup-bg: #14191f;
+    --theme-popup-border: #5c6773;
+    --theme-hover: #191f26;
+
+    --quote-bg: hsl(226, 15%, 17%);
+    --quote-border: hsl(226, 15%, 22%);
+
+    --warning-border: #ff8e00;
+
+    --table-border-color: hsl(210, 25%, 13%);
+    --table-header-bg: hsl(210, 25%, 28%);
+    --table-alternate-bg: hsl(210, 25%, 11%);
+
+    --searchbar-border-color: #848484;
+    --searchbar-bg: #424242;
+    --searchbar-fg: #fff;
+    --searchbar-shadow-color: #d4c89f;
+    --searchresults-header-fg: #666;
+    --searchresults-border-color: #888;
+    --searchresults-li-bg: #252932;
+    --search-mark-bg: #e3b171;
+
+    --color-scheme: dark;
+}
+
+.coal {
+    --bg: hsl(200, 7%, 8%);
+    --fg: #98a3ad;
+
+    --sidebar-bg: #292c2f;
+    --sidebar-fg: #a1adb8;
+    --sidebar-non-existant: #505254;
+    --sidebar-active: #3473ad;
+    --sidebar-spacer: #393939;
+
+    --scrollbar: var(--sidebar-fg);
+
+    --icons: #43484d;
+    --icons-hover: #b3c0cc;
+
+    --links: #2b79a2;
+
+    --inline-code-color: #c5c8c6;
+
+    --theme-popup-bg: #141617;
+    --theme-popup-border: #43484d;
+    --theme-hover: #1f2124;
+
+    --quote-bg: hsl(234, 21%, 18%);
+    --quote-border: hsl(234, 21%, 23%);
+
+    --warning-border: #ff8e00;
+
+    --table-border-color: hsl(200, 7%, 13%);
+    --table-header-bg: hsl(200, 7%, 28%);
+    --table-alternate-bg: hsl(200, 7%, 11%);
+
+    --searchbar-border-color: #aaa;
+    --searchbar-bg: #b7b7b7;
+    --searchbar-fg: #000;
+    --searchbar-shadow-color: #aaa;
+    --searchresults-header-fg: #666;
+    --searchresults-border-color: #98a3ad;
+    --searchresults-li-bg: #2b2b2f;
+    --search-mark-bg: #355c7d;
+
+    --color-scheme: dark;
+}
+
+.light {
+    --bg: hsl(0, 0%, 100%);
+    --fg: hsl(0, 0%, 0%);
+
+    --sidebar-bg: #fafafa;
+    --sidebar-fg: hsl(0, 0%, 0%);
+    --sidebar-non-existant: #aaaaaa;
+    --sidebar-active: #1f1fff;
+    --sidebar-spacer: #f4f4f4;
+
+    --scrollbar: #8F8F8F;
+
+    --icons: #747474;
+    --icons-hover: #000000;
+
+    --links: #20609f;
+
+    --inline-code-color: #301900;
+
+    --theme-popup-bg: #fafafa;
+    --theme-popup-border: #cccccc;
+    --theme-hover: #e6e6e6;
+
+    --quote-bg: hsl(197, 37%, 96%);
+    --quote-border: hsl(197, 37%, 91%);
+
+    --warning-border: #ff8e00;
+
+    --table-border-color: hsl(0, 0%, 95%);
+    --table-header-bg: hsl(0, 0%, 80%);
+    --table-alternate-bg: hsl(0, 0%, 97%);
+
+    --searchbar-border-color: #aaa;
+    --searchbar-bg: #fafafa;
+    --searchbar-fg: #000;
+    --searchbar-shadow-color: #aaa;
+    --searchresults-header-fg: #666;
+    --searchresults-border-color: #888;
+    --searchresults-li-bg: #e4f2fe;
+    --search-mark-bg: #a2cff5;
+
+    --color-scheme: light;
+}
+
+.navy {
+    --bg: hsl(226, 23%, 11%);
+    --fg: #bcbdd0;
+
+    --sidebar-bg: #282d3f;
+    --sidebar-fg: #c8c9db;
+    --sidebar-non-existant: #505274;
+    --sidebar-active: #2b79a2;
+    --sidebar-spacer: #2d334f;
+
+    --scrollbar: var(--sidebar-fg);
+
+    --icons: #737480;
+    --icons-hover: #b7b9cc;
+
+    --links: #2b79a2;
+
+    --inline-code-color: #c5c8c6;
+
+    --theme-popup-bg: #161923;
+    --theme-popup-border: #737480;
+    --theme-hover: #282e40;
+
+    --quote-bg: hsl(226, 15%, 17%);
+    --quote-border: hsl(226, 15%, 22%);
+
+    --warning-border: #ff8e00;
+
+    --table-border-color: hsl(226, 23%, 16%);
+    --table-header-bg: hsl(226, 23%, 31%);
+    --table-alternate-bg: hsl(226, 23%, 14%);
+
+    --searchbar-border-color: #aaa;
+    --searchbar-bg: #aeaec6;
+    --searchbar-fg: #000;
+    --searchbar-shadow-color: #aaa;
+    --searchresults-header-fg: #5f5f71;
+    --searchresults-border-color: #5c5c68;
+    --searchresults-li-bg: #242430;
+    --search-mark-bg: #a2cff5;
+
+    --color-scheme: dark;
+}
+
+.rust {
+    --bg: hsl(60, 9%, 87%);
+    --fg: #262625;
+
+    --sidebar-bg: #3b2e2a;
+    --sidebar-fg: #c8c9db;
+    --sidebar-non-existant: #505254;
+    --sidebar-active: #e69f67;
+    --sidebar-spacer: #45373a;
+
+    --scrollbar: var(--sidebar-fg);
+
+    --icons: #737480;
+    --icons-hover: #262625;
+
+    --links: #2b79a2;
+
+    --inline-code-color: #6e6b5e;
+
+    --theme-popup-bg: #e1e1db;
+    --theme-popup-border: #b38f6b;
+    --theme-hover: #99908a;
+
+    --quote-bg: hsl(60, 5%, 75%);
+    --quote-border: hsl(60, 5%, 70%);
+
+    --warning-border: #ff8e00;
+
+    --table-border-color: hsl(60, 9%, 82%);
+    --table-header-bg: #b3a497;
+    --table-alternate-bg: hsl(60, 9%, 84%);
+
+    --searchbar-border-color: #aaa;
+    --searchbar-bg: #fafafa;
+    --searchbar-fg: #000;
+    --searchbar-shadow-color: #aaa;
+    --searchresults-header-fg: #666;
+    --searchresults-border-color: #888;
+    --searchresults-li-bg: #dec2a2;
+    --search-mark-bg: #e69f67;
+
+    --color-scheme: light;
+}
+
+@media (prefers-color-scheme: dark) {
+    .light.no-js {
+        --bg: hsl(200, 7%, 8%);
+        --fg: #98a3ad;
+
+        --sidebar-bg: #292c2f;
+        --sidebar-fg: #a1adb8;
+        --sidebar-non-existant: #505254;
+        --sidebar-active: #3473ad;
+        --sidebar-spacer: #393939;
+
+        --scrollbar: var(--sidebar-fg);
+
+        --icons: #43484d;
+        --icons-hover: #b3c0cc;
+
+        --links: #2b79a2;
+
+        --inline-code-color: #c5c8c6;
+
+        --theme-popup-bg: #141617;
+        --theme-popup-border: #43484d;
+        --theme-hover: #1f2124;
+
+        --quote-bg: hsl(234, 21%, 18%);
+        --quote-border: hsl(234, 21%, 23%);
+
+        --warning-border: #ff8e00;
+
+        --table-border-color: hsl(200, 7%, 13%);
+        --table-header-bg: hsl(200, 7%, 28%);
+        --table-alternate-bg: hsl(200, 7%, 11%);
+
+        --searchbar-border-color: #aaa;
+        --searchbar-bg: #b7b7b7;
+        --searchbar-fg: #000;
+        --searchbar-shadow-color: #aaa;
+        --searchresults-header-fg: #666;
+        --searchresults-border-color: #98a3ad;
+        --searchresults-li-bg: #2b2b2f;
+        --search-mark-bg: #355c7d;
+    }
+}
diff --git a/docs/design.html b/docs/design.html
new file mode 100644
index 0000000..2e88a9f
--- /dev/null
+++ b/docs/design.html
@@ -0,0 +1,234 @@
+<!DOCTYPE HTML>
+<html lang="en" class="light" dir="ltr">
+    <head>
+        <!-- Book generated using mdBook -->
+        <meta charset="UTF-8">
+        <title>Design - Zcash Trailing Finality Layer - v0.1.0</title>
+
+
+        <!-- Custom HTML head -->
+        
+        <meta name="description" content="">
+        <meta name="viewport" content="width=device-width, initial-scale=1">
+        <meta name="theme-color" content="#ffffff">
+
+        <link rel="icon" href="favicon.svg">
+        <link rel="shortcut icon" href="favicon.png">
+        <link rel="stylesheet" href="css/variables.css">
+        <link rel="stylesheet" href="css/general.css">
+        <link rel="stylesheet" href="css/chrome.css">
+        <link rel="stylesheet" href="css/print.css" media="print">
+
+        <!-- Fonts -->
+        <link rel="stylesheet" href="FontAwesome/css/font-awesome.css">
+        <link rel="stylesheet" href="fonts/fonts.css">
+
+        <!-- Highlight.js Stylesheets -->
+        <link rel="stylesheet" href="highlight.css">
+        <link rel="stylesheet" href="tomorrow-night.css">
+        <link rel="stylesheet" href="ayu-highlight.css">
+
+        <!-- Custom theme stylesheets -->
+        <link rel="stylesheet" href="./css/mdbook-admonish.css">
+
+    </head>
+    <body class="sidebar-visible no-js">
+    <div id="body-container">
+        <!-- Provide site root to javascript -->
+        <script>
+            var path_to_root = "";
+            var default_theme = window.matchMedia("(prefers-color-scheme: dark)").matches ? "navy" : "light";
+        </script>
+
+        <!-- Work around some values being stored in localStorage wrapped in quotes -->
+        <script>
+            try {
+                var theme = localStorage.getItem('mdbook-theme');
+                var sidebar = localStorage.getItem('mdbook-sidebar');
+
+                if (theme.startsWith('"') && theme.endsWith('"')) {
+                    localStorage.setItem('mdbook-theme', theme.slice(1, theme.length - 1));
+                }
+
+                if (sidebar.startsWith('"') && sidebar.endsWith('"')) {
+                    localStorage.setItem('mdbook-sidebar', sidebar.slice(1, sidebar.length - 1));
+                }
+            } catch (e) { }
+        </script>
+
+        <!-- Set the theme before any content is loaded, prevents flash -->
+        <script>
+            var theme;
+            try { theme = localStorage.getItem('mdbook-theme'); } catch(e) { }
+            if (theme === null || theme === undefined) { theme = default_theme; }
+            var html = document.querySelector('html');
+            html.classList.remove('light')
+            html.classList.add(theme);
+            var body = document.querySelector('body');
+            body.classList.remove('no-js')
+            body.classList.add('js');
+        </script>
+
+        <input type="checkbox" id="sidebar-toggle-anchor" class="hidden">
+
+        <!-- Hide / unhide sidebar before it is displayed -->
+        <script>
+            var body = document.querySelector('body');
+            var sidebar = null;
+            var sidebar_toggle = document.getElementById("sidebar-toggle-anchor");
+            if (document.body.clientWidth >= 1080) {
+                try { sidebar = localStorage.getItem('mdbook-sidebar'); } catch(e) { }
+                sidebar = sidebar || 'visible';
+            } else {
+                sidebar = 'hidden';
+            }
+            sidebar_toggle.checked = sidebar === 'visible';
+            body.classList.remove('sidebar-visible');
+            body.classList.add("sidebar-" + sidebar);
+        </script>
+
+        <nav id="sidebar" class="sidebar" aria-label="Table of contents">
+            <div class="sidebar-scrollbox">
+                <ol class="chapter"><li class="chapter-item expanded "><a href="introduction.html"><strong aria-hidden="true">1.</strong> Introduction</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="introduction/a-path-to-pos-zcash.html"><strong aria-hidden="true">1.1.</strong> A Path to Proof-of-Stake Zcash</a></li><li class="chapter-item expanded "><a href="introduction/trailing-finality-layer-in-a-nutshell.html"><strong aria-hidden="true">1.2.</strong> Trailing Finality Layer in a Nutshell</a></li><li class="chapter-item expanded "><a href="introduction/status-and-next-steps.html"><strong aria-hidden="true">1.3.</strong> Status and Next Steps</a></li><li class="chapter-item expanded "><a href="introduction/motivating-finality.html"><strong aria-hidden="true">1.4.</strong> Motivating Finality</a></li><li class="chapter-item expanded "><a href="introduction/get-involved.html"><strong aria-hidden="true">1.5.</strong> Get Involved</a></li></ol></li><li class="chapter-item expanded "><a href="terminology.html"><strong aria-hidden="true">2.</strong> Terminology</a></li><li class="chapter-item expanded "><a href="design.html" class="active"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/design-at-a-glance.html"><strong aria-hidden="true">3.1.</strong> Design at a Glance</a></li><li class="chapter-item expanded "><a href="design/goals.html"><strong aria-hidden="true">3.2.</strong> Design Goals</a></li><li class="chapter-item expanded "><a href="design/crosslink.html"><strong aria-hidden="true">3.3.</strong> Crosslink</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/crosslink/the-arguments-for-bounded-availability-and-finality-overrides.html"><strong aria-hidden="true">3.3.1.</strong> The Arguments for Bounded Availability and Finality Overrides</a></li><li class="chapter-item expanded "><a href="design/crosslink/notes-on-snap-and-chat.html"><strong aria-hidden="true">3.3.2.</strong> Notes On Snap‑and‑Chat</a></li><li class="chapter-item expanded "><a href="design/crosslink/construction.html"><strong aria-hidden="true">3.3.3.</strong> The Crosslink 2 Construction</a></li><li class="chapter-item expanded "><a href="design/crosslink/security-analysis.html"><strong aria-hidden="true">3.3.4.</strong> Security Analysis of Crosslink 2</a></li><li class="chapter-item expanded "><a href="design/crosslink/questions.html"><strong aria-hidden="true">3.3.5.</strong> Questions About Crosslink</a></li><li class="chapter-item expanded "><a href="design/crosslink/potential-changes.html"><strong aria-hidden="true">3.3.6.</strong> Potential Changes to Crosslink</a></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">3.4.</strong> Model Code Architecture</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">3.5.</strong> Network Architecture</div></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">4.</strong> Protocol Specification</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">4.1.</strong> Abstract Protocol</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">4.1.1.</strong> Token Dynamics</div></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">4.2.</strong> Concrete Protocol</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">4.2.1.</strong> Token Dynamics</div></li></ol></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.</strong> Security</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.</strong> Abstract Analysis</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.1.</strong> Ebb-and-Flow analysis</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.2.</strong> Subprotocol Compromise Analysis</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.3.</strong> Model Code Architecture Analysis</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.4.</strong> Network Architecture Analysis</div></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.2.</strong> Concrete Analysis</div></li></ol></li><li class="chapter-item expanded "><a href="references.html"><strong aria-hidden="true">6.</strong> References</a></li><li class="chapter-item expanded "><a href="version-history.html"><strong aria-hidden="true">7.</strong> Version History</a></li></ol>
+            </div>
+            <div id="sidebar-resize-handle" class="sidebar-resize-handle"></div>
+        </nav>
+
+        <!-- Track and set sidebar scroll position -->
+        <script>
+            var sidebarScrollbox = document.querySelector('#sidebar .sidebar-scrollbox');
+            sidebarScrollbox.addEventListener('click', function(e) {
+                if (e.target.tagName === 'A') {
+                    sessionStorage.setItem('sidebar-scroll', sidebarScrollbox.scrollTop);
+                }
+            }, { passive: true });
+            var sidebarScrollTop = sessionStorage.getItem('sidebar-scroll');
+            sessionStorage.removeItem('sidebar-scroll');
+            if (sidebarScrollTop) {
+                // preserve sidebar scroll position when navigating via links within sidebar
+                sidebarScrollbox.scrollTop = sidebarScrollTop;
+            } else {
+                // scroll sidebar to current active section when navigating via "next/previous chapter" buttons
+                var activeSection = document.querySelector('#sidebar .active');
+                if (activeSection) {
+                    activeSection.scrollIntoView({ block: 'center' });
+                }
+            }
+        </script>
+
+        <div id="page-wrapper" class="page-wrapper">
+
+            <div class="page">
+                                <div id="menu-bar-hover-placeholder"></div>
+                <div id="menu-bar" class="menu-bar sticky">
+                    <div class="left-buttons">
+                        <label id="sidebar-toggle" class="icon-button" for="sidebar-toggle-anchor" title="Toggle Table of Contents" aria-label="Toggle Table of Contents" aria-controls="sidebar">
+                            <i class="fa fa-bars"></i>
+                        </label>
+                        <button id="theme-toggle" class="icon-button" type="button" title="Change theme" aria-label="Change theme" aria-haspopup="true" aria-expanded="false" aria-controls="theme-list">
+                            <i class="fa fa-paint-brush"></i>
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+                        <ul id="theme-list" class="theme-popup" aria-label="Themes" role="menu">
+                            <li role="none"><button role="menuitem" class="theme" id="light">Light</button></li>
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+                            <li role="none"><button role="menuitem" class="theme" id="navy">Navy</button></li>
+                            <li role="none"><button role="menuitem" class="theme" id="ayu">Ayu</button></li>
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+                        <button id="search-toggle" class="icon-button" type="button" title="Search. (Shortkey: s)" aria-label="Toggle Searchbar" aria-expanded="false" aria-keyshortcuts="S" aria-controls="searchbar">
+                            <i class="fa fa-search"></i>
+                        </button>
+                    </div>
+
+                    <h1 class="menu-title">Zcash Trailing Finality Layer - v0.1.0</h1>
+
+                    <div class="right-buttons">
+                        <a href="print.html" title="Print this book" aria-label="Print this book">
+                            <i id="print-button" class="fa fa-print"></i>
+                        </a>
+
+                    </div>
+                </div>
+
+                <div id="search-wrapper" class="hidden">
+                    <form id="searchbar-outer" class="searchbar-outer">
+                        <input type="search" id="searchbar" name="searchbar" placeholder="Search this book ..." aria-controls="searchresults-outer" aria-describedby="searchresults-header">
+                    </form>
+                    <div id="searchresults-outer" class="searchresults-outer hidden">
+                        <div id="searchresults-header" class="searchresults-header"></div>
+                        <ul id="searchresults">
+                        </ul>
+                    </div>
+                </div>
+
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+<h1 id="design-overview"><a class="header" href="#design-overview">Design Overview</a></h1>
+<p>This design augments the existing Zcash Proof‑of‑Work (PoW) network with a new consensus layer which provides <em>trailing finality</em>, called the <em>Trailing Finality Layer (TFL)</em>.</p>
+<p>This layer enables blocks produced via PoW to become <em>final</em> which ensures they may never be rolled back. This enables safer and simpler wallets and other infrastructure, and aids trust-minimized cross-chain bridges.</p>
+<p>This consensus layer uses a finalizing <em>Proof-of-Stake (PoS)</em> consensus protocol, and enables ZEC holders to earn protocol rewards for contributing to the security of the Zcash network. By integrating a PoS layer with the current PoW Zcash protocol, this design specifies a <em>hybrid consensus protocol</em>.</p>
+<p>The integration of the current PoW consensus with the TFL produces a new top-level consensus protocol referred to as <em>PoW+TFL</em>.</p>
+<p>In the following subchapters we introduce the <a href="./design/design-at-a-glance.html">Design at a Glance</a>, then provide an overview of the major components of the design.</p>
+<p>Following this overview chapter, we proceed into a detailed <a href="">Protocol Specification (TODO)</a>.</p>
+
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+                <ol class="chapter"><li class="chapter-item expanded "><a href="../introduction.html"><strong aria-hidden="true">1.</strong> Introduction</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../introduction/a-path-to-pos-zcash.html"><strong aria-hidden="true">1.1.</strong> A Path to Proof-of-Stake Zcash</a></li><li class="chapter-item expanded "><a href="../introduction/trailing-finality-layer-in-a-nutshell.html"><strong aria-hidden="true">1.2.</strong> Trailing Finality Layer in a Nutshell</a></li><li class="chapter-item expanded "><a href="../introduction/status-and-next-steps.html"><strong aria-hidden="true">1.3.</strong> Status and Next Steps</a></li><li class="chapter-item expanded "><a href="../introduction/motivating-finality.html"><strong aria-hidden="true">1.4.</strong> Motivating Finality</a></li><li class="chapter-item expanded "><a href="../introduction/get-involved.html"><strong aria-hidden="true">1.5.</strong> Get Involved</a></li></ol></li><li class="chapter-item expanded "><a href="../terminology.html"><strong aria-hidden="true">2.</strong> Terminology</a></li><li class="chapter-item expanded "><a href="../design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/design-at-a-glance.html"><strong aria-hidden="true">3.1.</strong> Design at a Glance</a></li><li class="chapter-item expanded "><a href="../design/goals.html"><strong aria-hidden="true">3.2.</strong> Design Goals</a></li><li class="chapter-item expanded "><a href="../design/crosslink.html" class="active"><strong aria-hidden="true">3.3.</strong> Crosslink</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/crosslink/the-arguments-for-bounded-availability-and-finality-overrides.html"><strong aria-hidden="true">3.3.1.</strong> The Arguments for Bounded Availability and Finality Overrides</a></li><li class="chapter-item expanded "><a href="../design/crosslink/notes-on-snap-and-chat.html"><strong aria-hidden="true">3.3.2.</strong> Notes On Snap‑and‑Chat</a></li><li class="chapter-item expanded "><a href="../design/crosslink/construction.html"><strong aria-hidden="true">3.3.3.</strong> The Crosslink 2 Construction</a></li><li class="chapter-item expanded "><a href="../design/crosslink/security-analysis.html"><strong aria-hidden="true">3.3.4.</strong> Security Analysis of Crosslink 2</a></li><li class="chapter-item expanded "><a href="../design/crosslink/questions.html"><strong aria-hidden="true">3.3.5.</strong> Questions About Crosslink</a></li><li class="chapter-item expanded "><a href="../design/crosslink/potential-changes.html"><strong aria-hidden="true">3.3.6.</strong> Potential Changes to Crosslink</a></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">3.4.</strong> Model Code Architecture</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">3.5.</strong> Network Architecture</div></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">4.</strong> Protocol Specification</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">4.1.</strong> Abstract Protocol</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">4.1.1.</strong> Token Dynamics</div></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">4.2.</strong> Concrete Protocol</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">4.2.1.</strong> Token Dynamics</div></li></ol></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.</strong> Security</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.</strong> Abstract Analysis</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.1.</strong> Ebb-and-Flow analysis</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.2.</strong> Subprotocol Compromise Analysis</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.3.</strong> Model Code Architecture Analysis</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.4.</strong> Network Architecture Analysis</div></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.2.</strong> Concrete Analysis</div></li></ol></li><li class="chapter-item expanded "><a href="../references.html"><strong aria-hidden="true">6.</strong> References</a></li><li class="chapter-item expanded "><a href="../version-history.html"><strong aria-hidden="true">7.</strong> Version History</a></li></ol>
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+<h1 id="crosslink"><a class="header" href="#crosslink">Crosslink</a></h1>
+<p>Crosslink is the proposed <a href="../terminology.html#definition-hybrid-construction">hybrid construction</a> for the Trailing Finality Layer. The current version is Crosslink 2.</p>
+<h2 id="contents"><a class="header" href="#contents">Contents</a></h2>
+<ul>
+<li><a href="./crosslink/the-arguments-for-bounded-availability-and-finality-overrides.html">The Arguments for Bounded Availability and Finality Overrides</a></li>
+<li><a href="./crosslink/notes-on-snap-and-chat.html">Notes On Snap‑and‑Chat</a></li>
+<li><a href="./crosslink/construction.html">The Crosslink 2 Construction</a></li>
+<li><a href="./crosslink/security-analysis.html">Security Analysis of Crosslink 2</a></li>
+<li><a href="./crosslink/questions.html">Questions About Crosslink</a></li>
+<li><a href="./crosslink/potential-changes.html">Potential Changes to Crosslink</a></li>
+</ul>
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+                <ol class="chapter"><li class="chapter-item expanded "><a href="../../introduction.html"><strong aria-hidden="true">1.</strong> Introduction</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../introduction/a-path-to-pos-zcash.html"><strong aria-hidden="true">1.1.</strong> A Path to Proof-of-Stake Zcash</a></li><li class="chapter-item expanded "><a href="../../introduction/trailing-finality-layer-in-a-nutshell.html"><strong aria-hidden="true">1.2.</strong> Trailing Finality Layer in a Nutshell</a></li><li class="chapter-item expanded "><a href="../../introduction/status-and-next-steps.html"><strong aria-hidden="true">1.3.</strong> Status and Next Steps</a></li><li class="chapter-item expanded "><a href="../../introduction/motivating-finality.html"><strong aria-hidden="true">1.4.</strong> Motivating Finality</a></li><li class="chapter-item expanded "><a href="../../introduction/get-involved.html"><strong aria-hidden="true">1.5.</strong> Get Involved</a></li></ol></li><li class="chapter-item expanded "><a href="../../terminology.html"><strong aria-hidden="true">2.</strong> Terminology</a></li><li class="chapter-item expanded "><a href="../../design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../design/design-at-a-glance.html"><strong aria-hidden="true">3.1.</strong> Design at a Glance</a></li><li class="chapter-item expanded "><a href="../../design/goals.html"><strong aria-hidden="true">3.2.</strong> Design Goals</a></li><li class="chapter-item expanded "><a href="../../design/crosslink.html"><strong aria-hidden="true">3.3.</strong> Crosslink</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../design/crosslink/the-arguments-for-bounded-availability-and-finality-overrides.html"><strong aria-hidden="true">3.3.1.</strong> The Arguments for Bounded Availability and Finality Overrides</a></li><li class="chapter-item expanded "><a href="../../design/crosslink/notes-on-snap-and-chat.html"><strong aria-hidden="true">3.3.2.</strong> Notes On Snap‑and‑Chat</a></li><li class="chapter-item expanded "><a href="../../design/crosslink/construction.html" class="active"><strong aria-hidden="true">3.3.3.</strong> The Crosslink 2 Construction</a></li><li class="chapter-item expanded "><a href="../../design/crosslink/security-analysis.html"><strong aria-hidden="true">3.3.4.</strong> Security Analysis of Crosslink 2</a></li><li class="chapter-item expanded "><a href="../../design/crosslink/questions.html"><strong aria-hidden="true">3.3.5.</strong> Questions About Crosslink</a></li><li class="chapter-item expanded "><a href="../../design/crosslink/potential-changes.html"><strong aria-hidden="true">3.3.6.</strong> Potential Changes to Crosslink</a></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">3.4.</strong> Model Code Architecture</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">3.5.</strong> Network Architecture</div></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">4.</strong> Protocol Specification</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">4.1.</strong> Abstract Protocol</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">4.1.1.</strong> Token Dynamics</div></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">4.2.</strong> Concrete Protocol</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">4.2.1.</strong> Token Dynamics</div></li></ol></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.</strong> Security</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.</strong> Abstract Analysis</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.1.</strong> Ebb-and-Flow analysis</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.2.</strong> Subprotocol Compromise Analysis</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.3.</strong> Model Code Architecture Analysis</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.4.</strong> Network Architecture Analysis</div></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.2.</strong> Concrete Analysis</div></li></ol></li><li class="chapter-item expanded "><a href="../../references.html"><strong aria-hidden="true">6.</strong> References</a></li><li class="chapter-item expanded "><a href="../../version-history.html"><strong aria-hidden="true">7.</strong> Version History</a></li></ol>
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+<h1 id="the-crosslink-2-construction"><a class="header" href="#the-crosslink-2-construction">The Crosslink 2 Construction</a></h1>
+<p>We are now ready to give a description of a protocol that takes into account the issues described in <a href="./notes-on-snap-and-chat.html">Notes on Snap‑and‑Chat</a>, and that implements <a href="./the-arguments-for-bounded-availability-and-finality-overrides.html#what-is-bounded-availability">bounded availability</a>. We call this the “Crosslink” construction; more precisely the version described here is “Crosslink 2”.</p>
+<p>This description will attempt to be self-contained, but <a href="https://eprint.iacr.org/2020/1091.pdf">[NTT2020]</a> (<a href="https://arxiv.org/pdf/2009.04987.pdf">arXiv version</a>) is useful background on the general model of Ebb-and-Flow protocols.</p>
+<h2 id="conventions"><a class="header" href="#conventions">Conventions</a></h2>
+<p>“<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>” is a metavariable for the name of a protocol. We also use it as a wildcard in protocol names of a particular type, for example <span style="white-space: nowrap">“<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bc”</span> for the name of some best‑chain protocol.</p>
+<p>Protocols are referred to as <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1757em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∗</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> for a <span style="white-space: nowrap">name “<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>”.</span> Where it is useful to avoid ambiguity, when referring to a concept defined by <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1757em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∗</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> we prefix it with <span style="white-space: nowrap">“<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>‑”.</span></p>
+<p>We do not take synchrony or partial synchrony as an implicit assumption of the communication model; that is, <em>unless otherwise specified</em>, messages between protocol participants can be arbitrarily delayed or dropped. A given message is received at most once, and messages are nonmalleably authenticated as originating from a given sender whenever needed by the applicable protocol. Particular subprotocols may require a stronger model.</p>
+<div id="admonition-background" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Background</p>
+<p><a class="admonition-anchor-link" href="#admonition-background"></a></p>
+</div>
+<div>
+<p>For an overview of communication models used to analyze distributed protocols, see <a href="https://decentralizedthoughts.github.io/2019-06-01-2019-5-31-models/">this blog post by Ittai Abraham</a>.</p>
+</div>
+</div>
+<details id="admonition-discussion-of-incorrect-applications-of-the-gst-formalization-of-partial-synchrony-to-continuously-operating-protocols" class="admonition admonish-info">
+<summary class="admonition-title">
+<p>Discussion of incorrect applications of the GST formalization of partial synchrony to continuously operating protocols.</p>
+<p><a class="admonition-anchor-link" href="#admonition-discussion-of-incorrect-applications-of-the-gst-formalization-of-partial-synchrony-to-continuously-operating-protocols"></a></p>
+</summary>
+<div>
+<p>The original context for the definition of the partially synchronous model in <a href="https://groups.csail.mit.edu/tds/papers/Lynch/jacm88.pdf">[DLS1988]</a> was for “one‑shot” Byzantine Agreement — called “the consensus problem” in that paper. The following argument is used to justify assuming that all messages from the Global Stabilization Time onward are delivered within the <span style="white-space: nowrap">upper time bound <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Δ</span></span></span></span>:</span></p>
+<blockquote>
+<p>Therefore, we impose an additional constraint: For each execution there is a global stabilization time (GST), unknown to the processors, such that the message system respects the upper bound <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Δ</span></span></span></span> from time GST onward.</p>
+<p>This constraint might at first seem too strong: In realistic situations, the upper bound cannot reasonably be expected to hold forever after GST, but perhaps only for a limited time. However, any good solution to the consensus problem in this model would have an <span style="white-space: nowrap">upper bound <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span></span> on the amount of time after GST required for consensus to be reached; in this case it is not really necessary that the bound <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Δ</span></span></span></span> hold forever after time GST, but only up to <span style="white-space: nowrap">time GST <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord">+</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">L</span></span></span></span>.</span> We find it technically convenient to avoid explicit mention of the interval length <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span> in the model, but will instead present the appropriate upper bounds on time for each of our algorithms.</p>
+</blockquote>
+<p>Several subsequent authors applying the partially synchronous model to block chains appear to have forgotten or neglected this context. In particular, the argument depends on the protocol completing soon after GST. Obviously a block‑chain protocol does not satisfy this assumption; it is not a “one‑shot” consensus problem.</p>
+<p>This assumption could be removed, but some authors of papers about block‑chain protocols have taken it to be an essential aspect of modelling partial synchrony. I believe this is contrary to the intent of <a href="https://groups.csail.mit.edu/tds/papers/Lynch/jacm88.pdf">[DLS1988]</a>:</p>
+<blockquote>
+<p>Instead of requiring that the consensus problem be solvable in the GST model, we might think of separating the correctness conditions into safety and termination properties. The safety conditions are that no two correct processors should ever reach disagreement, and that no correct processor should ever make a decision that is contrary to the specified validity conditions. The termination property is just that each correct processor should eventually make a decision. Then we might require an algorithm to satisfy the safety conditions no matter how asynchronously the message system behaves, that is, even if <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Δ</span></span></span></span> does not hold eventually. On the other hand, we might only require termination in case <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Δ</span></span></span></span> holds eventually. It is easy to see that these safety and termination conditions are [for the consensus problem] equivalent to our GST condition: If an algorithm solves the consensus problem when <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Δ</span></span></span></span> holds from time GST onward, then that algorithm cannot possibly violate a safety property even if the message system is completely asynchronous. This is because safety violations must occur at some finite point in time, and there would be some continuation of the violating execution in which <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Δ</span></span></span></span> eventually holds.</p>
+</blockquote>
+<p>This argument is correct as stated, i.e. for the one‑shot consensus problem. Subtly, essentially the same argument can be adapted to protocols with <em>safety</em> properties that need to be satisfied continuously. However, it cannot correctly be applied to <em>liveness</em> properties of non‑terminating protocols. The authors (Cynthia Dwork, Nancy Lynch, and Larry Stockmeyer) would certainly have known this: notice how they carefully distinguish “the GST model” from “partial synchrony”. They cannot plausibly have intended this GST formalization to be applied unmodified to analyze liveness in such protocols, which seems to be common in the block‑chain literature, including in the Ebb-and-Flow paper <a href="https://eprint.iacr.org/2020/1091.pdf">[NTT2020]</a> and the Streamlet paper <a href="https://eprint.iacr.org/2020/088.pdf">[CS2020]</a>.</p>
+<p>The Ebb-and-Flow paper acknowledges the issue by saying “Although in reality, multiple such periods of (a‑)synchrony could alternate, we follow the long‑standing practice in the BFT literature and study only a single such transition.” This is not adequate: “long‑standing practice” notwithstanding, it is not valid in general to infer that properties holding for the first transition to synchrony also apply to subsequent transitions (where the protocol can be in states that would not occur initially), and it is plausible that this inference could fail for real protocols. The Streamlet paper also refers to “periods of synchrony” which indicates awareness of the issue, but then it uses the unmodified GST model in the proofs.</p>
+<p>Informally, to solve this issue it is necessary to also prove that existing progress is maintained during periods of asynchrony, and that during such periods the protocol remains in states where it will be able to take advantage of a future period of synchrony to make further progress.</p>
+<p>This provides further motivation to avoid taking the GST formalization of partial synchrony as a basic assumption.</p>
+<p>Note that the recent result <a href="https://arxiv.org/pdf/2405.10249">[CGSW2024]</a> does not contradict anything we say here. Although the GST and Unknown Latency models are “equally demanding” in the sense of existence of protocols that satisfy a given goal, this result does not show that the models are equivalent for any specific protocol. In particular the requirements of the “clock‑slowing” technique fail in practice for any protocol involving Proof‑of‑Work.</p>
+</div>
+</details>
+<p><span style="white-space: nowrap">A <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>‑execution</span> is the complete set of events (message sends/receives and decisions by protocol participants) that occur in a particular <span style="white-space: nowrap">run of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1757em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∗</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span> from its initiation up to a given time. A prefix of a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>‑execution</span> is also a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>‑execution.</span> Since executions always start from protocol initiation, a strict suffix of a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>‑execution</span> is not a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>‑execution.</span></p>
+<p>Times are modelled as values of a totally ordered type <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathsf">Time</span></span></span></span></span> with minimum value <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>. For convenience, we consider all protocol executions to start at time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>.</p>
+<div id="admonition-remark" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Remark</p>
+<p><a class="admonition-anchor-link" href="#admonition-remark"></a></p>
+</div>
+<div>
+<p>Although protocols may be nondeterministic, an execution fixes the events that occur and times at which they occur, for the purpose of modeling.</p>
+<p>For simplicity, we assume that all events occur at global times in a total ordering. This assumption is not realistic in an asynchronous communication model, but it is not essential to the design or analysis and could be removed: we could use a partial happens-before ordering on events in place of a total ordering on times.</p>
+</div>
+</div>
+<p><span style="white-space: nowrap">A “<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>‑node”</span> is a participant in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1757em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∗</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> (the protocol may be implicit). <span style="white-space: nowrap">A <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>‑node</span> is <span style="white-space: nowrap">“honest at time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>”</span> in a given execution iff it has followed the protocol up to and including <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span></span> in that execution.</p>
+<p>A time series on <span style="white-space: nowrap">type <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span></span></span></span></span> is a function <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.1293em;"><span style="top:-2.6293em;"><span class="pstrut" style="height:2.5em;"></span><span class="mord"><span class="mord sizing reset-size6 size1">⦂</span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathsf">Time</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span></span></span></span> assigning a <span style="white-space: nowrap">value of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span></span></span></span></span> to each time in an execution. By convention, we will write the <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span></span> as a superscript: <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7936em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span>.</span></p>
+<p><span style="white-space: nowrap">A <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>‑chain</span> is a nonempty sequence of <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>‑blocks</span>, starting at the <span style="white-space: nowrap">“genesis block” <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1757em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∗</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>,</span> in which each subsequent block refers to its preceding or “parent block” by a collision‑resistant hash. The “tip” of <span style="white-space: nowrap">a <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>‑chain</span> is its last element.</p>
+<p>For convenience, we conflate <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>‑blocks</span> with <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>‑chains;</span> that is, we identify a chain with the block at its tip. This is justified because, assuming that the hash function used for parent links is collision‑resistant, there is exactly one <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>‑chain</span> corresponding to a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>‑block;</span> and conversely there is exactly one <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>‑block</span> at the tip of a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>‑chain.</span></p>
+<p>If <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span> is a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>‑chain,</span> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0991em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∗</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span> means <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span> with the last <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> blocks pruned, except that <span style="white-space: nowrap">if <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">len</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span>,</span> the result is the <span style="white-space: nowrap">genesis <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>‑chain</span> consisting only <span style="white-space: nowrap">of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1757em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∗</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</span></p>
+<p>The block at <span style="white-space: nowrap">depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">N</span></span></span></span></span> in a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>‑chain <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span></span> is defined to be the <span style="white-space: nowrap">tip of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0991em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∗</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span>.</span> Thus the block at <span style="white-space: nowrap">depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span></span> in a chain is the last one that cannot be affected by a rollback of <span style="white-space: nowrap">length <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span></span> (this also applies when <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">len</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> because the <span style="white-space: nowrap">genesis <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>‑chain</span> cannot roll back).</p>
+<div id="admonition-terminology" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Terminology</p>
+<p><a class="admonition-anchor-link" href="#admonition-terminology"></a></p>
+</div>
+<div>
+<p>Our usage of “depth” is different from <a href="https://eprint.iacr.org/2020/1091.pdf">[NTT2020]</a>, which uses “depth” to refer to what Bitcoin and Zcash call “height”. It also differs by <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> from the convention for confirmation depths in <code>zcashd</code>, where the tip is considered to be at <span style="white-space: nowrap">depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>,</span> rather <span style="white-space: nowrap">than <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>.</span></p>
+</div>
+</div>
+<p><span id="notation"></span>
+For <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>‑blocks <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span>:</span></p>
+<ul>
+<li>The notation <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1757em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∗</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span> means that the <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>‑chain</span> with <span style="white-space: nowrap">tip <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span></span> is a prefix of the one with <span style="white-space: nowrap">tip <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span>.</span> This includes the <span style="white-space: nowrap">case <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span>.</span></li>
+<li>The notation <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8193em;vertical-align:-0.136em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:-0.5em;"></span></span><span class="base"><span class="strut" style="height:0.786em;vertical-align:-0.15em;"></span><span class="mrel"><span class="mrel">⪰</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1757em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∗</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span> means that <span style="white-space: nowrap">either <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1757em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∗</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span> or <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1757em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∗</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span>.</span> That is, <span style="white-space: nowrap">“one of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span></span> is a prefix of the other”. This also includes the <span style="white-space: nowrap">case <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span>.</span></li>
+<li>The notation <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.5em;"></span><span class="mord"><span class="delimsizing size1">/</span></span><span class="mspace" style="margin-right:-1.06em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:-0.5em;"></span></span><span class="base"><span class="strut" style="height:0.786em;vertical-align:-0.15em;"></span><span class="mrel"><span class="mrel">⪰</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1757em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∗</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span> means that <span style="white-space: nowrap">both <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span></span><span class="base"><span class="strut" style="height:0.786em;vertical-align:-0.15em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1757em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∗</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span></span><span class="base"><span class="strut" style="height:0.786em;vertical-align:-0.15em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1757em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∗</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span>.</span> That is, <span style="white-space: nowrap">“neither of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span></span> is a prefix of the other”.</li>
+</ul>
+<p>A function <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.1293em;"><span style="top:-2.6293em;"><span class="pstrut" style="height:2.5em;"></span><span class="mord"><span class="mord sizing reset-size6 size1">⦂</span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">∗</span><span class="mord text"><span class="mord textsf">-block</span></span></span></span></span> is <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>‑linear</span> iff <span style="white-space: nowrap">for every <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8095em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">t</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">u</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.1293em;"><span style="top:-2.6293em;"><span class="pstrut" style="height:2.5em;"></span><span class="mord"><span class="mord sizing reset-size6 size1">⦂</span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span></span></span></span> where <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span></span> we have <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1757em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∗</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mopen">(</span><span class="mord mathnormal">u</span><span class="mclose">)</span></span></span></span>.</span> (This definition can be applied to time series <span style="white-space: nowrap">where <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathsf">Time</span></span></span></span></span>,</span> or to sequences of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>‑blocks where values of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span></span></span></span> are indices.)</p>
+<p><span id="linear-prefix-lemma"></span></p>
+<div id="admonition-lemma-linear-prefix" class="admonition admonish-success">
+<div class="admonition-title">
+<p>Lemma: Linear prefix</p>
+<p><a class="admonition-anchor-link" href="#admonition-lemma-linear-prefix"></a></p>
+</div>
+<div>
+<p>If <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1757em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∗</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1757em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∗</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span> then <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8193em;vertical-align:-0.136em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:-0.5em;"></span></span><span class="base"><span class="strut" style="height:0.786em;vertical-align:-0.15em;"></span><span class="mrel"><span class="mrel">⪰</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1757em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∗</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span>.</p>
+<p>Proof: The chain of ancestors of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span> is <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>-linear, and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> are both on that chain.</p>
+</div>
+</div>
+<p>The notation <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span><span class="mord text"><span class="mord"> for </span></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1757em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∗</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">]</span></span></span></span> means the sequence of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span></span></span> for each <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>‑block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span></span> in chain order from genesis up to and <span style="white-space: nowrap">including <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span>.</span> <span style="white-space: nowrap">(<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> is</span> a bound variable within this construct.)</p>
+<p><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="enclosing" style="background-color:yellow;color:black;"><span class="mord text"><span class="mord textsf">TODO:</span></span></span></span></span></span> remove this if not used:</p>
+<p>We use <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">log</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0307em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">log</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8362em;"><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> (without a subscript <span style="white-space: nowrap">on <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">⪯</span></span></span></span>)</span> to mean that the transaction ledger <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">log</span></span></span></span></span> is a prefix <span style="white-space: nowrap">of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0307em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">log</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8362em;"><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span>.</span> Similarly to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:-0.5em;"></span></span><span class="base"><span class="strut" style="height:0.786em;vertical-align:-0.15em;"></span><span class="mrel"><span class="mrel">⪰</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1757em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∗</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> above, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">log</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:-0.5em;"></span></span><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">⪰</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0307em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">log</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8362em;"><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> means that either <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">log</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0307em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">log</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8362em;"><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> or <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0307em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">log</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8362em;"><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">log</span></span></span></span></span>;</span> that is, <span style="white-space: nowrap">“one of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">log</span></span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0307em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">log</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8362em;"><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span> is a prefix of the other”.</p>
+<h2 id="views"><a class="header" href="#views">Views</a></h2>
+<p>In the simplest case, a block‑chain protocol <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1757em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∗</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> provides a single “view” that, <span style="white-space: nowrap">for a given <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>‑execution,</span> provides <span style="white-space: nowrap">each <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>‑node</span> with a time series <span style="white-space: nowrap">on <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>‑chains.</span> More generally a protocol may define several “views” that provide <span style="white-space: nowrap">each <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>‑node</span> with time series on potentially different chain types.</p>
+<p>We model a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>‑view</span> as a function <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.1293em;"><span style="top:-2.6293em;"><span class="pstrut" style="height:2.5em;"></span><span class="mord"><span class="mord sizing reset-size6 size1">⦂</span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathsf">Node</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathsf">Time</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">∗</span><span class="mord text"><span class="mord textsf">-chain</span></span></span></span></span>.</span> By convention, we will write the <span style="white-space: nowrap">node index <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span></span> as a subscript and the <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span></span> as a superscript: <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mopen">(</span><span class="mord mathnormal">i</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span>.</span></p>
+<p><span id="agreement-on-a-view"></span></p>
+<div id="admonition-definition-agreement-on-a-view" class="admonition admonish-success">
+<div class="admonition-title">
+<p>Definition: Agreement on a view</p>
+<p><a class="admonition-anchor-link" href="#admonition-definition-agreement-on-a-view"></a></p>
+</div>
+<div>
+<p>An execution of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Π</span></span></span></span> has <strong>Agreement</strong> on the view <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.1293em;"><span style="top:-2.6293em;"><span class="pstrut" style="height:2.5em;"></span><span class="mord"><span class="mord sizing reset-size6 size1">⦂</span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathsf">Node</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathsf">Time</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">∗</span><span class="mord text"><span class="mord textsf">-chain</span></span></span></span></span> iff <span style="white-space: nowrap">for all times <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span></span> and <span style="white-space: nowrap">all <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Π</span></span></span></span> nodes <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span></span> (potentially the same) such that <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span></span> and <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span>,</span> we have <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:-0.5em;"></span></span><span class="base"><span class="strut" style="height:0.786em;vertical-align:-0.15em;"></span><span class="mrel"><span class="mrel">⪰</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1757em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∗</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0781em;vertical-align:-0.3948em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4413em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3948em;"><span></span></span></span></span></span></span></span></span></span>.</span></p>
+</div>
+</div>
+<h2 id="subprotocols"><a class="header" href="#subprotocols">Subprotocols</a></h2>
+<p>As in Snap‑and‑Chat, we depend on a <span style="white-space: nowrap">BFT protocol <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>,</span> and a <span style="white-space: nowrap">best‑chain protocol <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>.</span></p>
+<div id="admonition-info" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Info</p>
+<p><a class="admonition-anchor-link" href="#admonition-info"></a></p>
+</div>
+<div>
+<p>See <a href="./the-arguments-for-bounded-availability-and-finality-overrides.html#terminology-note">this terminology note</a> for why we do not call <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> a “longest‑chain” protocol.</p>
+</div>
+</div>
+<p>We modify <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> <span style="white-space: nowrap">(resp. <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>)</span> to give <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> <span style="white-space: nowrap">(resp. <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>)</span> by adding structural elements, changing validity rules, and changing the specified behaviour of honest nodes.</p>
+<p>A Crosslink 2 node must participate in <span style="white-space: nowrap">both <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>;</span> that is, it must maintain a view of the state of each protocol. Acting in more specific roles such as bft‑proposer, bft‑validator, or bc‑block‑producer is optional, but we assume that all such actors are Crosslink 2 nodes.</p>
+<h3 id="model-for-bft-protocols-Πorigbftbft"><a class="header" href="#model-for-bft-protocols-Πorigbftbft">Model for BFT protocols (Π<sub>{origbft,bft}</sub>)</a></h3>
+<p><span style="white-space: nowrap">A <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑node’s view</span> includes a set of <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑block chains</span> each rooted at a fixed genesis <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</span> There is a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑block‑validity</span> rule (specified below), which depends only on the content of the block and its ancestors. A non‑genesis block can only be <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑block‑valid</span> if its parent is <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑block‑valid. <span style="white-space: nowrap">A <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑valid‑chain</span> is a chain of <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑block‑valid</span> blocks.</p>
+<p>Execution proceeds in a sequence of epochs. In each epoch, an honest proposer for that epoch may make a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑proposal</span>.</p>
+<p>A <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑proposal</span> refers to a parent <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑block,</span> and specifies the proposal’s epoch. The content of a proposal is signed by the proposer using a strongly unforgeable signature scheme. We consider the proposal to include this signature. There is a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑proposal‑validity</span> rule, depending only on the content of the proposal and its parent block, and the validity of the proposer’s signature.</p>
+<p>We extend the <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0991em;vertical-align:-0.25em;"></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span></span></span></span> notation to <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑proposals</span> in the obvious way: <span style="white-space: nowrap">if <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8304em;vertical-align:-0.136em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>,</span> <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span> is a <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑proposal</span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> its parent <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑block,</span> then <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1322em;vertical-align:-0.2831em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mrel">:=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1905em;vertical-align:-0.3013em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8892em;"><span style="top:-2.3987em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span><span style="top:-3.1031em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3013em;"><span></span></span></span></span></span></span></span></span></span>.</span></p>
+<div id="admonition-terminology-1" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Terminology</p>
+<p><a class="admonition-anchor-link" href="#admonition-terminology-1"></a></p>
+</div>
+<div>
+<p>We will shorten <span style="white-space: nowrap">“<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑block‑valid <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑block”</span> to <span style="white-space: nowrap">“<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑valid‑block”,</span> and <span style="white-space: nowrap">“<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑proposal‑valid <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑proposal”</span> to <span style="white-space: nowrap">“<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑valid‑proposal”.</span></p>
+</div>
+</div>
+<p>For each epoch, there is a fixed number of voting units distributed between the <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑nodes,</span> which they use to vote for a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑proposal.</span> We say that a voting unit has been cast for a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑proposal <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span></span> at a given time in a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑execution,</span> <span style="white-space: nowrap">if and only if</span> <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span> is <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑proposal‑valid</span> and a ballot <span style="white-space: nowrap">for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span></span> authenticated by the holder of the voting unit exists at that time.</p>
+<p>Using knowledge of ballots cast for a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑proposal <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span></span> that collectively satisfy a notarization rule at a given time in a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑execution,</span> and only with such knowledge, it is possible to obtain a valid <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑notarization‑proof <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9386em;vertical-align:-0.2441em;"></span><span class="mord"><span class="mord"><span class="mord mathsf" style="margin-right:0.06944em;">proof</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2342em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">P</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span></span></span></span>.</span> The notarization rule must require at least a two‑thirds absolute supermajority of voting units <span style="white-space: nowrap">in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span>’s epoch</span> to have been cast <span style="white-space: nowrap">for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span>.</span> It may also require other conditions.</p>
+<p>A voting unit is cast non‑honestly for an epoch’s proposal iff:</p>
+<ul>
+<li>it is cast other than by the holder of the unit (due to key compromise or any flaw in the voting protocol, for example); or</li>
+<li>it is double‑cast (i.e. there are at least two ballots casting it for distinct proposals); or</li>
+<li>the holder of the unit following the conditions for honest voting <span style="white-space: nowrap">in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>,</span> according to its view, should not have cast that vote.</li>
+</ul>
+<p>Note that a unit should be considered to be cast non-honestly in the case of key compromise, because it is then effectively under the control of an adversary. The key compromise may or may not be attributable to another flaw in the protocol, but such a flaw would not be a break of the consensus mechanism per se.</p>
+<p><span id="one-third-bound"></span></p>
+<div id="admonition-definition-onethird-bound-on-nonhonest-voting" class="admonition admonish-success">
+<div class="admonition-title">
+<p>Definition: One‑third bound on non‑honest voting</p>
+<p><a class="admonition-anchor-link" href="#admonition-definition-onethird-bound-on-nonhonest-voting"></a></p>
+</div>
+<div>
+<p>An execution of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has the <strong>one‑third bound on non‑honest voting</strong> property iff for every epoch, <em>strictly</em> fewer than one third of the total voting units for that epoch are ever cast non‑honestly.</p>
+</div>
+</div>
+<div id="admonition-info-1" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Info</p>
+<p><a class="admonition-anchor-link" href="#admonition-info-1"></a></p>
+</div>
+<div>
+<p>It may be the case that a ballot cast for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span> is not <a href="#in-honest-view">in honest view</a> when it is used to create a notarization proof for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span>. Since we are not assuming synchrony, it may also be the case that such a ballot is in honest view but that any given node has not received it (and perhaps will never receive it).</p>
+<p>There may be multiple distinct ballots or distinct ballot messages attempting to cast a given voting unit for the same proposal; this is undesirable for bandwidth usage, but it is not necessary to consider it to be non‑honest behaviour for the purpose of security analysis, as long as such ballots are not double‑counted toward the two‑thirds threshold.</p>
+</div>
+</div>
+<p><span id="one-third-bound-caveat"></span></p>
+<div id="admonition-security-caveat" class="admonition admonish-warning">
+<div class="admonition-title">
+<p>Security caveat</p>
+<p><a class="admonition-anchor-link" href="#admonition-security-caveat"></a></p>
+</div>
+<div>
+<p>The <strong>one‑third bound on non‑honest voting</strong> property considers all ballots cast in the entire execution. In particular, it is possible that a validator’s key is compromised and then used to cast its voting units for a proposal of an epoch long finished. If the number of voting units cast non-honestly for any epoch <em>ever</em> reaches one third of the total voting units for that epoch during an execution, then the <strong>one‑third bound on non‑honest voting</strong> property is violated for that execution.</p>
+<p>Therefore, validator keys of honest nodes must remain secret indefinitely. Whenever a key is rotated, the old key must be securely deleted. For further discussion and potential improvements, see <a href="https://github.com/Electric-Coin-Company/tfl-book/issues/140">tfl-book issue #140</a>.</p>
+</div>
+</div>
+<p>A <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑block</span> consists <span style="white-space: nowrap">of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord mathsf" style="margin-right:0.06944em;">proof</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2342em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">P</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span> re‑signed by the same proposer using a strongly unforgeable signature scheme. It is <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑block‑valid</span> iff:</p>
+<ul>
+<li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span> is <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑proposal‑valid</span>; and</li>
+<li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9386em;vertical-align:-0.2441em;"></span><span class="mord"><span class="mord"><span class="mord mathsf" style="margin-right:0.06944em;">proof</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2342em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">P</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span></span></span></span> is a valid proof that some subset of ballots cast for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span> are sufficient to satisfy the notarization rule; and</li>
+<li>the proposer’s outer signature <span style="white-space: nowrap">on <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord mathsf" style="margin-right:0.06944em;">proof</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2342em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">P</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span> is valid.</li>
+</ul>
+<p><span style="white-space: nowrap">A <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑proposal’s</span> parent reference hashes the entire <span style="white-space: nowrap">parent <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑block,</span> i.e. proposal, proof, and outer signature.</p>
+<div id="admonition-info-2" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Info</p>
+<p><a class="admonition-anchor-link" href="#admonition-info-2"></a></p>
+</div>
+<div>
+<p>Neither <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9386em;vertical-align:-0.2441em;"></span><span class="mord"><span class="mord"><span class="mord mathsf" style="margin-right:0.06944em;">proof</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2342em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">P</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span></span></span></span> nor the proposer’s outer signature is unique for a <span style="white-space: nowrap">given <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span>.</span> The proposer’s outer signature is however third‑party nonmalleable, by definition of a strongly unforgeable signature scheme. An <span style="white-space: nowrap">“honest <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑proposal”</span> is a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑proposal</span> made for a given epoch by a proposer who is honest in that epoch. Such a proposer will only create one proposal and only sign at most once for each epoch, and so there will be at most one <span style="white-space: nowrap">“honestly submitted”</span> <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑block</span> for each epoch.</p>
+<p>It is possible for there to be multiple <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑valid‑blocks</span> for the same proposal, with different notarization proofs and/or outer signatures, if the proposer is not honest. However, the property that there will be at most one <span style="white-space: nowrap">“honestly submitted”</span> <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑block</span> for each epoch is important for liveness, even though we cannot guarantee that any particular proposer for an epoch is honest.</p>
+<p><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="enclosing" style="background-color:yellow;color:black;"><span class="mord text"><span class="mord textsf">TODO:</span></span></span></span></span></span> check that we are correctly using this in the liveness analysis.</p>
+</div>
+</div>
+<p>There is an efficiently computable function <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">∗</span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.1293em;"><span style="top:-2.6293em;"><span class="pstrut" style="height:2.5em;"></span><span class="mord"><span class="mord sizing reset-size6 size1">⦂</span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">∗</span><span class="mord text"><span class="mord textsf">bft-block</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">∗</span><span class="mord text"><span class="mord textsf">bft-block</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∪</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord">⊥</span><span class="mclose">}</span></span></span></span>.</span> <span style="white-space: nowrap">For a <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑block‑valid</span> input <span style="white-space: nowrap">block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span>,</span> this function outputs the last ancestor of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span> that is final in the <span style="white-space: nowrap">context of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span>.</span></p>
+<div id="admonition-info-3" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Info</p>
+<p><a class="admonition-anchor-link" href="#admonition-info-3"></a></p>
+</div>
+<div>
+<p>The chain of ancestors is unambiguously determined because a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑proposal’s</span> parent reference hashes the entire parent <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑block;</span> each <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑block</span> commits to a proposal; and the parent hashes are collision‑resistant. This holds despite the caveat mentioned above that there may be multiple <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑valid‑blocks</span> for the same proposal.</p>
+</div>
+</div>
+<p><span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">∗</span><span class="mord text"><span class="mord textsf">bft-last-final</span></span></span></span></span></span> must satisfy all of the following:</p>
+<ul>
+<li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∗</span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7184em;vertical-align:-0.024em;"></span><span class="mord">⊥</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟺</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span> is not <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑block‑valid.</li>
+<li>If <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span> is <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑block‑valid, then:
+<ul>
+<li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∗</span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span> (and therefore it must also be <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑block‑valid);</span></li>
+<li>for all <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑valid‑blocks</span> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span> such that <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span>,</span> <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∗</span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∗</span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mclose">)</span></span></span></span>.</span></li>
+</ul>
+</li>
+<li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∗</span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</li>
+</ul>
+<div id="admonition-info-4" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Info</p>
+<p><a class="admonition-anchor-link" href="#admonition-info-4"></a></p>
+</div>
+<div>
+<p>It is correct to talk about the “last final block” of a given chain (that is, each <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑valid-block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span></span> unambiguously determines a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑valid-block</span> <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∗</span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mclose">)</span></span></span></span>)</span>, but it is not correct to refer to a given <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑block</span> as objectively <span style="white-space: nowrap">“<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑final”.</span></p>
+</div>
+</div>
+<p>A particular BFT protocol might need adaptations to fit it into this model <span style="white-space: nowrap">for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>,</span> <em>before</em> we apply the Crosslink 2 modifications to <span style="white-space: nowrap">obtain <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</span> Any such adaptions are necessarily protocol-specific. In particular:</p>
+<ul>
+<li>origbft‑proposal‑validity should correspond to the strongest property of an origbft‑proposal that is objectively and feasibly verifiable from the content of the proposal and its parent origbft‑block at the time the proposal is made. It must include verification of the proposer’s signature.</li>
+<li>origbft‑block‑validity should correspond to the strongest property of an origbft‑block that is objectively and feasibly verifiable from the content of the block and its ancestors at the time the block is added to an origbft‑chain. It should typically include all of the relevant checks from origbft‑proposal‑validity that apply to the created block (or equivalent checks). It must also include verification of the notarization proof and the proposer’s outer signature.</li>
+<li>If a node observes an origbft‑valid block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span>, then it should be infeasible for an adversary to cause a rollback in that node’s view <span style="white-space: nowrap">past <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">origbft-last-final</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mclose">)</span></span></span></span>,</span> and the view of the chain <span style="white-space: nowrap">up to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">origbft-last-final</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mclose">)</span></span></span></span></span> should agree with that of all other honest nodes. This is formalized in the next section.</li>
+</ul>
+<h4 id="safety-of-Πbft"><a class="header" href="#safety-of-Πbft">Safety of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></a></h4>
+<p>The intuition behind the following safety property is that:</p>
+<ul>
+<li>For <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> to be safe, it should never be the case that two honest nodes observe (at any time) <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑blocks <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span> respectively that they each consider final in some context, but <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8193em;vertical-align:-0.136em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:-0.5em;"></span></span><span class="base"><span class="strut" style="height:0.786em;vertical-align:-0.15em;"></span><span class="mrel"><span class="mrel">⪰</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span> does not hold.</li>
+<li>By definition, an honest node observes a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑block</span> to be final in the context of another <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span>,</span> <span style="white-space: nowrap">iff <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∗</span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mclose">)</span></span></span></span>.</span></li>
+</ul>
+<p><span id="in-honest-view"></span>
+We say that a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑block</span> is “in honest view” if a party observes it at some time at which that party is honest.</p>
+<p><span id="final-agreement"></span></p>
+<div id="admonition-definition-final-agreement" class="admonition admonish-success">
+<div class="admonition-title">
+<p>Definition: Final Agreement</p>
+<p><a class="admonition-anchor-link" href="#admonition-definition-final-agreement"></a></p>
+</div>
+<div>
+<p>An execution of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has <strong>Final Agreement</strong> iff for all <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑valid blocks <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span></span> in honest view at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span></span> and <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> in honest view</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span>,</span> we have <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∗</span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:-0.5em;"></span></span><span class="base"><span class="strut" style="height:0.786em;vertical-align:-0.15em;"></span><span class="mrel"><span class="mrel">⪰</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord">∗</span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>.</span></p>
+</div>
+</div>
+<p>Note that it is possible for this property to hold for an execution of a BFT protocol in an asynchronous communication model. As <a href="#one-third-bound-caveat">previously mentioned</a>, if the <a href="#one-third-bound"><strong>one‑third bound on non‑honest voting</strong></a> property is <em>ever</em> broken at any time in an execution, then it may not be possible to maintain <strong>Final Agreement</strong> from that point on.</p>
+<details id="admonition-adapting-the-streamlet-bft-protocol" class="admonition admonish-info">
+<summary class="admonition-title">
+<p>Adapting the Streamlet BFT protocol.</p>
+<p><a class="admonition-anchor-link" href="#admonition-adapting-the-streamlet-bft-protocol"></a></p>
+</summary>
+<div>
+<p>Streamlet as described in <a href="https://eprint.iacr.org/2020/088.pdf">[CS2020]</a> has three possible states of a block in a player’s view:</p>
+<ul>
+<li>“valid” (but not notarized or final);</li>
+<li>“notarized” (but not final);</li>
+<li>“final”.</li>
+</ul>
+<p>By “valid” the Streamlet paper means just that it satisfies the structural property of being part of a block chain with parent hashes. The role of <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑block‑validity</span> in our model corresponds roughly to Streamlet’s “notarized”. It turns out that with some straightforward changes relative to Streamlet, we can identify “origbft‑block‑valid” with “notarized” and consider an origbft‑valid‑chain to only consist of notarized blocks. This is not obvious, but is a useful simplification.</p>
+<p>Here is how the paper defines “notarized”:</p>
+<blockquote>
+<p>When a block gains votes from at least <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mord mathnormal">n</span><span class="mord">/3</span></span></span></span> distinct players, it becomes notarized. A chain is notarized if its constituent blocks are all notarized.</p>
+</blockquote>
+<p>This implies that blocks can be added to chains independently of notarization. However, the paper also says that an honest leader always proposes a block extending from a notarized chain. Therefore, only <em>notarized</em> chains really matter in the protocol.</p>
+<p>In unmodified Streamlet, the order in which a player sees signatures might cause it to view blocks as notarized out of order. Streamlet’s security analysis is in a synchronous model, and assumes for liveness that any vote will have been received by all players (Streamlet nodes) within two epochs.</p>
+<p>In Crosslink 2, however, we need origbft‑block‑validity to be an objectively and feasibly verifiable property. We also would prefer reliable message delivery within bounded time not to be a basic assumption of our communication model. (This does not dictate what assumptions about message delivery are made for particular security analyses.) If we did not make a modification to the protocol to take this into account, then some Crosslink 2 nodes might receive a two‑thirds absolute supermajority of voting messages and consider a BFT block to be notarized, while others might never receive enough of those messages.</p>
+<p>Obviously a <em>proposal</em> cannot include signatures on itself — but the block formed from it can include proofs about the proposal and signatures. We can therefore say that when a proposal gains a two‑thirds absolute supermajority of signatures, a block is created from it that contains a proof (such as an aggregate signature) that it had such a supermajority. For example, we can have the proposer itself make this proof once it has enough votes, sign the <span style="white-space: nowrap">resulting <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord mathsf" style="margin-right:0.06944em;">proof</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2342em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">P</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span> to create a block, then <em>submit</em> that block in a separate message. (The proposer has most incentive to do this in order to gain whatever reward attaches to a successful proposal; it can outsource the proving task if needed.) Then the origbft‑block‑validity rule can require a valid supermajority proof, which is objectively and feasibly verifiable. Players that see an origbft‑valid‑block can immediately consider it notarized.</p>
+<p>Note that for the liveness analysis to be unaffected, we need to assume that the combined latency of messages, of collecting and aggregating signatures, and of block submission is such that all adapted‑Streamlet nodes will receive a notarized block corresponding to a given proposal (rather than just all of the votes for the proposal) within two epochs. Alternatively we could re‑do the timing analysis.</p>
+<p>With this change, “origbft‑block‑valid” and “notarized” do not need to be distinguished.</p>
+<p>Streamlet’s finality rule is:</p>
+<blockquote>
+<p>If in any notarized chain, there are three adjacent blocks with consecutive epoch numbers, the prefix of the chain up to the second of the three blocks is considered final. When a block becomes final, all of its prefix must be final too.</p>
+</blockquote>
+<p>We can straightforwardly express this as an <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord textsf">origbft-last-final</span></span></span></span></span> function of a context block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span>, as required by the model:</p>
+<p>For an origbft‑valid‑block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">origbft-last-final</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mclose">)</span></span></span></span> is the last origbft‑valid‑block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span> such that either <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> or <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> is the second block of a group of three adjacent blocks with consecutive epoch numbers.</p>
+<p>Note that “When a block becomes final, all of its prefix must be final too.” is implicit in the model.</p>
+</div>
+</details>
+<h3 id="model-for-best-chain-protocols-Πorigbcbc"><a class="header" href="#model-for-best-chain-protocols-Πorigbcbc">Model for best-chain protocols (Π<sub>{origbc,bc}</sub>)</a></h3>
+<p>A node’s view in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> includes a set of <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bc‑block chains</span> each rooted at a fixed <span style="white-space: nowrap">genesis <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bc‑block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</span> There is a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bc‑block‑validity rule</span> (often described as a collection of <span style="white-space: nowrap">“consensus rules”),</span> depending only on the content of the block and its ancestors. A non‑genesis block can only be <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bc‑block‑valid</span> if its parent is <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bc‑block‑valid.</span> <span style="white-space: nowrap">By “<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bc‑valid‑chain”</span> we mean a chain of <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bc‑block‑valid blocks.</span></p>
+<div id="admonition-terminology-2" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Terminology</p>
+<p><a class="admonition-anchor-link" href="#admonition-terminology-2"></a></p>
+</div>
+<div>
+<p>The terminology commonly used in the block‑chain community does not distinguish between rules that are part of the consensus protocol proper, and rules required for validity of the economic computation supported by the block chain. Where it is necessary to distinguish, the former can be called “L0” consensus rules, and the latter “L1” consensus rules.</p>
+</div>
+</div>
+<p>The definition of <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bc‑block‑validity</span> is such that it is hard for a block producer to extend a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bc‑valid‑chain</span> unless they are selected by a random process that chooses a block producer in proportion to their resources with an approximately known and consistent time distribution, subject to some assumption about the total proportion of resources held by honest nodes.</p>
+<p>There is a function <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord"><span class="mord mathsf">score</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.1293em;"><span style="top:-2.6293em;"><span class="pstrut" style="height:2.5em;"></span><span class="mord"><span class="mord sizing reset-size6 size1">⦂</span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">∗</span><span class="mord text"><span class="mord textsf">bc-valid-chain</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathsf">Score</span></span></span></span></span>, with <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">&lt;</span></span></span></span> a <a href="https://proofwiki.org/wiki/Definition:Strict_Total_Ordering">strict total ordering</a></span> <span style="white-space: nowrap">on <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathsf">Score</span></span></span></span></span>.</span> An honest node will choose one of the <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bc‑valid‑chains</span> with highest score as the <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bc‑best‑chain</span> in its view. Any rule can be specified for breaking ties.</p>
+<p>The <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord"><span class="mord mathsf">score</span></span></span></span></span> function is required to satisfy <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0972em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord mathsf">score</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mspace mtight" style="margin-right:-0.1952em;"></span><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">score</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span></span></span> for any non‑genesis <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bc‑valid‑chain <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span>.</span></p>
+<div id="admonition-info-5" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Info</p>
+<p><a class="admonition-anchor-link" href="#admonition-info-5"></a></p>
+</div>
+<div>
+<p>For a Proof‑of‑Work protocol, the score of a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bc‑chain</span> should be its accumulated work.</p>
+</div>
+</div>
+<p>Unless an adversary is able to censor knowledge of other chains from a node’s view, it should be difficult to cause the node to switch to a chain with a last common ancestor more than <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> blocks</span> back from the tip of its previous <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bc‑best‑chain.</span></p>
+<p>Let <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathsf">ch</span></span></span></span></span> be a <a href="#views">view</a> such that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span> is <span style="white-space: nowrap">node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>’s <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bc‑best‑chain</span> at time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>. (This matches the notation used in <a href="https://eprint.iacr.org/2020/1091.pdf">[NTT2020]</a>.) We define <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1454em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8984em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span> to be <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</p>
+<p>A <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bc‑valid‑block</span> is assumed to commit to a collection (usually, a sequence) of <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bc‑transactions</span>. Unlike in Crosslink 1 or Snap-and-Chat, we do not need to explicitly model <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bc‑transaction</span> validity or impose any additional constraints on it. The consensus rules applying to <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bc‑transactions</span> are entirely unchanged, including any rules that depend on <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bc‑block</span> height or previous <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bc‑blocks.</span> This is because Crosslink 2 never reorders or selectively “sanitizes” transactions as Snap-and-Chat does. If a bc‑block is included in a Crosslink 2 block chain then its entire parent bc‑block chain is included just as it would have been in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> (only modified by the <a href="#structural-additions">structural additions</a> described later), so block heights are also preserved.</p>
+<p>A “coinbase transaction” is a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bc‑transaction</span> that only distributes newly issued funds and has no inputs.</p>
+<p>Define <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord textsf">is-coinbase-only-block</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.1293em;"><span style="top:-2.6293em;"><span class="pstrut" style="height:2.5em;"></span><span class="mord"><span class="mord sizing reset-size6 size1">⦂</span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">∗</span><span class="mord text"><span class="mord textsf">bc-block</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathsf">boolean</span></span></span></span></span> so that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">is-coinbase-only-block</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.5714em;"></span><span class="mord"><span class="mord mathsf">true</span></span></span></span></span> iff <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> has exactly one transaction that is a coinbase transaction.</p>
+<p>Each <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bc‑block</span> is summarized by a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bc‑header</span> that commits to the block. There is a notion of <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bc‑header‑validity</span> that is necessary, but not sufficient, for validity of the block. We will only make the distinction between <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bc‑headers</span> and <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bc‑blocks</span> when it is necessary to avoid ambiguity.</p>
+<details id="admonition-header-validity-for-proofofwork-protocols" class="admonition admonish-info">
+<summary class="admonition-title">
+<p>Header validity for Proof‑of‑Work protocols.</p>
+<p><a class="admonition-anchor-link" href="#admonition-header-validity-for-proofofwork-protocols"></a></p>
+</summary>
+<div>
+<p>In a Proof‑of‑Work protocol, it is normally possible to check the Proof‑of‑Work of a block using only the header. There is a difficulty adjustment function that determines the target difficulty for a block based on its parent chain. So, checking that the correct difficulty target has been used relies on knowing that the header’s parent chain is valid.</p>
+<p>Checking header validity before expending further resources on a purported block can be relevant to mitigating denial‑of‑service attacks that attempt to inflate validation cost.</p>
+</div>
+</details>
+<p>Typically, Bitcoin‑derived best chain protocols do not need much adaptation to fit into this model. The model still omits some details that would be important to implementing Crosslink 2, but distracting for this level of abstraction.</p>
+<h4 id="safety-of-Πbc"><a class="header" href="#safety-of-Πbc">Safety of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></a></h4>
+<p>We make an assumption on executions of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> that we will call <strong>Prefix Consistency</strong> (introduced in [<a href="https://eprint.iacr.org/2016/454.pdf">PSS2016</a>, section 3.3] as just “consistency”):</p>
+<p><span id="prefix-consistency"></span></p>
+<div id="admonition-definition-prefix-consistency" class="admonition admonish-success">
+<div class="admonition-title">
+<p>Definition: Prefix Consistency</p>
+<p><a class="admonition-anchor-link" href="#admonition-definition-prefix-consistency"></a></p>
+</div>
+<div>
+<p>An execution of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has <strong>Prefix Consistency</strong> at confirmation depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>, iff <span style="white-space: nowrap">for all times <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span></span> and <span style="white-space: nowrap">all nodes <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span></span> (potentially the same) such that <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span></span> and <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span>,</span> we have that <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.161em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mspace mtight" style="margin-right:-0.1952em;"></span><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1318em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span>.</span></p>
+</div>
+</div>
+<details id="admonition-explain-the-confusion-in-the-literature-about-what-variants-of-this-property-are-called" class="admonition admonish-info">
+<summary class="admonition-title">
+<p>Explain the confusion in the literature about what variants of this property are called.</p>
+<p><a class="admonition-anchor-link" href="#admonition-explain-the-confusion-in-the-literature-about-what-variants-of-this-property-are-called"></a></p>
+</summary>
+<div>
+<p>The literature uses the same name, <span style="white-space: nowrap">“common‑prefix property”,</span> for two different properties of very different strength.</p>
+<p>[<a href="https://eprint.iacr.org/2016/454.pdf">PSS2016</a>, section 3.3] introduced the stronger variant. That paper first describes the weaker variant, calling it the “common‑prefix property by Garay et al <a href="https://link.springer.com/chapter/10.1007/978-3-662-46803-6_10">[GKL2015]</a>.” Then it explains what is essentially a bug in that variant, and describes the stronger variant which it just calls “consistency”:</p>
+<blockquote>
+<p>The common‑prefix property by Garay et al <a href="https://link.springer.com/chapter/10.1007/978-3-662-46803-6_10">[GKL2015]</a>, which was already considered and studied by Nakamoto <a href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.221.9986">[Nakamoto2008]</a>, requires that in any round <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span></span></span></span>, the record chains of any two honest players <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> agree on all, but potentially the last <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span></span></span></span>, records. We note that this property (even in combination with the other two desiderata [of Chain Growth and Chain Quality]) provides quite weak guarantees: even if any two honest parties perfectly agree on the chains, the chain could be completely different on, say, even rounds and odd rounds. We here consider a stronger notion of consistency which additionally stipulates players should be consistent with their “future selves”.</p>
+<p>Let <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1606em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">consistent</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9106em;"><span style="top:-3.1323em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">view</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> iff for <span style="white-space: nowrap">all rounds <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span>,</span> and <span style="white-space: nowrap">all players <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span></span> (potentially the same) such that <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> is honest at <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7337em;"></span><span class="mord"><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">view</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7337em;"><span style="top:-3.1323em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span></span></span></span></span></span></span></span></span> and <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> is honest at <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0118em;"></span><span class="mord"><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">view</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0118em;"><span style="top:-3.1323em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8278em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span>,</span> we have that the prefixes of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0087em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.05834em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4413em;margin-left:-0.0583em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">view</span></span><span class="mclose">)</span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3373em;vertical-align:-0.3948em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.05834em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9425em;"><span style="top:-2.4413em;margin-left:-0.0583em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8278em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3948em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">view</span></span><span class="mclose">)</span></span></span></span> consisting of the first <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">ℓ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0087em;vertical-align:-0.2587em;"></span><span class="mord">∣</span><span class="mord"><span class="mord mathcal" style="margin-right:0.05834em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4413em;margin-left:-0.0583em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">view</span></span><span class="mclose">)</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span></span></span></span> records are identical.</p>
+</blockquote>
+<p>Unfortunately, <a href="https://eprint.iacr.org/2014/765.pdf">[GKL2020]</a>, which is a revised version of <a href="https://link.springer.com/chapter/10.1007/978-3-662-46803-6_10">[GKL2015]</a>, switches to the stronger variant <em>without changing the name</em>.</p>
+<p>(The <a href="https://eprint.iacr.org/archive/versions/2014/765">eprint version history</a> may be useful; the change was made in <a href="https://eprint.iacr.org/archive/2014/765/20181013:200033">version 20181013:200033</a>, page 17.)</p>
+<p>Note that [GKL2020] uses an adaptive‑corruption model, “meaning that the adversary is allowed to take control of parties on the fly”, and so their wording in Definition 3:</p>
+<blockquote>
+<p>... for any pair of <span style="white-space: nowrap">honest players <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span> adopting the <span style="white-space: nowrap">chains <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.05834em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0583em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.05834em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0583em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span> at rounds <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.786em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> in <span style="white-space: nowrap"><font style="font-variant: small-caps;">view</font><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3411em;vertical-align:-0.4296em;"></span><span class="mord"><span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9115em;"><span style="top:-2.4065em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">Π</span><span class="mpunct mtight">,</span><span class="mord mathcal mtight">A</span><span class="mpunct mtight">,</span><span class="mord mathcal mtight" style="margin-right:0.07944em;">Z</span></span></span></span><span style="top:-3.1809em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4296em;"><span></span></span></span></span></span></span></span></span></span></span> respectively, it holds that <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3111em;vertical-align:-0.2663em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.05834em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em;"><span style="top:-2.4337em;margin-left:-0.0583em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span><span style="top:-3.2198em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">⌈</span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2663em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.05834em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0583em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</span></p>
+</blockquote>
+<p>is intended to mean the same as our</p>
+<blockquote>
+<p>... <span style="white-space: nowrap">for all times <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span></span> and <span style="white-space: nowrap">all nodes <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span></span> (potentially the same) such that <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span></span> and <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span>,</span> we have that <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.161em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mspace mtight" style="margin-right:-0.1952em;"></span><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1318em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span>.</span></p>
+</blockquote>
+<p>The latter is closer to <a href="https://eprint.iacr.org/2016/454.pdf">[PSS2016]</a>.</p>
+<p>Incidentally, this property does <em>not</em> seem to be mentioned in <a href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.221.9986">[Nakamoto2008]</a>, contrary to the [PSS2016] authors’ assertion. Maybe implicitly, but it’s a stretch.</p>
+</div>
+</details>
+<details id="admonition-discussion-of-gkl2020s-communication-model-and-network-partition" class="admonition admonish-info">
+<summary class="admonition-title">
+<p>Discussion of [GKL2020]’s communication model and network partition.</p>
+<p><a class="admonition-anchor-link" href="#admonition-discussion-of-gkl2020s-communication-model-and-network-partition"></a></p>
+</summary>
+<div>
+<p>When <strong>Prefix Consistency</strong> is taken to hold of typical PoW-based block‑chain protocols like Bitcoin (as it often is), this implies that, in the relevant executions, the network of honest nodes is never <span style="white-space: nowrap">partitioned —</span> unless any partition lasts only for a short length of time relative to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> block times. <span style="white-space: nowrap">If node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span></span> is on one side of a full partition and <span style="white-space: nowrap">node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span></span> on the other, then after <span style="white-space: nowrap">node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>’s</span> best chain has been extended by more than <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> blocks,</span> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.161em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mspace mtight" style="margin-right:-0.1952em;"></span><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span> will contain information that has no way to get to <span style="white-space: nowrap">node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span>.</span> And even if the partition is incomplete, we cannot guarantee that the Prefix Consistency property will hold for any given pair of nodes.</p>
+<p>It might be possible to maintain Prefix Consistency if the honest nodes on one side of the partition knew that they should not continue building on their chain until the partition has healed, but it is unclear how that would be done in general without resorting to a BFT protocol (as opposed to in specific cases like a single node being unable to connect to the rest of the network). Certainly there is no mechanism to explicitly detect and respond to partitions in protocols derived from Bitcoin.</p>
+<p>And yet, <a href="https://eprint.iacr.org/2014/765.pdf">[GKL2020]</a> claims to <em>prove</em> Prefix Consistency from other assumptions. So we know that those assumptions must also rule out a long partition between honest nodes. In fact the required assumption is <em>implicit</em> in the communication model:</p>
+<ul>
+<li>A synchronous network cannot be partitioned.</li>
+<li>A partially synchronous network <span style="white-space: nowrap">—that is,</span> providing reliable delivery with bounded but <span style="white-space: nowrap">unknown delay—</span> cannot be partitioned for longer than the delay.</li>
+</ul>
+<p>We might be concerned that these implicit assumptions are stronger than we would like. In practice, the peer‑to‑peer network protocol of Bitcoin and Zcash attempts to flood blocks to all nodes. This protocol might have weaknesses, but it is not intended to (and plausibly does not) depend on all messages being received. (Incidentally, Streamlet also implicitly floods messages to all nodes.)</p>
+<p>Also, Streamlet and many other BFT protocols do <em>not</em> assume <em>for safety</em> that the network is not partitioned. That is, BFT protocols can be safe in a fully asynchronous communication model with unreliable messaging. That is why we avoid taking synchrony or partial synchrony as an implicit assumption of the communication model, or else we could end up with a protocol with weaker safety properties than <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> alone.</p>
+<p>This leaves the question of whether the <strong>Prefix Consistency</strong> property is still too strong, even if we do not rely on it for the analysis of safety when <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has not been subverted. In particular, if a particular <span style="white-space: nowrap">node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">h</span></span></span></span></span> is not well-connected to the rest of the network, then that will inevitably affect <span style="white-space: nowrap">node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">h</span></span></span></span>’s</span> security, but should not affect other honest nodes’ security.</p>
+<p>Fortunately, it is not the case that disconnecting a single <span style="white-space: nowrap">node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">h</span></span></span></span></span> from the network causes the security assumption to be voided. The solution is to view <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">h</span></span></span></span> as not honest in that case (even though it would follow the protocol if it could). This achieves the desired effect within the model, because other nodes can no longer rely on <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">h</span></span></span></span>’s honest input.</span> Although viewing <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">h</span></span></span></span> as potentially adversarial might seem conservative from the point of view of other nodes, bear in mind that an adversary could censor an arbitrary subset of incoming and outgoing messages from the node, and this may be best modelled by considering it to be effectively controlled by the adversary.</p>
+</div>
+</details>
+<p>Prefix Consistency compares the <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>-truncated chain</span> of <span style="white-space: nowrap">some node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span></span> with the <em>untruncated</em> chain of <span style="white-space: nowrap">node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span>.</span> For our analysis of safety of the derived ledgers, we will also need to make an assumption on executions of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> that at <span style="white-space: nowrap">any given time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>,</span> any two <span style="white-space: nowrap">honest nodes <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span></span> <strong>agree</strong> on their confirmed <span style="white-space: nowrap">prefixes —</span> with only the caveat that one may have observed more of the chain than the other. <span style="white-space: nowrap">That is:</span></p>
+<p><span id="prefix-agreement"></span></p>
+<div id="admonition-definition-prefix-agreement" class="admonition admonish-success">
+<div class="admonition-title">
+<p>Definition: Prefix Agreement</p>
+<p><a class="admonition-anchor-link" href="#admonition-definition-prefix-agreement"></a></p>
+</div>
+<div>
+<p>An execution of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has <strong>Prefix Agreement</strong> at confirmation <span style="white-space: nowrap">depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span></span> iff it has <a href="#agreement-on-a-view"><strong>Agreement</strong></a> on the <span style="white-space: nowrap">view <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">i</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">↦</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.161em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mspace mtight" style="margin-right:-0.1952em;"></span><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span>.</span></p>
+</div>
+</div>
+<details id="admonition-why-are-this-property-and-prefix-consistency-above-stated-as-unconditional-properties-of-protocol-executions-rather-than-as-probabilistic-assumptions" class="admonition admonish-info">
+<summary class="admonition-title">
+<p>Why are this property, and Prefix Consistency above, stated as unconditional properties of protocol <em>executions</em>, rather than as probabilistic assumptions?</p>
+<p><a class="admonition-anchor-link" href="#admonition-why-are-this-property-and-prefix-consistency-above-stated-as-unconditional-properties-of-protocol-executions-rather-than-as-probabilistic-assumptions"></a></p>
+</summary>
+<div>
+<p>Our security arguments that depend on these properties will all be of the form “in an execution where ⟨safety properties⟩ are not violated, ⟨undesirable thing⟩ cannot happen”.</p>
+<p>It is not necessary to involve probability in arguments of this form. Any probabilistic reasoning can be done separately.</p>
+<p>In particular, if a statement of this form holds, and ⟨safety properties⟩ are violated with probability <span style="white-space: nowrap">at most <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span></span></span></span></span> under certain conditions, then it immediately follows that under those conditions ⟨undesirable thing⟩ happens with probability <span style="white-space: nowrap">at most <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span></span></span></span>.</span> Furthermore, ⟨undesirable thing⟩ can only happen <em>after</em> ⟨safety properties⟩ have been violated, because the execution up to that point has been an execution in which ⟨safety properties⟩ are <em>not</em> violated.</p>
+<p>With few exceptions, involving probability in a security argument is best done only to account for nondeterministic choices in the protocol itself. This is opinionated advice, but a lot of security proofs would likely be simpler if inherently probabilistic arguments were more distinctly separated from unconditional ones.</p>
+<p>In the case of the Prefix Agreement property, an alternative approach would be to prove that Prefix Agreement holds with some probability given Prefix Consistency and some other chain properties. This is what <a href="https://eprint.iacr.org/2020/1091.pdf">[NTT2020]</a> does in its <span style="white-space: nowrap">Theorem 2,</span> which essentially says that under certain conditions Prefix Agreement holds except with <span style="white-space: nowrap">probability <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9271em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9271em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">Ω</span><span class="mopen mtight">(</span><span class="mord sqrt mtight"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8059em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mtight" style="padding-left:0.833em;"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span><span style="top:-2.7659em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail mtight" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z'/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2341em;"><span></span></span></span></span></span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span></span></span></span>.</span></p>
+<p>The conclusions that can be obtained from this approach are necessarily probabilistic, and depending on the techniques used, the proof may not be tight; that is, the proof may obtain a bound on the probability of failure that is (either asymptotically or concretely) higher than needed. This is the case for <span>[<a href="https://eprint.iacr.org/2020/1091.pdf">NTT2020</a>, Theorem 2];</span> <span style="white-space: nowrap">footnote 10</span> in that paper points out that the expression for the probability can be asymptotically improved:</p>
+<blockquote>
+<p>Using the recursive bootstrapping argument developed in <span>[<a href="https://eprint.iacr.org/2020/601.pdf">DKT+2020</a>, Section 4.2],</span> it is possible to bring the error <span style="white-space: nowrap">probability <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9271em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9271em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">Ω</span><span class="mopen mtight">(</span><span class="mord sqrt mtight"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8059em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mtight" style="padding-left:0.833em;"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span><span style="top:-2.7659em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail mtight" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z'/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2341em;"><span></span></span></span></span></span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span></span></span></span></span> as close to an exponential decay as possible. In this context, <span style="white-space: nowrap">for any <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">ϵ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>,</span> it is possible to find <span style="white-space: nowrap">constants <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">ϵ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">ϵ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span> such that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">p</span><span class="mclose">)</span></span></span></span> is secure after <span style="white-space: nowrap">
+<em>C</em><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mop">max</span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">GST</span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathsf">GAT</span></span><span class="mclose">))</span></span></span></span></span> with confirmation time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> except with <span style="white-space: nowrap">probability <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1369em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">ϵ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9869em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight"><span class="mord mathnormal mtight">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1645em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">ϵ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mspace mtight" style="margin-right:0.1952em;"></span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight">ϵ</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span>.</span></p>
+</blockquote>
+<p>(Here <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span></span></span></span> is the probability that any given node gets to produce a block in any given time slot.)</p>
+<p>In fact none of the proofs of security properties for Snap‑and‑Chat depend on the particular expression <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9271em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9271em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">Ω</span><span class="mopen mtight">(</span><span class="mord sqrt mtight"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8059em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mtight" style="padding-left:0.833em;"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span><span style="top:-2.7659em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail mtight" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z'/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2341em;"><span></span></span></span></span></span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span></span></span></span>;</span> for example in the proofs of <span style="white-space: nowrap">Lemma 5</span> and <span style="white-space: nowrap">Theorem 1,</span> this probability just “passes through” the proof from the premisses to the conclusion, because the argument is not probabilistic. The same will be true of our safety arguments.</p>
+<p>Talking about what is possible in particular executions has further advantages:</p>
+<ul>
+<li>It sidesteps the issue of how to interpret results in the GST model of partial synchrony, when we do not know what <span style="white-space: nowrap">
+<em>C</em><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mop">max</span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">GST</span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathsf">GAT</span></span><span class="mclose">))</span></span></span></span> is.</span> See also the critique of applying the GST model to block‑chain protocols under <a href="#admonition-discussion-of-gkl2020s-communication-model-and-network-partition">“Discussion of [GKL2020]’s communication model and network partition”</a> above. (This is not an inherent problem with analyzing the protocol in the partially synchronous setting, but only with inappropriate use of the GST model of that setting.)</li>
+<li>We do not require <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> to be a Nakamoto‑style Proof‑of‑Work block chain protocol. Some other kind of protocol could potentially satisfy Prefix Consistency and Prefix Agreement.</li>
+<li>It is not clear whether a <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9271em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9271em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">Ω</span><span class="mopen mtight">(</span><span class="mord sqrt mtight"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8059em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mtight" style="padding-left:0.833em;"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span><span style="top:-2.7659em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail mtight" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z'/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2341em;"><span></span></span></span></span></span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span></span></span></span> probability of failure would be concretely adequate. That would depend on the value of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> and the constant hidden by the <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Ω</span></span></span></span> notation. The asymptotic property using <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Ω</span></span></span></span> tells us whether a sufficiently large <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> <em>could</em> be chosen, but we are more interested in what needs to be assumed for a given concrete choice of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>.</li>
+<li>If a violation of a required safety property occurs <em>in a given execution</em>, then the safety argument for Crosslink that depended on the property fails for that execution, regardless of what the probability of that occurrence was. This approach therefore more precisely models the consequences of such violations.</li>
+</ul>
+</div>
+</details>
+<details id="admonition-why-intuitively-should-we-believe-that-prefix-agreement-and-prefix-consistency-for-a-large-enough-confirmation-depth-hold-with-high-probability-for-executions-of-a-powbased-bestchain-protocol" class="admonition admonish-info">
+<summary class="admonition-title">
+<p>Why, intuitively, should we believe that Prefix Agreement and Prefix Consistency for a large enough confirmation depth hold with high probability for executions of a PoW‑based best‑chain protocol?</p>
+<p><a class="admonition-anchor-link" href="#admonition-why-intuitively-should-we-believe-that-prefix-agreement-and-prefix-consistency-for-a-large-enough-confirmation-depth-hold-with-high-probability-for-executions-of-a-powbased-bestchain-protocol"></a></p>
+</summary>
+<div>
+<p>Roughly speaking, the intuition behind both properties is as follows:</p>
+<p>Honest nodes are collectively able to find blocks faster than an adversary, and communication between honest nodes is sufficiently reliable that they act as a combined network racing against that adversary. Then by the argument in <a href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.221.9986">[Nakamoto2008]</a>, modified by <a href="https://arxiv.org/pdf/1702.02867.pdf">[GP2020]</a> to correct an error in the concrete analysis, a private mining attack that attempts to cause a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>‑block</span> rollback will, with high probability, fail for <span style="white-space: nowrap">large enough <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>.</span> A private mining attack is optimal by the argument in <a href="https://arxiv.org/pdf/2005.10484.pdf">[DKT+2020]</a>.</p>
+<p>Any further analysis of the conditions under which these properties hold should be done in the context of a <span style="white-space: nowrap">particular <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</span></p>
+</div>
+</details>
+<details id="admonition-why-is-the-quantification-in-prefix-agreement-over-two-different-times-t-and-t" class="admonition admonish-info">
+<summary class="admonition-title">
+<p>Why is the quantification in Prefix Agreement over two <em>different</em> times <em>t</em> and <em>t′</em>?</p>
+<p><a class="admonition-anchor-link" href="#admonition-why-is-the-quantification-in-prefix-agreement-over-two-different-times-t-and-t"></a></p>
+</summary>
+<div>
+<p>This strengthens the security property, relative to quantifying over a single time. The question can then be split into several parts:</p>
+<ol>
+<li><em><strong>What does the strengthened property mean, intuitively?</strong></em>
+Consider the full tree of <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bc‑valid-blocks</span> that honest nodes have considered to be part of their <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bc‑best-chain</span> at any times during the execution. This property holds iff, when we strip off all branches of length up to and including <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> blocks,</span> the resulting tree <span style="white-space: nowrap">is <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bc‑linear.</span></li>
+<li><em><strong>Why is the strengthening needed?</strong></em>
+Suppose that time were split into periods such that honest nodes agreed on one chain in odd periods, and a completely different chain in even periods. This would obviously not satisfy the intent, but it would satisfy a version of the property that did not quantify over different <span style="white-space: nowrap">times <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span>.</span></li>
+<li><em><strong>Why should we expect the strengthened property to hold?</strong></em>
+If <span style="white-space: nowrap">node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span></span> were far ahead, <span style="white-space: nowrap">i.e. <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">u</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≫</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>,</span> then it is obvious that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.161em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mspace mtight" style="margin-right:-0.1952em;"></span><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1331em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mspace mtight" style="margin-right:-0.1952em;"></span><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span> should hold. Conversely, if node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> were far ahead then it is obvious that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1331em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mspace mtight" style="margin-right:-0.1952em;"></span><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.161em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mspace mtight" style="margin-right:-0.1952em;"></span><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span> should hold. The case where <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span> is the same as quantifying over a single time. By considering intermediate cases where <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span></span> converge from the extremes or where they diverge from being equal, you should be able to convince yourself that the property holds for any relative values of <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span>,</span> in executions of a reasonable best‑chain protocol.</li>
+</ol>
+</div>
+</details>
+<h2 id="definition-of-crosslink-2"><a class="header" href="#definition-of-crosslink-2">Definition of Crosslink 2</a></h2>
+<h3 id="parameters"><a class="header" href="#parameters">Parameters</a></h3>
+<p>Crosslink 2 is parameterized by a <span style="white-space: nowrap">bc‑confirmation‑depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7713em;"></span><span class="mord"><span class="mord mathbb">N</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7713em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">+</span></span></span></span></span></span></span></span></span></span></span></span> (as in Snap‑and‑Chat), and also a finalization gap bound <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7224em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">L</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7713em;"></span><span class="mord"><span class="mord mathbb">N</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7713em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">+</span></span></span></span></span></span></span></span></span></span></span> with <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span> significantly greater <span style="white-space: nowrap">than <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>.</span></p>
+<p>Each node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> always uses the fixed confirmation <span style="white-space: nowrap">depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span></span> to obtain its view of the finalized chain <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.1293em;"><span style="top:-2.6293em;"><span class="pstrut" style="height:2.5em;"></span><span class="mord"><span class="mord sizing reset-size6 size1">⦂</span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord textsf">bc-valid-chain</span></span></span></span></span>.</span> Unlike in Snap‑and‑Chat or Crosslink 1, this is just a block chain; because we do not need sanitization, there is no need to express it as a log of transactions rather than blocks.</p>
+<p>Each node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> chooses a potentially different <span style="white-space: nowrap">bc‑confirmation‑depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">N</span></span></span></span> where <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6835em;vertical-align:-0.0391em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8304em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span></span> to obtain its view of the bounded‑available ledger at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span></span>, <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0797em;vertical-align:-0.2861em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ba</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.1293em;"><span style="top:-2.6293em;"><span class="pstrut" style="height:2.5em;"></span><span class="mord"><span class="mord sizing reset-size6 size1">⦂</span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord textsf">bc-valid-chain</span></span></span></span></span>.</span> (We make the restriction <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8304em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> because there is no reason to choose a <span style="white-space: nowrap">larger <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span></span></span></span>.)</span></p>
+<div id="admonition-security-caveat-1" class="admonition admonish-warning">
+<div class="admonition-title">
+<p>Security caveat</p>
+<p><a class="admonition-anchor-link" href="#admonition-security-caveat-1"></a></p>
+</div>
+<div>
+<p>Choosing <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span></span> is at the node’s own risk and may increase the risk of rollback attacks against <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0797em;vertical-align:-0.2861em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ba</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span> (it does not <span style="white-space: nowrap">affect <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span>).</span> Using small values of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span></span></span></span> is not recommended. The default should <span style="white-space: nowrap">be <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>.</span></p>
+</div>
+</div>
+<h3 id="stalled-mode"><a class="header" href="#stalled-mode">Stalled Mode</a></h3>
+<p>Consider, roughly speaking, the number of bc‑blocks that are not yet finalized at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span></span> (a more precise definition will be given in the section on <a href="#%CE%A0bc-changes-from-%CE%A0origbc"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> changes from <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span></a>). We call this the “finality gap” at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>.</span> Under an assumption about the distribution of bc‑block intervals, if this gap stays roughly constant then it corresponds to the approximate time that transactions take to be finalized after being included in a bc‑block (if they are finalized at all) just prior to <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>.</span></p>
+<p>As explained in detail by <a href="./the-arguments-for-bounded-availability-and-finality-overrides.html">The Arguments for Bounded Availability and Finality Overrides</a>, if this bound exceeds a threshold <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span>, then it likely signals an exceptional or emergency condition, in which it is undesirable to keep accepting user transactions that spend funds into new bc‑blocks. In practice, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span> should be <em>at least</em> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>.</p>
+<p>The condition that the network enters in such cases will be called “Stalled Mode”. For a given higher‑level transaction protocol, we can define a policy for which bc‑blocks will be accepted in Stalled Mode. This will be modelled by a predicate <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord textsf">is-stalled-block</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.1293em;"><span style="top:-2.6293em;"><span class="pstrut" style="height:2.5em;"></span><span class="mord"><span class="mord sizing reset-size6 size1">⦂</span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord textsf">bc-block</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathsf">boolean</span></span></span></span></span>.</span> A bc‑block for which <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord textsf">is-stalled-block</span></span></span></span></span> returns <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5714em;"></span><span class="mord"><span class="mord mathsf">true</span></span></span></span></span> is called a “stalled block”.</p>
+<div id="admonition-caution" class="admonition admonish-warning">
+<div class="admonition-title">
+<p>Caution</p>
+<p><a class="admonition-anchor-link" href="#admonition-caution"></a></p>
+</div>
+<div>
+<p>A bc‑block producer is only constrained to produce stalled blocks while, roughly speaking, its view of the finalization point is not advancing. In particular an adversary that has subverted the BFT protocol in a way that does <em>not</em> keep the finalization point from advancing, can always avoid being constrained by Stalled Mode.</p>
+</div>
+</div>
+<p>The desired properties of stalled blocks and a possible Stalled Mode policy for Zcash are discussed in the <a href="./the-arguments-for-bounded-availability-and-finality-overrides.html#how-to-block-hazards">How to block hazards</a> section of <a href="./the-arguments-for-bounded-availability-and-finality-overrides.html">The Arguments for Bounded Availability and Finality Overrides</a>.</p>
+<p>In practice a node's view of the finalized chain, <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span>,</span> is likely to lag only a few blocks behind <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ba</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span> (depending on the latency overhead imposed <span style="white-space: nowrap">by <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>),</span> <em>unless</em> the chain has entered Stalled Mode. <span style="white-space: nowrap">So when <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>,</span> the main factor influencing the choice of a given application to use <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span> or <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0797em;vertical-align:-0.2861em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ba</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span> is not the average latency, but rather the desired behaviour in the case of a finalization stall: i.e. stall immediately, or keep processing user transactions until <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span> blocks</span> have passed.</p>
+<h3 id="structural-additions"><a class="header" href="#structural-additions">Structural additions</a></h3>
+<ol>
+<li>Each bc‑header has, in addition to origbc‑header fields, a <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">context_bft</span></span></span></span></span> field that commits to a bft‑block.</li>
+<li>Each bft‑proposal has, in addition to origbft‑proposal fields, a <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">headers_bc</span></span></span></span></span> field containing a sequence of exactly <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> bc‑headers</span> (zero‑indexed, deepest first).</li>
+<li>Each non‑genesis bft‑block has, in addition to origbft‑block fields, a <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">headers_bc</span></span></span></span></span> field containing a sequence of exactly <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> bc‑headers</span> (zero-indexed, deepest first). The genesis bft‑block has <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">headers_bc</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6633em;vertical-align:-0.0817em;"></span><span class="mord amsrm">∅</span></span></span></span>.</span></li>
+</ol>
+<p>For a bft‑block or bft‑proposal <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span>, define <span class="katex-display fleqn"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.75em;"><span class="pstrut" style="height:3.75em;"></span><span class="mord"><span class="mord"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mord" style="color:transparent;"><span class="mord" style="color:transparent;"><span class="mord mathsf" style="color:transparent;">LF</span></span><span class="mopen" style="color:transparent;">(</span><span class="mord mathnormal" style="margin-right:0.08125em;color:transparent;">H</span><span class="mclose" style="color:transparent;">)</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.75em;"><span class="pstrut" style="height:3.75em;"></span><span class="mord"><span class="mspace" style="margin-right:-0.1667em;"></span><span class="mspace" style="margin-right:-0.1667em;"></span><span class="mspace" style="margin-right:-0.1667em;"></span><span class="mspace" style="margin-right:-0.1667em;"></span><span class="mrel">:=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">headers_bc</span></span><span class="mopen">[</span><span class="mord">0</span><span class="mclose">]</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mpunct">,</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord">if </span></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">headers_bc</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord amsrm">∅</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord">otherwise.</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span></span></span></span></span>
+For a bc‑block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span>, define <span class="katex-display fleqn"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mord" style="color:transparent;"><span class="mord" style="color:transparent;"><span class="mord mathsf" style="color:transparent;">snapshot</span></span><span class="mopen" style="color:transparent;">(</span><span class="mord mathnormal" style="margin-right:0.05017em;color:transparent;">B</span><span class="mclose" style="color:transparent;">)</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">candidate</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mspace" style="margin-right:-0.1667em;"></span><span class="mspace" style="margin-right:-0.1667em;"></span><span class="mspace" style="margin-right:-0.1667em;"></span><span class="mspace" style="margin-right:-0.1667em;"></span><span class="mrel">:=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">context_bft</span></span><span class="mclose">)</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mspace" style="margin-right:-0.1667em;"></span><span class="mspace" style="margin-right:-0.1667em;"></span><span class="mspace" style="margin-right:-0.1667em;"></span><span class="mspace" style="margin-right:-0.1667em;"></span><span class="mrel">:=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord text"><span class="mord textsf">last-common-ancestor</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.1em;"></span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span></span></span></span></span></p>
+<p>When <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> is the tip of a node’s bc‑best‑chain, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">candidate</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span></span></span> will give the candidate finalization point, subject to a condition described below that prevents local rollbacks.</p>
+<details id="admonition-use-of-the-headers_bc-field-and-its-relation-to-the-ch-field-in-snapandchat" class="admonition admonish-info">
+<summary class="admonition-title">
+<p>Use of the headers_bc field, and its relation to the ch field in Snap‑and‑Chat.</p>
+<p><a class="admonition-anchor-link" href="#admonition-use-of-the-headers_bc-field-and-its-relation-to-the-ch-field-in-snapandchat"></a></p>
+</summary>
+<div>
+<p>For a bft‑proposal or <span style="white-space: nowrap">bft‑block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span>,</span> the role of the bc‑chain snapshot referenced by <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1641em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">headers_bc</span></span><span class="mopen">[</span><span class="mord">0</span><span class="mclose">]</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span> is comparable to the <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> snapshot</span> referenced by <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">ch</span></span></span></span></span> in the Snap‑and‑Chat construction from <a href="https://eprint.iacr.org/2020/1091.pdf">[NTT2020]</a>. The motivation for the additional headers is to demonstrate, to any party that sees a bft‑proposal (resp. bft‑block), that the snapshot had been confirmed when the proposal (resp. the block’s proposal) was made.</p>
+<p>Typically, a node that is validating an honest bft‑proposal or bft‑block will have seen at least the snapshotted bc‑block (and possibly some of the subsequent bc‑blocks in the <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">headers_bc</span></span></span></span></span> chain) before. For this not to be the case, the validator’s bc‑best‑chain would have to be more than <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> bc‑blocks</span> behind the honest proposer’s bc‑best‑chain at a given time, which would violate the <strong>Prefix Consistency</strong> property <span style="white-space: nowrap">of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</span></p>
+<p>If the headers do not connect to any bc‑valid‑chain known to the validator, then the validator should be suspicious that the proposer might not be honest. It can assign a lower priority to validating the proposal in this case, or simply drop it. The latter option could drop a valid proposal, but this does not in practice cause a problem as long as a sufficient number of validators are properly synced (so that <strong>Prefix Consistency</strong> holds for them).</p>
+<p>If the headers <em>do</em> connect to a known bc‑valid‑chain, it could still be the case that the whole header chain up to and including <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">headers_bc</span></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">]</span></span></span></span> is not a bc‑valid‑chain. Therefore, to limit denial‑of‑service attacks the validator should first check the Proofs‑of‑Work and difficulty adjustment <span style="white-space: nowrap">—which it can do locally using only the headers—</span> before attempting to download and validate any bc‑blocks that it has not already seen. This is why we include the full headers rather than just the block hashes. Nodes may “trim” (i.e. not explicitly store) headers in a bft‑block that overlap with those referred to by its <span style="white-space: nowrap">ancestor bft‑block(s).</span></p>
+</div>
+</details>
+<details id="admonition-why-is-a-distinguished-value-needed-for-the-headers_bc-field-in-the-genesis-bftblock" class="admonition admonish-info">
+<summary class="admonition-title">
+<p>Why is a distinguished value needed for the headers_bc field in the genesis bft‑block?</p>
+<p><a class="admonition-anchor-link" href="#admonition-why-is-a-distinguished-value-needed-for-the-headers_bc-field-in-the-genesis-bftblock"></a></p>
+</summary>
+<div>
+<p>It would be conceptually nice for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0972em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">headers</span></span><span class="mopen">[</span><span class="mord">0</span><span class="mclose">]</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span> to refer to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>, as well as <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">context_bft</span></span></span></span></span> being <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> so that <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1em;vertical-align:-0.35em;"></span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">context_bft</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</span> That reflects the fact that we know <span style="white-space: nowrap">“from the start”</span> that neither genesis block can be rolled back.</p>
+<p>This is not literally implementable using block hashes because it would involve a hash cycle, but we achieve the same effect by defining a <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span></span></span></span> function that allows us to “patch” <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> <span style="white-space: nowrap">to be <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</span> We do it this way around rather than “patching” the link from a bc‑block to a bft‑block, because the genesis bft‑block already needs a special case since there are not <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> bc‑headers</span> available.</p>
+</div>
+</details>
+<details id="admonition-why-is-the-context_bft-field-needed-why-not-use-a-final_bft-field-to-refer-directly-to-the-last-final-bftblock-before-the-context-block" class="admonition admonish-info">
+<summary class="admonition-title">
+<p>Why is the context_bft field needed? Why not use a final_bft field to refer directly to the last final bft‑block before the context block?</p>
+<p><a class="admonition-anchor-link" href="#admonition-why-is-the-context_bft-field-needed-why-not-use-a-final_bft-field-to-refer-directly-to-the-last-final-bftblock-before-the-context-block"></a></p>
+</summary>
+<div>
+<p>The finality of some bft‑block is only defined in the context of another bft‑block. One possible design would be for a bc‑block to have both <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">final_bft</span></span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">context_bft</span></span></span></span></span> fields, so that the finality of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">final_bft</span></span></span></span></span> could be checked objectively in the context <span style="white-space: nowrap">of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">context_bft</span></span></span></span></span>.</span></p>
+<p>However, specifying just the context block is sufficient information to determine its last final ancestor. There would never be any need to give a context block and a final ancestor that is not the last one. The <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord textsf">bft-last-final</span></span></span></span></span> function can be computed efficiently for typical BFT protocols. Therefore, having just the <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">context_bft</span></span></span></span></span> field is sufficient.</p>
+</div>
+</details>
+<h3 id="locally-finalized-chain"><a class="header" href="#locally-finalized-chain">Locally finalized chain</a></h3>
+<p>Each node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> keeps track of a “locally finalized” <span style="white-space: nowrap">bc‑chain <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span></span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>.</span> Each node’s locally finalized bc‑chain starts <span style="white-space: nowrap">at <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0728em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</span> However, this chain state should not be exposed to clients of the node until it <a href="#syncing-and-checkpoints">has synced</a>.</p>
+<p><span id="local-finalization-linearity"></span></p>
+<div id="admonition-definition-local-finalization-linearity" class="admonition admonish-success">
+<div class="admonition-title">
+<p>Definition: Local finalization linearity</p>
+<p><a class="admonition-anchor-link" href="#admonition-definition-local-finalization-linearity"></a></p>
+</div>
+<div>
+<p>Node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> has <strong>Local finalization linearity</strong> up to time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span> iff the time series of <span style="white-space: nowrap">bc‑blocks <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.162em;vertical-align:-0.2769em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8852em;"><span style="top:-2.4231em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.14em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span><span class="mspace mtight" style="margin-right:0.1952em;"></span><span class="mrel mtight">≤</span><span class="mspace mtight" style="margin-right:0.1952em;"></span><span class="mord mathnormal mtight">t</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2769em;"><span></span></span></span></span></span></span></span></span></span></span> is <span style="white-space: nowrap">
+<a href="#notation">bc‑linear</a>.</span></p>
+</div>
+</div>
+<p>When <span style="white-space: nowrap">node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>’s bc‑best‑chain view</span> is updated <span style="white-space: nowrap">from <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9957em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">s</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span> to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span>,</span> <span style="white-space: nowrap">the node’s <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span></span> will become <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1279em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">candidate</span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> if and only if this is a descendant <span style="white-space: nowrap">of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9531em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">s</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span>.</span> <span style="white-space: nowrap">Otherwise <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span> will stay at <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9531em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">s</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span>.</span> This guarantees <strong>Local finalization linearity</strong> by construction.</p>
+<p>If when making this update, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2279em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">candidate</span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.5em;"></span><span class="mord"><span class="delimsizing size1">/</span></span><span class="mspace" style="margin-right:-1.06em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:-0.5em;"></span></span><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">⪰</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9531em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">s</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span> (i.e. <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1279em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">candidate</span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9531em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">s</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span> are on different forks), then the node should record a finalization safety hazard. This can only happen if global safety assumptions are violated. Note that <strong>Local finalization linearity</strong> on each node is <em>not</em> sufficient for <a href="#assured-finality"><strong>Assured Finality</strong></a>, but it is necessary.</p>
+<p>This can be expressed by the following state update algorithm, where <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">s</span></span></span></span> is the time of the last update and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span> is the time of the current update:</p>
+<p><span class="katex-display fleqn"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1279em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">update-fin</span></span><span class="mopen">(</span><span class="mord text"><span class="mord">mut </span></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">s</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mspace" style="margin-right:2em;"></span><span class="mord text"><span class="mord">let </span></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1279em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">candidate</span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.8304em;vertical-align:-0.136em;"></span><span class="mspace" style="margin-right:2em;"></span><span class="mord text"><span class="mord">if </span></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪰</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9614em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">s</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord text"><span class="mord">:</span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:1.0906em;vertical-align:-0.247em;"></span><span class="mspace" style="margin-right:2em;"></span><span class="mspace" style="margin-right:2em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8436em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">←</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mspace" style="margin-right:2em;"></span><span class="mord text"><span class="mord">else:</span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:1.0906em;vertical-align:-0.247em;"></span><span class="mspace" style="margin-right:2em;"></span><span class="mspace" style="margin-right:2em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8436em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">←</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9614em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">s</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mspace" style="margin-right:2em;"></span><span class="mspace" style="margin-right:2em;"></span><span class="mord text"><span class="mord">if </span></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span></span><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9614em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">s</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord text"><span class="mord">:</span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mspace" style="margin-right:2em;"></span><span class="mspace" style="margin-right:2em;"></span><span class="mspace" style="margin-right:2em;"></span><span class="mord text"><span class="mord">record finalization safety hazard</span></span></span></span></span></span></p>
+<p>A safety hazard record should include <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span> and the history of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> updates including and since the last one that was an ancestor of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span></span></span></span>.</p>
+<p><span id="local-fin-depth"></span></p>
+<div id="admonition-lemma-local-findepth" class="admonition admonish-success">
+<div class="admonition-title">
+<p>Lemma: Local fin‑depth</p>
+<p><a class="admonition-anchor-link" href="#admonition-lemma-local-findepth"></a></p>
+</div>
+<div>
+<p>In any execution of Crosslink 2, for any <span style="white-space: nowrap">node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span></span> that is honest at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>,</span> there exists a <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span></span> <span style="white-space: nowrap">such that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0331em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span>.</span></p>
+<p>Proof: By the definition of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathsf">candidate</span></span></span></span></span> we have <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">candidate</span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0331em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span> for all <span style="white-space: nowrap">times <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span>.</span> <span style="white-space: nowrap">Let <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span></span> be the last time at which <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span> changed,</span> or the <span style="white-space: nowrap">genesis time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span> if it has never changed. <span style="white-space: nowrap">Then for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span> we have <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9531em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">candidate</span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0331em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span>,</span> and <span style="white-space: nowrap">for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span> we have <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9531em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0331em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span></span> <span style="white-space: nowrap">(because <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1454em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8984em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>,</span> and <span style="white-space: nowrap">truncating <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span> always <span style="white-space: nowrap">yields <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>).</span></p>
+</div>
+</div>
+<details id="admonition-why-does-fini-need-to-be-maintained-using-local-state" class="admonition admonish-info">
+<summary class="admonition-title">
+<p>Why does fin<sub>i</sub> need to be maintained using local state?</p>
+<p><a class="admonition-anchor-link" href="#admonition-why-does-fini-need-to-be-maintained-using-local-state"></a></p>
+</summary>
+<div>
+<p>When a node’s view of the bc‑best‑chain reorgs to a different fork (even if the reorg is shorter than <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> blocks), it may be the case that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1279em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">candidate</span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> rolls back. If <a href="#final-agreement"><strong>Final Agreement</strong></a> of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> holds up to time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>, the new snapshot should in that case be an ancestor of the old one. If all is well then this snapshot will subsequently roll forward along the same path. However, we do not want applications using the node to see the temporary rollback.</p>
+</div>
+</details>
+<p><span id="assured-finality"></span>
+<a href="../../terminology.html#definition-assured-finality">Assured Finality</a> is our main safety goal for Crosslink 2. It is essentially the same goal as <strong>Final Agreement</strong> but applied to nodes’ locally finalized bc‑chains; intuitively it means that honest nodes never see conflicting locally finalized chains. We intend to prove that this goal holds under reasonable assumptions about <em>either</em> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> <em>or</em> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>.</p>
+<div id="admonition-definition-assured-finality" class="admonition admonish-success">
+<div class="admonition-title">
+<p>Definition: Assured Finality</p>
+<p><a class="admonition-anchor-link" href="#admonition-definition-assured-finality"></a></p>
+</div>
+<div>
+<p>An execution of Crosslink 2 has <strong>Assured Finality</strong> iff <span style="white-space: nowrap">for all times <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span></span> and <span style="white-space: nowrap">all nodes <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span></span> (potentially the same) such that <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span></span> and <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span>,</span> we have <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:-0.5em;"></span></span><span class="base"><span class="strut" style="height:0.786em;vertical-align:-0.15em;"></span><span class="mrel"><span class="mrel">⪰</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0892em;vertical-align:-0.3948em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3948em;"><span></span></span></span></span></span></span></span></span></span>.</span></p>
+</div>
+</div>
+<p>Note that if an execution of Crosslink 2 has <strong>Assured Finality</strong>, then all nodes that are honest for that execution have <a href="#local-finalization-linearity"><strong>Local finalization linearity</strong></a>. That is because the restriction of <strong>Assured Finality</strong> to the <span style="white-space: nowrap">case <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span></span> is equivalent to <strong>Local finalization linearity</strong> <span style="white-space: nowrap">for node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span></span> up to any time at which <span style="white-space: nowrap">node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> is honest.</span></p>
+<details id="admonition-why-do-we-need-to-use-candidateh-rather-than-snapshotlfh" class="admonition admonish-info">
+<summary class="admonition-title">
+<p>Why do we need to use candidate(H) rather than snapshot(LF(H))?</p>
+<p><a class="admonition-anchor-link" href="#admonition-why-do-we-need-to-use-candidateh-rather-than-snapshotlfh"></a></p>
+</summary>
+<div>
+<p>This ensures that the candidate is at least <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>‑confirmed.</span></p>
+<p>In practice <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">candidate</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span></span></span> will rarely differ from <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.1em;"></span><span class="mclose">)</span></span></span></span>, but using the former patches over a potential gap in the safety proof. The <a href="#last-final-snapshot-rule"><strong>Last Final Snapshot rule</strong></a> specified later will guarantee that <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.1em;"></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span>,</span> and this ensures that <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.1em;"></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:-0.5em;"></span></span><span class="base"><span class="strut" style="height:0.786em;vertical-align:-0.15em;"></span><span class="mrel"><span class="mrel">⪰</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0331em;vertical-align:-0.2831em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span>.</span> However, the depth of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.1em;"></span><span class="mclose">)</span></span></span></span> relative to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> is not guaranteed to be <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>. For the proof we will need <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">candidate</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0331em;vertical-align:-0.2831em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span>,</span> so that we can use the <a href="#local-fin-depth"><strong>Local fin‑depth lemma</strong></a> together with <a href="#prefix-agreement"><strong>Prefix Agreement</strong></a> of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> at confirmation depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> to prove <a href="#assured-finality"><strong>Assured Finality</strong></a>.</p>
+<p>An alternative would be to change the <a href="#last-final-snapshot-rule"><strong>Last Final Snapshot rule</strong></a> to directly require <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.1em;"></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0331em;vertical-align:-0.2831em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span>.</span></p>
+<p><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="enclosing" style="background-color:yellow;color:black;"><span class="mord text"><span class="mord textsf">TODO:</span></span></span></span></span></span> Choose between these options based on what works well for the security proofs and finalization latency.</p>
+</div>
+</details>
+<h3 id="locally-boundedavailable-chain"><a class="header" href="#locally-boundedavailable-chain">Locally bounded‑available chain</a></h3>
+<p>Define the locally bounded‑available chain on node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> for bc‑confirmation‑depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span></span></span></span>, as <span class="katex-display fleqn"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1297em;vertical-align:-0.2861em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ba</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8436em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em;"><span style="top:-2.3987em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.1809em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3013em;"><span></span></span></span></span></span></span><span class="mpunct">,</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mpunct">,</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord">if </span></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em;"><span style="top:-2.3987em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.1809em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3013em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord">otherwise.</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>
+<p>Like the locally finalized bc‑chain, this chain state should not be exposed to clients of the node until it <a href="#syncing-and-checkpoints">has synced</a>.</p>
+<p><span id="ledger-prefix-property"></span></p>
+<div id="admonition-theorem-ledger-prefix-property" class="admonition admonish-success">
+<div class="admonition-title">
+<p>Theorem: Ledger prefix property</p>
+<p><a class="admonition-anchor-link" href="#admonition-theorem-ledger-prefix-property"></a></p>
+</div>
+<div>
+<p>For any node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> that is honest at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>,</span> and any confirmation <span style="white-space: nowrap">depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span></span></span></span>,</span> <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0797em;vertical-align:-0.2861em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ba</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span>.</span></p>
+<p>Proof: By construction of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0797em;vertical-align:-0.2861em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ba</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span>.</span></p>
+</div>
+</div>
+<p><span id="local-ba-depth"></span></p>
+<div id="admonition-lemma-local-badepth" class="admonition admonish-success">
+<div class="admonition-title">
+<p>Lemma: Local ba‑depth</p>
+<p><a class="admonition-anchor-link" href="#admonition-lemma-local-badepth"></a></p>
+</div>
+<div>
+<p>In any execution of Crosslink 2, for any <span style="white-space: nowrap">confirmation depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8304em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span></span> and any <span style="white-space: nowrap">node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span></span> that is honest at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>,</span> there exists a <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span></span> <span style="white-space: nowrap">such that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0797em;vertical-align:-0.2861em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ba</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0836em;vertical-align:-0.3013em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em;"><span style="top:-2.3987em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.1809em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3013em;"><span></span></span></span></span></span></span></span></span></span>.</span></p>
+<p>Proof: Either <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0797em;vertical-align:-0.2861em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ba</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span>, in which case the result follows by the <a href="#local-fin-depth"><strong>Local fin‑depth lemma</strong></a> since <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8304em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>, or <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0797em;vertical-align:-0.2861em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ba</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1792em;vertical-align:-0.3013em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em;"><span style="top:-2.3987em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.1809em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3013em;"><span></span></span></span></span></span></span></span></span></span> in which case it follows trivially with <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>.</p>
+</div>
+</div>
+<p>Our security goal for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ba</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> will be <a href="#agreement-on-a-view"><strong>Agreement</strong></a> on <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ba</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">μ</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span></span> as already defined.</p>
+<details id="admonition-why-is-the-otherwise-case-in-the-definition-of-baμit-necessary" class="admonition admonish-info">
+<summary class="admonition-title">
+<p>Why is the ‘otherwise’ case in the definition of (ba<sub>μ</sub>)<sub>i</sub><sup>t</sup> necessary?</p>
+<p><a class="admonition-anchor-link" href="#admonition-why-is-the-otherwise-case-in-the-definition-of-baμit-necessary"></a></p>
+</summary>
+<div>
+<p>Assume for this discussion that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> uses PoW.</p>
+<p>Depending on the value of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>, the timestamps of bc‑blocks, and the difficulty adjustment rule, it can be the case that if <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> switches to a different fork, the difficulty on that fork is greater than on the chain of the previous snapshot. Then, the new bc‑chain could reach a higher score than the previous chain in fewer than <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> blocks</span> from the fork point, and so <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1792em;vertical-align:-0.3013em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em;"><span style="top:-2.3987em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.1809em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3013em;"><span></span></span></span></span></span></span></span></span></span> might not be a descendant of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span> (which is more likely <span style="white-space: nowrap">if <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>).</span> This can occur even when all safety assumptions are satisfied.</p>
+<p>For <a href="https://zips.z.cash/protocol/protocol.pdf#diffadjustment">Zcash’s difficulty adjustment algorithm</a>, the difficulty of each block is adjusted based on the median timestamps and difficulty target thresholds over a range of <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">PoWAveragingWindow</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">17</span></span></span></span> blocks,</span> where each median is taken over <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathsf">PoWMedianBlockSpan</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">11</span></span></span></span> blocks.</span> Other damping factors and clamps are applied in order to prevent instability and to reduce the influence that adversarially chosen timestamps can have on difficulty adjustment. This makes it unlikely that an adversary could gain a significant advantage by manipulating the difficulty adjustment. So it is safe to use <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span> in this case: even though it does not have <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span></span></span></span> confirmations</span> relative to <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span>,</span> it does have at least the required amount of work “on top” of it.</p>
+<p>Defining <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0797em;vertical-align:-0.2861em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ba</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span> this way also has the advantage of making the proof of the <strong>Ledger prefix property</strong> trivial.</p>
+</div>
+</details>
+<h3 id="syncing-and-checkpoints"><a class="header" href="#syncing-and-checkpoints">Syncing and checkpoints</a></h3>
+<p>It is recommended that node implementations “bake in” a checkpointed <span style="white-space: nowrap">bft‑block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathsf mtight">checkpoint</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span></span> to each released version, and that node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> should only expose <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0797em;vertical-align:-0.2861em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ba</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span></span> to its clients once it is “synced”, that is:</p>
+<ul>
+<li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathsf mtight">checkpoint</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1279em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>;  and</li>
+<li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathsf mtight">checkpoint</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span>;  and</li>
+<li>the timestamp of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span> is within some threshold of the current time.</li>
+</ul>
+<h3 id="Πbft-changes-from-Πorigbft"><a class="header" href="#Πbft-changes-from-Πorigbft">Π<sub>bft</sub> changes from Π<sub>origbft</sub></a></h3>
+<h4 id="Πbft-proposal-and-block-validity"><a class="header" href="#Πbft-proposal-and-block-validity">Π<sub>bft</sub> proposal and block validity</a></h4>
+<p><span id="genesis-bft-block-rule"></span>
+<strong>Genesis bft‑block rule:</strong> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> is bft‑block‑valid.</p>
+<p>A bft‑proposal (resp. non‑genesis bft‑block) <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> is bft‑proposal‑valid (resp. bft‑block‑valid) iff all of the following hold:</p>
+<ul>
+<li><span id="inherited-origbft-rules"></span>
+<strong>Inherited origbft rules:</strong> The corresponding origbft‑proposal‑validity (resp. origbft‑block‑validity) rules hold for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span>.</li>
+<li><span id="linearity-rule"></span>
+<strong>Linearity rule:</strong> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0972em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span>.</li>
+<li><span id="tail-confirmation-rule"></span>
+<strong>Tail Confirmation rule:</strong> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">headers_bc</span></span></span></span></span> form the <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>‑block tail of a bc‑valid‑chain.</li>
+</ul>
+<p><span id="parent-rule"></span>
+The “corresponding validity rules” are assumed to include the <strong>Parent rule</strong> that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span>’s parent is bft‑valid.</p>
+<p>Note: origbft‑block‑validity rules may be different to origbft‑proposal‑validity rules. For example, in adapted Streamlet, a origbft‑block needs evidence that it was voted for by a supermajority, and an origbft‑proposal doesn’t. Such differences also apply to bft‑block‑validity vs bft‑proposal‑validity.</p>
+<details id="admonition-why-have-validity-rules-been-separated-from-the-honest-voting-condition-below" class="admonition admonish-info">
+<summary class="admonition-title">
+<p>Why have validity rules been separated from the honest voting condition below?</p>
+<p><a class="admonition-anchor-link" href="#admonition-why-have-validity-rules-been-separated-from-the-honest-voting-condition-below"></a></p>
+</summary>
+<div>
+<p>The reason to separate the validity rules from the honest voting condition, is that the validity rules are objective: they don’t depend on an observer’s view of the bc‑best‑chain. Therefore, they can be checked independently of validator signatures. Even a proposal voted for by 100% of validators will not be considered bft‑proposal‑valid by other nodes unless it satisfies the above rules. If more than two thirds of voting units are cast for an <strong>invalid</strong> proposal, something is seriously <em>and visibly</em> wrong; in any case, the block will not be accepted as a bft‑valid‑block. Importantly, a purportedly valid bft‑block will not be recognized as such by <em>any</em> honest Crosslink 2 node even if it includes a valid notarization proof, if it does not meet other bft‑block‑validity rules.</p>
+<p>This is essential to making the finalized chain safe against a flaw in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> or its security assumptions (even, say, a complete break of the validator signature algorithm), as long as <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> remains safe.</p>
+</div>
+</details>
+<details id="admonition-what-does-the-linearity-rule-do" class="admonition admonish-info">
+<summary class="admonition-title">
+<p>What does the <strong>Linearity rule</strong> do?</p>
+<p><a class="admonition-anchor-link" href="#admonition-what-does-the-linearity-rule-do"></a></p>
+</summary>
+<div>
+<p>This rule is key to combining simplicity with strong security properties in Crosslink 2. It essentially says that, in a given bft‑valid‑chain, the snapshots pointed to by blocks in that chain cannot roll back.</p>
+<p>This allows the informal safety argument for Crosslink 2 to be rather intuitive.</p>
+<p>Informally, if <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has <a href="#final-agreement"><strong>Final Agreement</strong></a>, then all nodes see only one consistent <a href="#notation">bft‑linear</a> chain (restricting to bft‑blocks that are final in the context of some bft‑block <a href="#in-honest-view">in honest view</a>). Within such a bft‑chain, the <a href="#linearity-rule"><strong>Linearity rule</strong></a> ensures <em>by construction</em> that the sequence of referenced bc‑chain snapshots is <a href="#notation">bc‑linear</a>. This implies <a href="#assured-finality"><strong>Assured Finality</strong></a>, without needing to assume any safety property of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</p>
+<p>We will <em>also</em> be able to prove safety of the finalized snapshots based only on safety of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> (for a confirmation depth of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>), without needing to assume any safety property of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>. Informally, that is because each node sees each candidate final snapshot at a given time as a <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>-confirmed prefix of its bc‑best‑chain at that time (this can be proven based on the <a href="#%CE%A0bc-changes-from-%CE%A0origbc"><strong>Last Final Snapshot rule</strong></a> and the fact that a snapshot includes <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> subsequent headers), and <a href="#prefix-agreement"><strong>Prefix Agreement</strong></a> implies that honest nodes agree on this prefix. We will leave a more detailed argument until after we have presented the <span style="white-space: nowrap">
+<a href="#%CE%A0bc-changes-from-%CE%A0origbc"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> changes from <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span></a>.</span></p>
+<p>The <a href="#linearity-rule"><strong>Linearity rule</strong></a> replaces the “Increasing Score rule” used in Crosslink 1. The Increasing Score rule required that each snapshot in a bft‑valid‑chain either be the same snapshot, or a higher-scoring snapshot to that of its parent block. Since scores strictly increase within a bc‑valid‑chain, the <a href="#linearity-rule"><strong>Linearity rule</strong></a> <em>implies</em> the Increasing Score rule. It retains the same or stronger positive effects:</p>
+<ul>
+<li>It prevents potential attacks that rely on proposing a bc‑valid‑chain that forks from a much earlier block. This is necessary because the difficulty (or stake threshold) at that point could have been much lower.</li>
+<li>It limits the extent of disruption an adversary can feasibly cause to the bounded‑available <span style="white-space: nowrap">chain <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>,</span> <em>even if</em> it has <span style="white-space: nowrap">subverted <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</span> Informally, because the finalized chain <em>is</em> a <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> chain, its safety is no worse than <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> alone for a rollback of <em>any</em> depth.</li>
+<li>It ensures that either progress is made (the snapshot advances relative to that of the parent bft‑block), <em>or</em> there is no further validation that needs to be done for the snapshot because it was already validated.</li>
+</ul>
+<p>Note that the adversary could take advantage of an “accidental” fork and start its attack from the base of that fork, so that not all of this work is done by it alone. This is also possible in the case of a standard “private mining” attack, and is not so much of a problem in practice because accidental forks are expected to be short. In any case, <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> should</span> be chosen to take it into account.</p>
+<p>The <a href="#linearity-rule"><strong>Linearity rule</strong></a> is also critical to removing the need for one of the most complex elements of Snap‑and‑Chat and Crosslink 1, “sanitization”. In those protocols, because bc‑chain snapshots could be unrelated to each other, it was necessary to sanitize the chain formed from these snapshots to remove transactions that were contextually invalid (e.g. because they double‑spend). The negative consequences of this are described in <a href="./notes-on-snap-and-chat.html">Notes on Snap‑and‑Chat</a>; avoiding it is much simpler.</p>
+<p>The linearity property is intentionally always relative to the snapshot of the parent bft‑block, even if it is not final in the context of the current bft‑block. This is because the rule needs to hold if and when it becomes final in the context of some descendant bft‑block.</p>
+<p><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="enclosing" style="background-color:yellow;color:black;"><span class="mord text"><span class="mord textsf">TODO:</span></span></span></span></span></span> PoS Desideratum: we want leader selection with good security / performance properties that will be relevant to this rule. (Suggested: <a href="https://arxiv.org/pdf/2010.08154.pdf">PoSAT</a>.)</p>
+</div>
+</details>
+<details id="admonition-why-does-the-linearity-rule-allow-keeping-the-same-snapshot-as-the-parent" class="admonition admonish-info">
+<summary class="admonition-title">
+<p>Why does the <strong>Linearity rule</strong> allow keeping the same snapshot as the parent?</p>
+<p><a class="admonition-anchor-link" href="#admonition-why-does-the-linearity-rule-allow-keeping-the-same-snapshot-as-the-parent"></a></p>
+</summary>
+<div>
+<p>This is necessary in order to preserve liveness of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> relative <span style="white-space: nowrap">to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>.</span> Liveness of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> might require honest proposers to make proposals at a minimum rate. That requirement could be consistently violated if it were not always possible to make a valid proposal. But given that it is allowed to repeat the same snapshot as in the parent bft‑block, neither the <a href="#linearity-rule"><strong>Linearity rule</strong></a> nor the <a href="#tail-confirmation-rule"><strong>Tail Confirmation rule</strong></a> can prevent making a valid proposal — and all other rules of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> affecting the ability to make valid proposals are the same as <span style="white-space: nowrap">in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>.</span> <span style="white-space: nowrap">(In principle,</span> changes to voting in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> could also affect its liveness; we’ll discuss that in the liveness proof later.)</p>
+<p>For example, Streamlet requires three notarized blocks <em>in consecutive epochs</em> in order to finalize <span style="white-space: nowrap">a block [<a href="https://eprint.iacr.org/2020/088.pdf">CS2020</a>, section 1.1].</span> Its proof of liveness depends on the assumption that in each epoch for which the leader is honest, that leader will make a proposal, and that during a “period of synchrony” this proposal will be received by <span style="white-space: nowrap">every node [<a href="https://eprint.iacr.org/2020/088.pdf">CS2020</a>, section 3.6].</span> This argument can also be extended to adapted‑Streamlet.</p>
+<p>We could alternatively have allowed to always make a “null” proposal, rather than to always make a proposal with the same snapshot as the parent. We prefer the latter because the former would require specifying the rules for null proposals <span style="white-space: nowrap">in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>.</span></p>
+<p>As a clarification, no BFT protocol that uses leader election can <em>require</em> a proposal in each epoch, because the leader might be dishonest. The above issue concerns liveness of the protocol when assumptions about the attacker’s share of bft‑validators or stake are met, so that it can be assumed that sufficiently long periods with enough honest leaders to make progress <span style="white-space: nowrap">(5 consecutive</span> epochs in the case of Streamlet), will occur with significant probability.</p>
+</div>
+</details>
+<h4 id="Πbft-block-finality-in-context"><a class="header" href="#Πbft-block-finality-in-context">Π<sub>bft</sub> block finality in context</a></h4>
+<p>The finality rule for bft‑blocks in a given context is unchanged from origbft‑finality. That is, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord textsf">bft-last-final</span></span></span></span></span> is defined in the same way as <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord textsf">origbft-last-final</span></span></span></span></span> (modulo referring to bft‑block‑validity and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>).</p>
+<h4 id="Πbft-honest-proposal"><a class="header" href="#Πbft-honest-proposal">Π<sub>bft</sub> honest proposal</a></h4>
+<p>An honest proposer of a bft‑proposal <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span> chooses <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">headers_bc</span></span></span></span></span> as the <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>‑block tail of its bc‑best‑chain, provided that it is consistent with the <a href="#linearity-rule"><strong>Linearity rule</strong></a>. If it would not be consistent with that rule, it sets <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">headers_bc</span></span></span></span></span> to the same <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">headers_bc</span></span></span></span></span> field as <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span>’s parent bft‑block. It does not make proposals until its bc‑best‑chain is at least <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> blocks long.</p>
+<details id="admonition-why-σ--1" class="admonition admonish-info">
+<summary class="admonition-title">
+<p>Why σ + 1?</p>
+<p><a class="admonition-anchor-link" href="#admonition-why-σ--1"></a></p>
+</summary>
+<div>
+<p>If the length were less than <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> blocks, it would be impossible to construct the <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">headers_bc</span></span></span></span></span> field of the proposal.</p>
+<p>Note that when the length of the proposer’s bc‑best‑chain is exactly <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> blocks, the snapshot must be of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">.</span></span></span></span> But this does not violate the <a href="#linearity-rule"><strong>Linearity rule</strong></a>, because <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> matches the previous snapshot by <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</p>
+</div>
+</details>
+<details id="admonition-how-is-it-possible-that-the-linearity-rulelinearity-rule-would-not-be-satisfied-by-choosing-headers-from-an-honest-proposers-bcbestchain" class="admonition admonish-info">
+<summary class="admonition-title">
+<p>How is it possible that the <a href="#linearity-rule"><strong>Linearity rule</strong></a> would not be satisfied by choosing headers from an honest proposer’s bc‑best‑chain?</p>
+<p><a class="admonition-anchor-link" href="#admonition-how-is-it-possible-that-the-linearity-rulelinearity-rule-would-not-be-satisfied-by-choosing-headers-from-an-honest-proposers-bcbestchain"></a></p>
+</summary>
+<div>
+<p>As in the answer to <a href="#admonition-why-is-the-otherwise-case-in-the-definition-of-bai%CE%BCt-necessary">Why is the ‘otherwise’ case in the definition of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0797em;vertical-align:-0.2861em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ba</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span> necessary?</a> above, after a reorg on the bc‑chain, the <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>-confirmed block on the new chain might not be a descendant of the <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>-confirmed block on the old chain, which could break the <a href="#linearity-rule"><strong>Linearity rule</strong></a>.</p>
+</div>
+</details>
+<h4 id="Πbft-honest-voting"><a class="header" href="#Πbft-honest-voting">Π<sub>bft</sub> honest voting</a></h4>
+<p>An honest validator considering a proposal <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span>, first updates its view of both subprotocols with the bc‑headers given in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">headers_bc</span></span></span></span></span>, downloading bc‑blocks for these headers and checking their bc‑block‑validity.</p>
+<p>For each downloaded bc‑block, the bft‑chain referenced by its <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">context_bft</span></span></span></span></span> field might need to be validated if it has not been seen before.</p>
+<details id="admonition-wait-what-how-much-validation-is-that" class="admonition admonish-info">
+<summary class="admonition-title">
+<p>Wait what, how much validation is that?</p>
+<p><a class="admonition-anchor-link" href="#admonition-wait-what-how-much-validation-is-that"></a></p>
+</summary>
+<div>
+<p>In general the entire referenced bft‑chain needs to be validated, not just the referenced block — and for each bft‑block, the bc‑chain in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">headers_bc</span></span></span></span></span> needs to be validated, and so on recursively. If this sounds overwhelming, note that:</p>
+<ul>
+<li>We should check the requirement that a bft‑valid‑block must have been voted for by a two‑thirds absolute supermajority of validators, and any other <em>non‑recursive</em> bft‑validity rules, <em>first</em>.</li>
+<li>Before validating a bc‑chain referenced by a <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">headers_bc</span></span></span></span></span> field, we check that it connects to an already-validated bc‑chain and that the Proofs‑of‑Work are valid. This implies that the amount of bc‑block validation is constrained by how fast the network can find valid Proofs‑of‑Work.</li>
+<li>The <a href="#linearity-rule"><strong>Linearity rule</strong></a> reduces the worst‑case validation effort, by ensuring that only one bc‑chain needs to be validated for any bft‑chain. Assuming safety of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> and that the adversary does not have an overwhelming advantage in computing the Proof‑of‑Work, this is effectively only one bc‑chain overall with, at most, short side branches.</li>
+</ul>
+<p>In summary, the order of validation is important to avoid denial‑of‑service — but it already is in Bitcoin and Zcash.</p>
+</div>
+</details>
+<p>After updating its view, the validator will vote for a proposal <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span> <em>only if</em>:</p>
+<ul>
+<li><span id="valid-proposal-criterion"></span>
+<strong>Valid proposal criterion:</strong> it is bft‑proposal‑valid, and</li>
+<li><span id="confirmed-best-chain-criterion"></span>
+<strong>Confirmed best‑chain criterion:</strong> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mclose">)</span></span></span></span> is part of the validator’s bc‑best‑chain at a bc‑confirmation‑depth of at least <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>.</li>
+</ul>
+<details id="admonition-blocks-in-a-bcbestchain-are-by-definition-bcblockvalid-if-were-checking-the-confirmed-bestchain-criterion-why-do-we-need-to-have-separately-checked-that-the-blocks-referenced-by-the-headers-are-bcblockvalid" class="admonition admonish-info">
+<summary class="admonition-title">
+<p>Blocks in a bc‑best‑chain are by definition bc‑block‑valid. If we’re checking the <strong>Confirmed best‑chain criterion</strong>, why do we need to have separately checked that the blocks referenced by the headers are bc‑block‑valid?</p>
+<p><a class="admonition-anchor-link" href="#admonition-blocks-in-a-bcbestchain-are-by-definition-bcblockvalid-if-were-checking-the-confirmed-bestchain-criterion-why-do-we-need-to-have-separately-checked-that-the-blocks-referenced-by-the-headers-are-bcblockvalid"></a></p>
+</summary>
+<div>
+<p>The <strong>Confirmed best‑chain criterion</strong> is quite subtle. It ensures that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1641em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">headers_bc</span></span><span class="mopen">[</span><span class="mord">0</span><span class="mclose">]</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span> is bc‑block‑valid and has <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> bc‑block‑valid blocks after it in the validator’s bc‑best‑chain. However, it need not be the case that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">headers_bc</span></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">]</span></span></span></span> is part of the validator’s bc‑best‑chain after it updates its view. That is, the chain could fork after <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mclose">)</span></span></span></span>.</p>
+<p>The bft‑proposal‑validity rule must be objective; it can’t depend on what the validator’s bc‑best‑chain is. The validator’s bc‑best‑chain <em>may</em> have been updated to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">headers_bc</span></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">]</span></span></span></span> (if it has the highest score), but it also may not.</p>
+<p>However, if the validator’s bc‑best‑chain <em>was</em> updated, that makes it more likely that it will be able to vote for the proposal.</p>
+<p>In any case, if the validator does not check that all of the blocks referenced by the headers are bc‑block‑valid, then its vote may be invalid.</p>
+</div>
+</details>
+<details id="admonition-how-does-this-compare-to-snapandchat" class="admonition admonish-info">
+<summary class="admonition-title">
+<p>How does this compare to Snap‑and‑Chat?</p>
+<p><a class="admonition-anchor-link" href="#admonition-how-does-this-compare-to-snapandchat"></a></p>
+</summary>
+<div>
+<p>Snap‑and‑Chat already had the voting condition:</p>
+<blockquote>
+<p>An honest node only votes for a proposed BFT block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> if it views <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">ch</span></span></span></span></span> as confirmed.</p>
+</blockquote>
+<p>but it did <em>not</em> give the headers potentially needed to update the validator’s view, and it did not require a proposal to be for an <em>objectively confirmed</em> snapshot as a matter of validity.</p>
+<p>If a Crosslink‑like protocol were to require an objectively confirmed snapshot but without including the bc‑headers in the proposal, then validators would not immediately know which bc‑blocks to download to check its validity. This would increase latency, and would be likely to lead proposers to be more conservative and only propose blocks that they think will <em>already</em> be in at least a two‑thirds absolute supermajority of validators’ best chains.</p>
+<p>That is, showing <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">headers_bc</span></span></span></span></span> to all of the validators is advantageous to the proposer, because the proposer does not have to guess what blocks the validators might have already seen. It is also advantageous for the protocol goals in general, because it improves the trade‑off between finalization latency and security.</p>
+</div>
+</details>
+<h3 id="Πbc-changes-from-Πorigbc"><a class="header" href="#Πbc-changes-from-Πorigbc">Π<sub>bc</sub> changes from Π<sub>origbc</sub></a></h3>
+<h4 id="Πbc-block-validity"><a class="header" href="#Πbc-block-validity">Π<sub>bc</sub> block validity</a></h4>
+<p><span id="genesis-bc-block-rule"></span>
+<strong>Genesis bc‑block rule:</strong> For the genesis bc‑block we must have <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">context_bft</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>,</span> and therefore <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1em;vertical-align:-0.35em;"></span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">context_bft</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</span></p>
+<p>A bc‑block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> is bc‑block‑valid iff all of the following hold:</p>
+<ul>
+<li><span id="inherited-origbc-rules"></span>
+<strong>Inherited origbc rules:</strong> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> satisfies the corresponding origbc‑block‑validity rules.</li>
+<li><span id="valid-context-rule"></span>
+<strong>Valid context rule:</strong> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">context_bft</span></span></span></span></span> is bft‑block‑valid.</li>
+<li><span id="extension-rule"></span>
+<strong>Extension rule:</strong> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0972em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span></span></span>.</li>
+<li><span id="last-final-snapshot-rule"></span>
+<strong>Last Final Snapshot rule:</strong> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.1em;"></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span>.</li>
+<li><span id="finality-depth-rule"></span>
+<strong>Finality depth rule:</strong> Define: <span class="katex-display fleqn"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">finality-depth</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">height</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">height</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.1em;"></span><span class="mclose">))</span></span></span></span></span> Then either <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">finality-depth</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span> or <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">is-stalled-block</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span></span></span>.</span></li>
+</ul>
+<details id="admonition-explain-the-definition-of-finalitydepth" class="admonition admonish-info">
+<summary class="admonition-title">
+<p>Explain the definition of finality‑depth.</p>
+<p><a class="admonition-anchor-link" href="#admonition-explain-the-definition-of-finalitydepth"></a></p>
+</summary>
+<div>
+<p>The finality depth must be objectively defined, since it is used in a consensus rule. Therefore it should measure the height of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> relative to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.1em;"></span><span class="mclose">)</span></span></span></span>, which is an objectively defined function of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span>, rather than relative to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span>. (These will only differ for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span> when node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> has just reorged, and only then in corner cases.)</p>
+<p>Note that the <a href="#last-final-snapshot-rule"><strong>Last Final Snapshot rule</strong></a> ensures that it is meaningful to simply use the difference in heights, since <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.1em;"></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span>.</p>
+</div>
+</details>
+<h4 id="Πbc-contextual-validity"><a class="header" href="#Πbc-contextual-validity">Π<sub>bc</sub> contextual validity</a></h4>
+<p>The consensus rule changes above are all non-contextual. Modulo these changes, contextual validity <span style="white-space: nowrap">in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span> is the same as <span style="white-space: nowrap">in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>.</span></p>
+<h4 id="Πbc-honest-block-production"><a class="header" href="#Πbc-honest-block-production">Π<sub>bc</sub> honest block production</a></h4>
+<p>An honest producer of a <span style="white-space: nowrap">bc‑block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span></span> must follow the consensus rules under <a href="#%CE%A0bc-block-validity"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> block validity</a> above. In particular, it must produce a stalled block if required to do so by the <a href="#finality-depth-rule"><strong>Finality depth rule</strong></a>.</p>
+<p>To <span style="white-space: nowrap">choose <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">context_bft</span></span></span></span></span>,</span> the producer considers a subset of the tips of bft‑valid‑chains in its view: <span class="katex-display fleqn"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="mord text"><span class="mord"> is bft‑block‑valid and </span></span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="mclose">)}</span></span></span></span></span> It chooses one of the longest of these chains, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span>, breaking ties by maximizing <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">score</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mclose">)))</span></span></span></span>,</span> and if there is still a tie then by taking <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span> with the smallest hash.</p>
+<p>The honest block producer then sets <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">context_bft</span></span></span></span></span> to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span>.</span></p>
+<div id="admonition-attention" class="admonition admonish-warning">
+<div class="admonition-title">
+<p>Attention</p>
+<p><a class="admonition-anchor-link" href="#admonition-attention"></a></p>
+</div>
+<div>
+<p>An honest bc‑block‑producer must not use information from the BFT protocol, other than the specified consensus rules, to decide which bc‑valid‑chain to follow. The specified consensus rules that depend on <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> have been carefully constructed to preserve safety of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> relative <span style="white-space: nowrap">to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>.</span> Imposing any additional constraints could potentially allow an adversary that is able to <span style="white-space: nowrap">subvert <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>,</span> to influence the evolution of the bc‑best‑chain in ways that are not considered in the safety argument.</p>
+</div>
+</div>
+<details id="admonition-why-not-choose-t--such-that-h-1bc--context_bft--bft--bftlastfinalt-" class="admonition admonish-info">
+<summary class="admonition-title">
+<p>Why not choose <em>T</em>  such that <em>H</em> ⌈<sup>1</sup><sub>bc</sub> . context_bft  ⪯<sub>bft</sub>  bft‑last‑final(<em>T</em> )?</p>
+<p><a class="admonition-anchor-link" href="#admonition-why-not-choose-t--such-that-h-1bc--context_bft--bft--bftlastfinalt-"></a></p>
+</summary>
+<div>
+<p>The effect of this would be to tend to more often follow the last bft‑block seen by the producer of the parent bc‑block, if there is a choice. It is not always possible to do so, though: the resulting set of candidates <span style="white-space: nowrap">for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span></span> might be empty.</p>
+<p>Also, it is not clear that giving the parent bc‑block‑producer the chance to “guide” what bft‑block should be chosen next is beneficial, since that producer might be adversarial and the resulting incentives are difficult to reason about.</p>
+</div>
+</details>
+<details id="admonition-why-choose-the-longest-c-rather-than-the-longest-bftlastfinalc-" class="admonition admonish-info">
+<summary class="admonition-title">
+<p>Why choose the longest <em>C</em>, rather than the longest bft‑last‑final(<em>C</em> )?</p>
+<p><a class="admonition-anchor-link" href="#admonition-why-choose-the-longest-c-rather-than-the-longest-bftlastfinalc-"></a></p>
+</summary>
+<div>
+<p>We could have instead chosen <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span> to maximize the length <span style="white-space: nowrap">of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mclose">)</span></span></span></span>.</span> The rule we chose follows Streamlet, which builds on the longest notarized chain, not the longest finalized chain. This may call for more analysis specific to the chosen BFT protocol.</p>
+</div>
+</details>
+<details id="admonition-why-this-tiebreaking-rule" class="admonition admonish-info">
+<summary class="admonition-title">
+<p>Why this tie‑breaking rule?</p>
+<p><a class="admonition-anchor-link" href="#admonition-why-this-tiebreaking-rule"></a></p>
+</summary>
+<div>
+<p>Choosing the bft‑chain that has the last final snapshot with the highest score, tends to inhibit an adversary’s ability to finalize its own chain if it has a lesser score. (If it has a greater score, then it has already won a hash race and we cannot stop the adversary chain from being finalized.)</p>
+</div>
+</details>
+<p>For discussion of potentially unifying the roles of bc‑block producer and bft‑proposer, see <a href="./potential-changes.html#what-about-making-the-bcblockproducer-the-bftproposer">What about making the bc‑block‑producer the bft‑proposer?</a> in <a href="./potential-changes.html">Potential changes to Crosslink</a>.</p>
+<p>At this point we have completed the definition of Crosslink 2. In <a href="./security-analysis.html">Security Analysis of Crosslink 2</a>, we will prove it secure.</p>
+
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+<h1 id="notes-on-snapandchat"><a class="header" href="#notes-on-snapandchat">Notes on Snap‑and‑Chat</a></h1>
+<p>The discussion in <a href="./the-arguments-for-bounded-availability-and-finality-overrides.html">The Argument for Bounded Availability and Finality Overrides</a> is at an abstract level, applying to any Ebb‑and‑Flow-like protocol.</p>
+<p>This document considers specifics of the Snap‑and‑Chat construction proposed in <a href="https://eprint.iacr.org/2020/1091.pdf">[NTT2020]</a> (<a href="https://arxiv.org/pdf/2009.04987.pdf">arXiv version</a>).</p>
+<div id="admonition-terminology" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Terminology</p>
+<p><a class="admonition-anchor-link" href="#admonition-terminology"></a></p>
+</div>
+<div>
+<p>We are trying to be precise in this document about use of the terms “Ebb‑and‑Flow”, which is the security model and goal introduced in <a href="https://eprint.iacr.org/2020/1091.pdf">[NTT2020]</a>, vs “Snap‑and‑Chat”, which is the construction proposed in the same paper to achieve that goal. There are other ways to design an Ebb‑and‑Flow protocol that don’t run into the difficulties described in this section (or that run into different difficulties).</p>
+</div>
+</div>
+<h3 id="effect-on-consensus"><a class="header" href="#effect-on-consensus">Effect on consensus</a></h3>
+<p>A general problem with the Snap‑and‑Chat construction is that it does not follow, from enforcement of the original consensus rules on blocks produced in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>, that the properties they are intended to enforce hold for the <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span> or <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">da</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span> ledgers. Less obviously, the converse also does not follow: enforcing unmodified consensus rules on <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> blocks is <strong>both</strong> too lax and too strict.</p>
+<p>Recall from the paper how <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">da</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span> are constructed (starting at the end of page 8 of <a href="https://eprint.iacr.org/2020/1091.pdf">[NTT2020]</a>):</p>
+<blockquote>
+<ol start="3">
+<li><em>Ledger extraction:</em> Finally, how honest nodes compute <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">da</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span> from <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">Ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span> is illustrated in Figure 6. Recall that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">Ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span> is an ordering of snapshots, i.e., a chain of chains of LC blocks. First, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">Ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span> is flattened, i.e. the chains of blocks are concatenated as ordered to arrive at a single sequence of LC blocks. Then, all but the first occurrence of each block are removed (sanitized) to arrive at the finalized ledger <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span> of LC blocks. To form the available ledger <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">da</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span>, which is a sequence of LC blocks, is appended to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span> and the result again sanitized.</li>
+</ol>
+</blockquote>
+<p>This says that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">da</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span> are sequences of transactions, <em>not</em> sequences of blocks. Therefore, <a href="https://zips.z.cash/protocol/protocol.pdf#blockheader">consensus rules defined at the block level</a> are not applicable.</p>
+<div id="admonition-zcash-specific" class="admonition admonish-warning">
+<div class="admonition-title">
+<p>Zcash-specific</p>
+<p><a class="admonition-anchor-link" href="#admonition-zcash-specific"></a></p>
+</div>
+<div>
+<p>Most of these rules are Proof‑of‑Work‑related checks that can be safely ignored at this level. Some are related to the <code>hashBlockCommitments</code> field intended for use by the FlyClient protocol. It is not at all clear how to make FlyClient (or other uses of this commitment) work with the Snap‑and‑Chat construction. In particular, the <code>hashEarliest{Sapling,Orchard}Root</code>, <code>hashLatest{Sapling,Orchard}Root</code>, and <code>n{Sapling,Orchard}TxCount</code> fields don’t make sense in this context since they could only reflect the values in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span>, which have no relation in general to those for any subrange of transactions in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">da</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span>. This problem occurs as a result of sanitization and so will be avoided by Crosslink 2, which does not need sanitization.</p>
+</div>
+</div>
+<p>Since <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">da</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span> does not have blocks, it is not well-defined whether it has “coinbase‑only blocks” when in Stalled Mode. That by itself is not so much of a problem because it would be sufficient for it to have only coinbase transactions in that mode.</p>
+<h3 id="effect-on-issuance"><a class="header" href="#effect-on-issuance">Effect on issuance</a></h3>
+<p>The issuance schedule of Zcash was designed under the assumption that blocks only come from a single chain, and that the difficulty adjustment algorithm keeps the rate of block mining roughly constant over time.</p>
+<p>For Snap‑and‑Chat, if there is a rollback longer than <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> blocks in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>, additional coinbase transactions from the rolled-back chain will be included in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</p>
+<p>We can argue that this will happen rarely enough not to cause any significant problem for the overall issuance schedule. However, it does mean that issuance is less predictable, because the block subsidies will be computed according to their depth in the <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> chain on which they were mined. So it would no longer be the case that coinbase transactions issue a deterministic, non-increasing sequence of block subsidies. (Again, this problem will be avoided by Crosslink 2.)</p>
+<h3 id="effect-on-transaction-ordering"><a class="header" href="#effect-on-transaction-ordering">Effect on transaction ordering</a></h3>
+<p>The order of transactions in any particular <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span> is <em>not</em> in general preserved in either <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> or <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">da</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>. This <em>is</em> considered in the paper (middle of the left column on page 10) but it is very easy to miss it:</p>
+<blockquote>
+<p>Thus, snapshots taken by different nodes or at different times can conflict. However, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> is still safe and thus orders these snapshots linearly. Any transactions invalidated by conflicts are sanitized during ledger extraction.</p>
+</blockquote>
+<p>That is, a transaction from one snapshot might double-spend an output already spent in a different transaction of a different snaphot earlier in the flattening order. If it is omitted, then later transactions could depend on the outputs of the omitted one. The paper is saying that each transaction is only included if (in Bitcoin and Zcash terminology) it satisfies contextual checks for double-spending and existence of inputs at the point in the ledger where it would be added.</p>
+<p>Since nullifiers for shielded spends are public, it is possible to do this even for shielded transactions. Each node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> will construct commitment trees in the order given by <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">da</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mord">.</span></span></span></span></p>
+<div id="admonition-info" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Info</p>
+<p><a class="admonition-anchor-link" href="#admonition-info"></a></p>
+</div>
+<div>
+<p>This means that if <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> is extended by a block that is not the next block in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">da</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> after the finalization point (and that has different note commitments), then <em>all</em> shielded transactions from that point onward in the previous <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">da</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> will be invalidated. It could be possible to do better at the expense of a more complicated note commitment tree structure. In any case, this situation is expected to be rare, because it can only occur if there is a rollback of more than <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> blocks in the <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> consensus chain or a failure of BFT safety.</p>
+</div>
+</div>
+<h3 id="subtlety-in-the-definition-of-sanitization"><a class="header" href="#subtlety-in-the-definition-of-sanitization">Subtlety in the definition of sanitization</a></h3>
+<p>There are two possible ways to interpret how <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0496em;vertical-align:-0.3552em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.5198em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">{</span><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mtight"><span class="mord mathrm mtight">da</span></span><span class="mclose mtight">}</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3552em;"><span></span></span></span></span></span></span></span></span></span> are constructed in Snap‑and‑Chat:</p>
+<ol>
+<li>Concatenate the transactions from each final BFT block snapshot of an LC chain, and sanitize the resulting transaction sequence by including each transaction iff it is contextually valid.</li>
+<li>Concatenate the <em>blocks</em> from each final BFT block snapshot of an LC chain, remove duplicate <em>blocks</em>, and only then sanitize the resulting transaction sequence by including each transaction iff it is contextually valid.</li>
+</ol>
+<p>These are equivalent in the setting considered by <a href="https://eprint.iacr.org/2020/1091.pdf">[NTT2020]</a>, but the argument for their equivalence is not obvious. We definitely want them to be equivalent: in practice there will be many duplicate blocks from chain prefixes in the input to sanitization, and so a literal implementation of the first variant would have to recheck all duplicate transactions for contextual validity. That would have at least <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> complexity (more likely <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span>) in the length <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> of the block chain, because the length of each final snapshot grows with <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span>.</p>
+<p>Suppose that, in a particular <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>, the only reasons for a transaction to be contextually invalid are double-spends and missing inputs. In that case the argument for equivalence is:</p>
+<ul>
+<li>If a transaction is omitted due to a double-spend, then any subsequent time it is checked, that input will still have been double-spent.</li>
+<li>If a transaction is omitted due to a missing input, this can only be because an earlier transaction in the input to sanitization was omitted. So the structure of omitted transactions forms a DAG in which parent links must be to <em>earlier</em> omitted transactions. The roots of the DAG are at double-spending transactions, which cannot be reinstated. A child cannot be reinstated until its parents have been reinstated. Therefore, no transactions are reinstated.</li>
+</ul>
+<p>Note that any <em>other</em> reason for transactions to be contextually invalid might interfere with this argument. Therefore, strictly speaking Snap‑and‑Chat should require of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> that there is no such other reason. This does not seem to be explicitly stated anywhere in <a href="https://eprint.iacr.org/2020/1091.pdf">[NTT2020]</a>.</p>
+<div id="admonition-zcash-specific-1" class="admonition admonish-warning">
+<div class="admonition-title">
+<p>Zcash-specific</p>
+<p><a class="admonition-anchor-link" href="#admonition-zcash-specific-1"></a></p>
+</div>
+<div>
+<p>In Zcash a transaction can also be contextually invalid because it has expired, or because it has a missing anchor. Expiry can be handled by extending the above argument as follows:</p>
+<ul>
+<li>If a transaction is omitted due to having expired, then any subsequent time it is checked, it will still be expired.</li>
+</ul>
+<p>It is not obvious how to extend it to handle missing anchors, because it <em>is</em> technically possible for a duplicate transaction that was invalid because of a missing anchor to <em>become</em> valid in a subsequent block. That situation would require careful manipulation of the commitment trees, but there does not seem to be anything preventing it from being provoked intentionally. The argument that was used above for missing inputs does not work here, because there is no corresponding DAG formed by the transactions with missing anchors: the same commitment treestate can be produced by two unrelated transactions.</p>
+</div>
+</div>
+<h3 id="spending-finalized-outputs"><a class="header" href="#spending-finalized-outputs">Spending finalized outputs</a></h3>
+<p>Transactions in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span> need to be able to spend outputs that are not necessarily from any previous transaction in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span>. This is because, from the point of view of a user of node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> at time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>, the block chain includes all transactions in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">da</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span>. All of the transactions after the finalization point are guaranteed to also be in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span>, but the ones before the finalization point (i.e. in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span>) are not, because they could be from some other <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1318em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span> for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal">u</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span><span class="mrel">=</span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> (intuitively, from some long chain fork that was once considered confirmed by enough nodes).</p>
+<div id="admonition-info-1" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Info</p>
+<p><a class="admonition-anchor-link" href="#admonition-info-1"></a></p>
+</div>
+<div>
+<p>Honest nodes only ever vote for confirmed snapshots, that is, prefixes of their best <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> chain truncated by the confirmation depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>. Obviously the whole point of having the BFT protocol is that chain forks longer than <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> <strong>can</strong> occur in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> — otherwise we'd just use <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> and have done. So it is not that we expect this case to be common, but if it happens then it will never fix itself: the consensus chain in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> will continue on without ever including the transactions from <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span> that were obtained from a snapshot of another fork.</p>
+</div>
+</div>
+<p>A user must be able to spend outputs for which they hold the spending key from <strong>any</strong> finalized transaction, otherwise there would be no point to the finalization.</p>
+<p>The authors of <a href="https://eprint.iacr.org/2020/1091.pdf">[NTT2020]</a> probably just missed this: the paper only has evidence that they simulated their construction, rather than implementing it for Bitcoin or any other concrete block chain as <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>. Let’s try to repair it.</p>
+<p>Suppose that node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> is trying to determine whether <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span> is a consensus-valid chain, which is necessary for deterministic consensus in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>. It cannot decide whether to allow transactions in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span> to spend outputs not in the history of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span> on the basis of its <em>own</em> finalized view <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mpunct">,</span></span></span></span> because <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span> are not in general the same.</p>
+<p>Of course, we hope that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span> are consistent, i.e. one is a prefix of the other. But even if they are consistent, they are not necessarily the same length. In particular, if <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span> is <em>shorter</em> than <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mpunct">,</span></span></span></span> then node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> does not have enough information to fill in the gap — and so it may incorrectly view a transaction in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span> as spending an output that does not exist, when actually it does exist in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∖</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mord">.</span></span></span></span> Conversely if <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span> were longer and node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> were to allow spending an output in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∖</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mpunct">,</span></span></span></span> that would be using information that is not necessarily available to other nodes, and so node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> could diverge from consensus.</p>
+<p>Consensus validity of the block at the tip of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span> can only be a deterministic function of the block itself and its ancestors in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span>. It is crucial to be able to <em>eventually</em> spend outputs from the finalized chain. We are forced to conclude that the chain <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span> <strong>must include</strong> the information needed to calculate <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1318em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">s</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span> for some <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">s</span></span></span></span> not too far behind <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>. That is, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> must be modified to ensure that this is the case. This leads us to strengthen the required properties of an Ebb‑and‑Flow protocol to include another property, “finalization availability”.</p>
+<h2 id="finalization-availability"><a class="header" href="#finalization-availability">Finalization Availability</a></h2>
+<p>In the absence of security flaws and under the security assumptions required by the finality layer, the finalization point will not be seen by any honest node to roll back. However, that does not imply that all nodes will see the same finalized height — which is impossible given network delays and unreliable messaging.</p>
+<p>Both in order to optimize the availability of applications that require finality, and in order to solve the technical issue of spending finalized outputs described in the previous section, we need to consider availability of the information needed to finalize the chain up to a particular point.</p>
+<p>Note that in Bitcoin-like consensus protocols, we don’t generally consider it to be an availability flaw that a block header only <em>commits</em> to the previous block hash and to the Merkle tree of transactions in the block, rather than including them directly. These commitments allow nodes to check that they have the correct information, which can then be requested separately.</p>
+<p>Suppose, then, that each block header in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> commits to the <strong>L</strong>ast <strong>F</strong>inal BFT block known by the <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> block producer. For an LC block chain with block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> at its tip, we will refer to this commitment as <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span></span></span>. We refer to the parent block of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> as <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0972em;vertical-align:-0.2831em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span> (this is a special case of a notation that will be defined in <a href="./construction.html">The Crosslink 2 Construction</a>).</p>
+<div id="admonition-consensus-rule-extension" class="admonition admonish-success">
+<div class="admonition-title">
+<p>Consensus rule: Extension</p>
+<p><a class="admonition-anchor-link" href="#admonition-consensus-rule-extension"></a></p>
+</div>
+<div>
+<p>We require, as a consensus rule, that if <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> is not the genesis block header, then this BFT block either descends from or is the same as the final BFT block committed to by the <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> block’s parent. <span style="white-space: nowrap">That is, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0972em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span></span></span>.</span></p>
+</div>
+</div>
+<p>This Extension rule will be preserved into Crosslink 2.</p>
+<p>The Extension rule does not prevent the BFT chain from rolling back, if the security assumptions of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> were violated. However, it means that if <span style="white-space: nowrap">a node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span></span> does not observe a rollback in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> at confirmation <span style="white-space: nowrap">depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>,</span> then it will also not observe any instability <span style="white-space: nowrap">in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>,</span> <em>even if</em> the security assumptions of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> are violated. This property holds <strong>by construction</strong>, and in fact regardless <span style="white-space: nowrap">of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</span></p>
+<div id="admonition-info-2" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Info</p>
+<p><a class="admonition-anchor-link" href="#admonition-info-2"></a></p>
+</div>
+<div>
+<p>In the Snap‑and‑Chat construction, we also have BFT block proposals committing to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> snapshots (top of right column of [<a href="https://eprint.iacr.org/2020/1091.pdf">NTT2020</a>, page 7]):</p>
+<blockquote>
+<p>In addition, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span> is used as side information in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> to boycott the finalization of invalid snapshots proposed by the adversary.</p>
+</blockquote>
+<p>This does not cause any circularity, because each protocol only commits to <em>earlier</em> blocks of the other. In fact, BFT validators have to listen to transmission of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> block headers anyway, so that <em>could</em> be also the protocol over which they get the information needed to make and broadcast their own signatures or proposals. (A possible reason not to broadcast individual signatures to all nodes is that with large numbers of validators, the proof that a sufficient proportion of validators/stake has signed can use an aggregate signature, which could be much smaller. Also, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> nodes only need to know about <em>successful</em> BFT block proposals.)</p>
+</div>
+</div>
+<p>Now suppose that, in a Snap‑and‑Chat protocol, the BFT consensus finalizes a <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> snapshot that does not extend the snapshot in the previous block (which can happen if either <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> is unsafe, or <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> suffers a rollback longer than <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> blocks). In that case we will initially not be able to spend outputs from the old snapshot in the new chain. But eventually for some node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> that sees the header <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> at the tip of its best chain at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>,</span> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span></span></span> will be such that from then on (i.e. at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal">u</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>),</span> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1318em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span> includes the output that we want to spend. This assumes <em>liveness</em> of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> and <em>safety</em> <span style="white-space: nowrap">of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</span></p>
+<p>That is, including a reference to a recent final BFT block in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> block headers both incentivizes nodes to propagate this information, <em>and</em> can be used to solve the “spending finalized outputs” problem.</p>
+<p>Optionally, we could incentivize the block producer to include the latest information it has, for example by burning part of the block reward or by giving the producer some limited mining advantage that depends on how many <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> blocks back the finalization information is.</p>
+<p>This raises the question of how we measure how far ahead a given block is relative to the finalization information it provides. As we said before, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span> is a sequence of transactions, not blocks. The transactions will in general be in a different order, and also some transactions from <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span> may have been omitted from <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span> (and even <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">da</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span>) because they were not contextually valid.</p>
+<div id="admonition-info-3" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Info</p>
+<p><a class="admonition-anchor-link" href="#admonition-info-3"></a></p>
+</div>
+<div>
+<p>In <a href="./construction.html">Crosslink 2</a>, we will sidestep this problem by avoiding the need for sanitization — that is, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span> will correspond exactly to a chain of blocks that is a prefix of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span>. Actually we use the notation <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span> to reflect the fact that it is a bc‑chain, not a sequence of bc‑transactions. This invariant is maintained statefully on each node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>: any rollback past <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span> will be ignored. If a new <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span> would conflict with the old one, the node will refuse to use it. This allows each node to straightforwardly measure how many blocks <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span> is ahead of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span> as the difference in heights. Since this document is intended to explain the development of Crosslink from Snap‑and‑Chat, here we describe the more complicated approach that we originally came up with for Crosslink 1 — which also serves to motivate the simplification in Crosslink 2.</p>
+</div>
+</div>
+<p>Assume that a block unambiguously specifies its ancestor chain. For a block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span>, define:<span class="katex-display fleqn"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">tailhead</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">last-common-ancestor</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.1em;"></span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">finality-depth</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">height</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">height</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">tailhead</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">))</span></span></span></span></span></p>
+<p>Here <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span></span></span> is the BFT block we are providing information for, and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.1em;"></span><span class="mclose">)</span></span></span></span> is the corresponding <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> snapshot. For a node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> that sees <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span></span></span> as the most recent final BFT block at time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span> will definitely contain transactions from blocks up to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">tailhead</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span></span></span>, but usually will not contain subsequent transactions on <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span>’s fork.</p>
+<div id="admonition-info-4" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Info</p>
+<p><a class="admonition-anchor-link" href="#admonition-info-4"></a></p>
+</div>
+<div>
+<p>Strictly speaking, it is possible that a previous BFT block took a snapshot <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> that is between <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">tailhead</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span>. This can only happen if there have been at least two rollbacks longer than <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> blocks (i.e. we went more than <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> blocks down <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span>’s fork from <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">tailhead</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span></span></span>, then reorged to more than <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> blocks down <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.1em;"></span><span class="mclose">)</span></span></span></span>’s fork, then reorged again to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span>’s fork). In that case, the finalized ledger would already have the non-conflicting transactions from blocks between <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">tailhead</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> — and it could be argued that the correct definition of finality depth in such cases is the depth of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> relative to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span>, not of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">tailhead</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span></span></span> relative to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span>.</p>
+<p>However,</p>
+<ul>
+<li>The definition above is simpler and easier to compute.</li>
+<li>The effect of overestimating the finality depth in such corner cases would only cause us to enforce Stalled Mode slightly sooner, which seems fine (and even desirable) in any situation where there have been at least two rollbacks longer than <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> blocks.</li>
+</ul>
+<p>By the way, the “tailhead” of a tailed animal is the area where the posterior of the tail joins the rump (also called the “dock” in some animals).</p>
+</div>
+</div>
+<p>We could alternatively just rely on the fact that some proportion of block producers are honest and will include the latest information they have. However, it turns out that having a definition of finality depth will also be useful to enforce going into Stalled Mode.</p>
+<p>Specifically, if we accept the above definition of finality depth, then the security property we want is</p>
+<div id="admonition-definition" class="admonition admonish-success">
+<div class="admonition-title">
+<p>Definition</p>
+<p><a class="admonition-anchor-link" href="#admonition-definition"></a></p>
+</div>
+<div>
+<p><strong>Bounded hazard-freeness</strong> for a finality gap bound of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span> blocks: There is never, for any node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> at time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>, observed to be a more-available ledger <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">da</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span> with a hazardous transaction that comes from block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span> such that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">finality-depth</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span>.</p>
+</div>
+</div>
+<div id="admonition-info-5" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Info</p>
+<p><a class="admonition-anchor-link" href="#admonition-info-5"></a></p>
+</div>
+<div>
+<p>This assumes that transactions in the non-finalized suffix  <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">da</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∖</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span> come from blocks in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span>. In Snap‑and‑Chat they do by definition, but ideally we wouldn’t depend on that. The difficulty in finding a more general security definition is due to the ledgers in an Ebb‑and‑Flow protocol being specified as sequences of transactions, so that a depth in the ledger would have only a very indirect correspondence to time. We could instead base a definition on timestamps, but that could run into difficulties in ensuring timestamp accuracy.</p>
+<p>Another possibility would be to count the number of coinbase transactions in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">da</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∖</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span> before the hazardous transaction. This would still be somewhat ad hoc (it depends on the fact that coinbase transactions happen once per block and cannot conflict with any other distinct transaction).</p>
+<p>In any case, if <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord textsf">finality-depth</span></span></span></span></span> sometimes overestimates the depth, that cannot weaken this security definition.</p>
+</div>
+</div>
+<p>Note that a node that is validating a chain <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span> must fetch all the chains referenced by BFT blocks reachable from it (back to an ancestor that it has seen before). In theory, there could be a partition that causes there to be multiple disjoint snapshots that get added to the BFT chain in quick succession. However, in practice we expect such long rollbacks to be rare if <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> is meeting its security goals.</p>
+<p>Going into Stalled Mode if there is a long finalization stall helps to reduce the cost of validation when the stall resolves. That is, if there is a partition and nodes build on several long chains, then in unmodified Snap‑and‑Chat, it could be necessary to validate an arbitrary number of transactions on each chain when the stall resolves. Having only coinbase transactions after a certain point in each chain would significantly reduce the concrete validation costs in this situation.</p>
+<p>Nodes should not simply trust that the BFT blocks are correct; they should check validator signatures (or aggregate signatures) and finalization rules. Similarly, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> snapshots should not be trusted just because they are referenced by BFT blocks; they should be fully validated, including the proofs-of-work.</p>
+<p>It is also possible for a snapshot reference to include the subsequent <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> block headers, which are guaranteed to be available for a confirmed snapshot. Having all nodes validate the proofs-of-work in these headers is likely to significantly increase the work that an attacker would need to perform to cause disruption under a partial failure of either <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> or <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>’s security properties.</p>
+<div id="admonition-info-6" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Info</p>
+<p><a class="admonition-anchor-link" href="#admonition-info-6"></a></p>
+</div>
+<div>
+<p>Note that <a href="https://eprint.iacr.org/2020/1091.pdf">[NTT2020]</a> (bottom of right column, page 9) makes a safety assumption about <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> in order to prove the consistency of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> with the output of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>:</p>
+<blockquote>
+<p>As indicated by Algorithm 1, a snapshot of the output of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> becomes final as part of a BFT block only if that snapshot is seen as confirmed by at least one honest node. However, since <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> is safe [i.e., does not roll back further than the confirmation depth <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>],</span> the fact that one honest node sees that snapshot as confirmed implies that every honest node sees the same snapshot as confirmed.</p>
+</blockquote>
+<p>We claim that, while this may be a reasonable assumption to make for parts of the security analysis, in practice we should always require any adversary to do the relevant amount of Proof‑of‑Work to construct block headers that are plausibly confirmed. This is useful even though we cannot require, for every possible attack, that it had those headers at the time they should originally have appeared.</p>
+</div>
+</div>
+<h2 id="enforcing-finalization-availability-and-stalled-mode"><a class="header" href="#enforcing-finalization-availability-and-stalled-mode">Enforcing Finalization Availability and Stalled Mode</a></h2>
+<p>The following idea for enforcing finalization availability <em>and</em> a bound on the finality gap was originally conceived before we had switched to advocating the Stalled Mode approach. It’s simpler to explain first in that variant.</p>
+<p>Suppose that for an <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span>-block availability bound,</span> we required each block header to include the information necessary for a node to finalize to <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span> blocks back.</span> This would automatically enforce a chain stall after the availability bound without any further explicit check, because it would be impossible to produce a block after the bound.</p>
+<p>Note that if full nodes have access to the BFT chain, knowing <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span></span></span> is sufficient to tell whether the correct version of any given BFT block in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span></span></span>’s ancestor chain has been obtained.</p>
+<p>Suppose that the finality gap bound is <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span> blocks.</span> Having already defined <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord textsf">finality-depth</span></span></span></span></span>, the necessary <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> consensus rule is attractively simple:</p>
+<div id="admonition-consensus-rule" class="admonition admonish-success">
+<div class="admonition-title">
+<p>Consensus rule</p>
+<p><a class="admonition-anchor-link" href="#admonition-consensus-rule"></a></p>
+</div>
+<div>
+<p>For every <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">finality-depth</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span>.</p>
+</div>
+</div>
+<p>To adapt this approach to enforce Stalled Mode instead of stalling the chain, we can allow the alternative of producing a block that follows the Stalled Mode restrictions:</p>
+<div id="admonition-consensus-rule-1" class="admonition admonish-success">
+<div class="admonition-title">
+<p>Consensus rule</p>
+<p><a class="admonition-anchor-link" href="#admonition-consensus-rule-1"></a></p>
+</div>
+<div>
+<p>For every <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span>, <font color="#4040f0">either</font> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">finality-depth</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span><font color="#4040f0">, or <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> follows the Stalled Mode restrictions</font>.</p>
+</div>
+</div>
+<p>Note that Stalled Mode will be exited automatically as soon as the finalization point catches up to within <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span> blocks (if it does without an intentional rollback). Typically, after recovery from whatever was causing the finalization stall, the validators will be able to obtain consensus on the same chain as <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">da</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>, and so there will be no rollback (or at least not a long one) of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">da</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</p>
+<div id="admonition-info-7" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Info</p>
+<p><a class="admonition-anchor-link" href="#admonition-info-7"></a></p>
+</div>
+<div>
+<p>An earlier iteration of this idea required the finalization information to be included in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> block headers. This is not necessary when we assume that full nodes have access to the BFT chain and can obtain arbitrary BFT blocks. This also sidesteps any need to relax the rule in order to bound the size of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> block headers. <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> block producers are still incentivized to make the relevant BFT blocks available, because without them the above consensus rule cannot be checked, and so their <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> blocks would not be accepted.</p>
+<p>There is, however, a potential denial-of-service attack by claiming the existence of a BFT block that is very far ahead of the actual BFT chain tip. This attack is not very serious as long as nodes limit the number of BFT blocks they will attempt to obtain in parallel before having checked validator signatures.</p>
+</div>
+</div>
+<h3 id="comment-on-security-assumptions"><a class="header" href="#comment-on-security-assumptions">Comment on security assumptions</a></h3>
+<p>Consider Lemma 5:</p>
+<blockquote>
+<p>Moreover, for a BFT block to become final in the view of an honest node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> under <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathcal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-2.4519em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2481em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.07944em;">Z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0794em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>, at least one vote from an honest node is required, and honest nodes only vote for a BFT block if they view the referenced LC block as confirmed.</p>
+</blockquote>
+<p>The stated assumptions are:</p>
+<blockquote>
+<p><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathcal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.07944em;">Z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0794em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> formalizes the model of <strong>P2</strong>, a synchronous network under dynamic participation, with respect to a bound <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span></span></span></span> on the fraction of awake nodes that are adversarial:</p>
+<ul>
+<li>At all times, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> is required to deliver all messages sent between honest nodes in at most <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Δ</span></span></span></span> slots.</li>
+<li>At all times, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> determines which honest nodes are awake/asleep and when, subject to the constraint that at all times at most fraction <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span></span></span></span> of awake nodes are adversarial and at least one honest node is awake.</li>
+</ul>
+</blockquote>
+<p><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathcal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-2.4519em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2481em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.07944em;">Z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0794em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> is defined as <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathcal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.07944em;">Z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0794em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">))</span></span></span></span>.</p>
+<p>Now consider this statement and figure:</p>
+<blockquote>
+<p>Even if <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> is unsafe (Figure 9c), finalization of a snapshot requires at least one honest vote, and thus only valid snapshots become finalized.</p>
+<p><img src="./NTT2020-Figure-9.png" alt="Figure 9 of [NTT2020]" /></p>
+</blockquote>
+<p>This argument is technically correct but has to be interpreted with care. It only applies when the number of malicious nodes <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span> is such that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">n</span><span class="mord">/3</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">n</span><span class="mord">/2</span></span></span></span>. What we are trying to do with Crosslink is to ensure that a similar conclusion holds even if <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> is completely subverted, i.e. the adversary has 100% of validators (but only &lt; 50% of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> hash rate).</p>
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+                <ol class="chapter"><li class="chapter-item expanded "><a href="../../introduction.html"><strong aria-hidden="true">1.</strong> Introduction</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../introduction/a-path-to-pos-zcash.html"><strong aria-hidden="true">1.1.</strong> A Path to Proof-of-Stake Zcash</a></li><li class="chapter-item expanded "><a href="../../introduction/trailing-finality-layer-in-a-nutshell.html"><strong aria-hidden="true">1.2.</strong> Trailing Finality Layer in a Nutshell</a></li><li class="chapter-item expanded "><a href="../../introduction/status-and-next-steps.html"><strong aria-hidden="true">1.3.</strong> Status and Next Steps</a></li><li class="chapter-item expanded "><a href="../../introduction/motivating-finality.html"><strong aria-hidden="true">1.4.</strong> Motivating Finality</a></li><li class="chapter-item expanded "><a href="../../introduction/get-involved.html"><strong aria-hidden="true">1.5.</strong> Get Involved</a></li></ol></li><li class="chapter-item expanded "><a href="../../terminology.html"><strong aria-hidden="true">2.</strong> Terminology</a></li><li class="chapter-item expanded "><a href="../../design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../design/design-at-a-glance.html"><strong aria-hidden="true">3.1.</strong> Design at a Glance</a></li><li class="chapter-item expanded "><a href="../../design/goals.html"><strong aria-hidden="true">3.2.</strong> Design Goals</a></li><li class="chapter-item expanded "><a href="../../design/crosslink.html"><strong aria-hidden="true">3.3.</strong> Crosslink</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../design/crosslink/the-arguments-for-bounded-availability-and-finality-overrides.html"><strong aria-hidden="true">3.3.1.</strong> The Arguments for Bounded Availability and Finality Overrides</a></li><li class="chapter-item expanded "><a href="../../design/crosslink/notes-on-snap-and-chat.html"><strong aria-hidden="true">3.3.2.</strong> Notes On Snap‑and‑Chat</a></li><li class="chapter-item expanded "><a href="../../design/crosslink/construction.html"><strong aria-hidden="true">3.3.3.</strong> The Crosslink 2 Construction</a></li><li class="chapter-item expanded "><a href="../../design/crosslink/security-analysis.html"><strong aria-hidden="true">3.3.4.</strong> Security Analysis of Crosslink 2</a></li><li class="chapter-item expanded "><a href="../../design/crosslink/questions.html"><strong aria-hidden="true">3.3.5.</strong> Questions About Crosslink</a></li><li class="chapter-item expanded "><a href="../../design/crosslink/potential-changes.html" class="active"><strong aria-hidden="true">3.3.6.</strong> Potential Changes to Crosslink</a></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">3.4.</strong> Model Code Architecture</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">3.5.</strong> Network Architecture</div></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">4.</strong> Protocol Specification</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">4.1.</strong> Abstract Protocol</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">4.1.1.</strong> Token Dynamics</div></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">4.2.</strong> Concrete Protocol</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">4.2.1.</strong> Token Dynamics</div></li></ol></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.</strong> Security</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.</strong> Abstract Analysis</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.1.</strong> Ebb-and-Flow analysis</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.2.</strong> Subprotocol Compromise Analysis</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.3.</strong> Model Code Architecture Analysis</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.4.</strong> Network Architecture Analysis</div></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.2.</strong> Concrete Analysis</div></li></ol></li><li class="chapter-item expanded "><a href="../../references.html"><strong aria-hidden="true">6.</strong> References</a></li><li class="chapter-item expanded "><a href="../../version-history.html"><strong aria-hidden="true">7.</strong> Version History</a></li></ol>
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+<h1 id="potential-changes-to-crosslink"><a class="header" href="#potential-changes-to-crosslink">Potential Changes to Crosslink</a></h1>
+<p>This page documents suggestions that have not had the same attention to security analysis as the <a href="./construction.html">Crosslink 2 construction</a>. Some of them are broken. Some of them also increase the complexity of the protocol (while some simplify it or have a mixed effect on complexity), and so we need to consider the security/complexity trade‑off of each suggestion before we could include it.</p>
+<div id="admonition-this-page-is-out-of-date" class="admonition admonish-warning">
+<div class="admonition-title">
+<p>This page is out-of-date</p>
+<p><a class="admonition-anchor-link" href="#admonition-this-page-is-out-of-date"></a></p>
+</div>
+<div>
+<p>This page has not yet been updated for the changes from Crosslink 1 to Crosslink 2.</p>
+</div>
+</div>
+<h2 id="attempts-to-improve-safety-or-to-simplify-the-protocol"><a class="header" href="#attempts-to-improve-safety-or-to-simplify-the-protocol">Attempts to improve safety or to simplify the protocol</a></h2>
+<h3 id="recommended-recording-more-info-about-the-bftchain-in-bcblocks"><a class="header" href="#recommended-recording-more-info-about-the-bftchain-in-bcblocks">[Recommended] Recording more info about the bft‑chain in bc‑blocks</a></h3>
+<p>We can allow honest bc‑block‑producers to record information about every proposed and notarized bft‑block, rather than just the one in the <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">context_bft</span></span></span></span></span> field.</p>
+<p>Duplicate information that has already been given in an ancestor bc‑block would be omitted.</p>
+<p>This would automatically expose the following shenanigans to public view (as long as enough bc‑block‑proposers are honest, which is already assumed):</p>
+<ul>
+<li>any attempt to double‑propose in the same epoch;</li>
+<li>any successful attempt to double‑notarize.</li>
+</ul>
+<p>We could also expose attempts to double‑vote.</p>
+<p>Note that double‑proposal and double‑voting could be a sign that a proposer or validator’s private key is compromised, rather than that it belongs to the adversary per se. However, the security analysis must treat such a proposer/validator as non‑honest in any case.</p>
+<h3 id="changing-the-increasing-score-rule-to-require-the-score-of-the-tip-rather-than-the-score-of-the-snapshot-to-increase"><a class="header" href="#changing-the-increasing-score-rule-to-require-the-score-of-the-tip-rather-than-the-score-of-the-snapshot-to-increase">Changing the Increasing Score rule to require the score of the tip (rather than the score of the snapshot) to increase</a></h3>
+<p>The current <a href="./construction.html#%CE%A0bft-proposal-and-block-validity"><strong>Increasing Score rule</strong></a> concerns the score of the snapshot:</p>
+<p><strong>Increasing Snapshot Score rule:</strong> Either <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0972em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord mathsf">score</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">))</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">score</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">))</span></span></span></span> or <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0972em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span>.</p>
+<p>We could instead require the score of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">tip</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">headers_bc</span></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">]</span></span></span></span> to increase:</p>
+<p><strong>Increasing Tip Score rule:</strong> Either <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0972em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord mathsf">score</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">tip</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">))</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">score</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">tip</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">))</span></span></span></span> or <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0972em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord mathsf">tip</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">tip</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span>.</p>
+<p>Pros:</p>
+<ul>
+<li>This more directly reflects the fork‑choice rule in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</li>
+<li>In Crosslink 1, an <a href="./construction.html#%CE%A0bft-honest-proposal">honest bft‑proposer</a> uses its bc‑best‑chain tip with the highest score <em>provided that</em> it is consistent with the <strong>Increasing Snapshot Score rule</strong>. This change removes the caveat, simplifying honest bft‑proposer behaviour.</li>
+<li>As a result of removing that caveat, we always know about an honest bft‑proposer’s bc‑best‑chain.</li>
+</ul>
+<p>Con:</p>
+<ul>
+<li>The score of the snapshot would not necessarily increase. As a result, it is technically possible that the new snapshot can be an ancestor of a previous snapshot. Whether this can actually happen depends on the value of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> and the difficulty adjustment rule. This causes no particular harm other than adding a corner case in ledger sanitization, but is inelegant.
+<ul>
+<li>This con is removed if we either use the <strong>Combined Increasing Score rule</strong> variant described below, <em>or</em> we also apply the “<a href="#Making-bc-rollbacks-more-difficult">Making bc‑rollbacks more difficult</a>” change in the next section.</li>
+</ul>
+</li>
+</ul>
+<p>Apart from the above con, the original motivations for the <strong>Increasing Snapshot Score rule</strong> also apply to the <strong>Increasing Tip Score rule</strong>. In particular,</p>
+<ul>
+<li>it still prevents potential attacks that rely on proposing a bc‑valid‑chain that forks from a much earlier block;</li>
+<li>it still limits the extent of disruption an adversary can feasibly cause to \LOG_{\bda}};</li>
+<li>it still always allows a proposal to be made, which may be needed to preserve liveness of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> relative to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>;</li>
+<li>it still prevents potential validation cost DoS attacks due to switching between snapshots with the same score.</li>
+</ul>
+<p>If we switch to using the Increasing Tip Score rule, then it would be more consistent for block producers to also change the tie‑breaking rule for choosing <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">context_bft</span></span></span></span></span> to use the tip score, <span style="white-space: nowrap">i.e. <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">score</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">tip</span></span><span class="mopen">(</span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mclose">)))</span></span></span></span>.</span></p>
+<p>A variation on this suggestion effectively keeps both the <strong>Increasing Snapshot Score rule</strong> and the <strong>Increasing Tip Score rule</strong>:</p>
+<p><strong>Combined Increasing Score rule:</strong> Either (<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0972em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord mathsf">score</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">))</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">score</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">))</span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0972em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord mathsf">score</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">tip</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">))</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">score</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">tip</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">))</span></span></span></span>), or <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0972em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord mathsf">tip</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">tip</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span>.</p>
+<p>Note that if <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0972em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord mathsf">tip</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">tip</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span>, both scores are necessarily equal.</p>
+<p>This variation does not simplify honest bft‑proposer behaviour.</p>
+<h3 id="making-exploitation-of-bcrollbacks-more-difficult"><a class="header" href="#making-exploitation-of-bcrollbacks-more-difficult">Making exploitation of bc‑rollbacks more difficult</a></h3>
+<p>Basic idea: Detect the case where the bc‑snapshot is rolling back, and impose a longer confirmation depth to switch to the new bc‑chain. Also temporarily stall finalization of the existing bc‑chain until the conflict has been resolved.</p>
+<p>Let <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">baseline_snapshot</span></span></span></span></span> be the Crosslink 1 definition of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span></span></span></span>, i.e. <span class="katex-display fleqn"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">baseline_snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">headers_bc</span></span><span class="mopen">[</span><span class="mord">0</span><span class="mclose">]</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mpunct">,</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord">if </span></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">headers_bc</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord amsrm">∅</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord">otherwise.</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>
+<p>When <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1641em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord"><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span><span class="mrel">⪯</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.242em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathsf">baseline_snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span>, we want to go into a mode where we require a longer confirmation <span style="white-space: nowrap">depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Σ</span></span></span></span>,</span> <span style="white-space: nowrap">say <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>.</span> Because we don’t know in this situation whether the old bc‑chain or the new bc‑chain will win, we stop finalizing both until a winner is clear.</p>
+<p>The simplest option is to record the state saying that we are in this mode explicitly, and add a consensus rule requiring it to be correct. That is, add an <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">bc_is_forked</span></span></span></span></span> field to bft‑proposals and bft‑blocks, and add a bft‑proposal and bft‑block validity rule as follows:</p>
+<ul>
+<li><strong>Is Forked rule:</strong>
+<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">bc_is_forked</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1em;vertical-align:-0.35em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">enter_forked</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span> or <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1641em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">bc_is_forked</span></span><span class="mclose">)</span></span></span></span> and not <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">exit_forked</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span></li>
+</ul>
+<p>where:</p>
+<ul>
+<li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">enter_forked</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1641em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord"><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span><span class="mrel">⪯</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.242em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathsf">baseline_snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span></li>
+<li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">exit_forked</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1244em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord mathsf">score</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">tip</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">Σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0972em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord mathsf">score</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">))</span></span></span></span>.</li>
+</ul>
+<p>It is intentional that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">exit_forked</span></span></span></span></span> takes precedence over <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">enter_forked</span></span></span></span></span>.</p>
+<p>Then redefine <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span></span></span></span> as follows: <span class="katex-display fleqn"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mpunct">,</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">baseline_snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mpunct">,</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord">if </span></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">bc_is_forked</span></span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord">otherwise.</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>
+<p>Since <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">bc_is_forked</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathsf">false</span></span></span></span></span>, the recursion will terminate.</p>
+<p>Note that there is an interaction between the <strong>Increasing Snapshot Score rule</strong> and this change: the <strong>Increasing Snapshot Score rule</strong> should arguably use <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">baseline_snapshot</span></span></span></span></span> instead of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span></span></span></span>. The <strong>Increasing Tip Score rule</strong>, on the other hand, works fine as‑is, and so it makes sense to use both of these changes together. The combination of both changes also fixes the con discussed above for the <strong>Increasing Tip Score rule</strong>; it ensures that the score of the snapshot must increase.</p>
+<p>Pros:</p>
+<ul>
+<li>If the winning chain is the chain that was first snapshotted, then there ends up being no disruption whatsoever <span style="white-space: nowrap">to \LOG_{\bda}}.</span></li>
+<li>It becomes extremely difficult for an adversary with less than 50% hash power to get the finalized snapshot to switch between two competing chains more than once.</li>
+</ul>
+<p>Cons:</p>
+<ul>
+<li>It is potentially easier to cause a temporary finalization stall. An adversary could try to provoke this situation on the honest chain, either as a DoS attack, or so that its own chain that is not so encumbered can finalize more quickly than the honest chain.
+<ul>
+<li>This does not seem like a practical attack, because such a stall can only happen when the adversary can cause a <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>‑block rollback or has <span style="white-space: nowrap">subverted <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</span></li>
+</ul>
+</li>
+<li>The definition of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span></span></span></span> becomes more complicated, and there is a risk of this complexity introducing problems.</li>
+<li>It can be argued that, unless <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> is</span> chosen too small, an adversary that can cause a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>‑block rollback</span> likely has 50% of hash power, and therefore can cause a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Σ</span></span></span></span>‑block rollback.</span> That is, increasing confirmation depth does not help beyond a certain point, and therefore (it could be argued) this change will also not help.
+<ul>
+<li>This argument may hold for private mining attacks, but does not necessarily hold for partitioning attacks, as discussed in the next section.</li>
+</ul>
+</li>
+</ul>
+<h3 id="using-tip-information-to-detect-rollbacks-and-partitions"><a class="header" href="#using-tip-information-to-detect-rollbacks-and-partitions">Using tip information to detect rollbacks and partitions</a></h3>
+<p>In the case of a private mining attack, the adversary will typically conceal the existence of the overtaking chain until it can be used to cause a rollback <span style="white-space: nowrap">in \LOG_{\bda}}.</span> So the approach used in the previous section seems to be all we can do against such an attack.</p>
+<p>In the case of a partitioning attack, on the other hand, the adversary relies on honest nodes to do mining work on each side of the partition. This relies on the successful miners on each side knowing about their chain, but not the chain on the other side. Subtly, it does not rely on a perfect network partition. An adversary could, for example, attempt to create partitions <em>around</em> the most successful mining pools. Occasional leaks of information across a partition also do not necessarily foil the attack unless that information gets to a successful miner. Therefore, measures that constrain the adversary’s ability to make use of an <em>incomplete</em> partition can be useful.</p>
+<p>This also has the benefit of making the protocol more robust against <em>non‑malicious</em> incomplete partitions.</p>
+<p>Given that in such an attack the competing chains may be visible to some proposers, there is the possibility of detecting a potential rollback even before it gets snapshotted, by using the fact that previous bft‑blocks created by honest bft‑proposers have been recording the bc‑best‑chain tip <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> blocks ahead.</span> Also, depending on what proportion of validators an adversary has, they may rely on honest validators on each side to ensure that a snapshot of each chain appears in a bft‑valid block; in that case, including information about competing chains in validators' votes (see the next subsection) may be useful.</p>
+<p>It is still possible that if an adversary has several consecutive proposal slots, it can get its chain snapshotted. However, if there is an intervening slot with an honest proposer, we can potentially compare its tip with the adversary’s tip and anticipate the need to go into <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">bc_is_forked</span></span></span></span></span> mode.</span></p>
+<p>In order to get this to work, we need to propose a definition to identify bc‑chains that are competing with the current best chain, such that there is some risk of a “long” rollback to a competing chain. <span style="white-space: nowrap">Let <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03785em;">δ</span></span></span></span> be</span> a measure of how close (in terms of bc‑blocks) a competing chain’s score needs to be to that of the bc‑best‑chain, and <span style="white-space: nowrap">let <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03785em;">δ</span></span></span></span> be</span> a lower bound on the rollback depth we would consider significant if the competing chain were to immediately catch up. <span style="white-space: nowrap">(The condition <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03785em;">δ</span></span></span></span></span> is necessary to avoid false positives that might only be a single‑block fork.)</p>
+<p>A node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> identifies <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03785em;">δ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">μ</span><span class="mclose">)</span></span></span></span>‑competing chains</span> as follows based on its current view at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>:</span></p>
+<ul>
+<li>A bc‑block is a tip if it has no known descendants.</li>
+<li>Let <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span> be <span style="white-space: nowrap">node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>’s bc‑best‑chain.</span></li>
+<li>Identify all of the <span style="white-space: nowrap">tips <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span></span></span></span></span> such that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">score</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.161em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord mathsf">score</span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03785em;">δ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> and <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1279em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">last-common-ancestor</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1792em;vertical-align:-0.3013em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em;"><span style="top:-2.3987em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.1809em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3013em;"><span></span></span></span></span></span></span></span></span></span>.</span></li>
+</ul>
+<p><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="enclosing" style="background-color:yellow;color:black;"><span class="mord text"><span class="mord textsf">TODO:</span></span></span></span></span></span> Details, including how to modify the <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">enter_forked</span></span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">exit_forked</span></span></span></span></span> conditions.</p>
+<p><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="enclosing" style="background-color:yellow;color:black;"><span class="mord text"><span class="mord textsf">TODO:</span></span></span></span></span></span> For now we will assume that all of the competing chain information in a bft‑block has to be checked as bc‑block‑valid in order for that block to be bft‑block‑valid. This might introduce validation DoS attacks and needs to be considered more carefully.</p>
+<h3 id="allowing-validators-to-signal-the-existence-of-a-competing-chain-in-their-votes"><a class="header" href="#allowing-validators-to-signal-the-existence-of-a-competing-chain-in-their-votes">Allowing validators to signal the existence of a competing chain in their votes</a></h3>
+<p>This complements the above idea by letting a validator that has seen a competing chain signal it in its signed vote. Then, as long as the adversary is reliant on some votes from honest validators that are signalling the existence of competing chains, we would go into <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">bc_is_forked</span></span></span></span></span> mode</span> without relying on honest proposers to have an intervening slot.</p>
+<p>The <a href="./construction.html#%CE%A0bft-proposal-and-block-validity">notarization proof</a> that appears in a bft‑block would need to be modified to preserve these signals. More precisely, it is necessary for a <span style="white-space: nowrap">bft‑block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span></span> to preserve at least:</p>
+<ul>
+<li>the best bc‑chain that credibly competes <span style="white-space: nowrap">with <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">tip</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span>,</span> if any.</li>
+<li>the best bc‑chain that does not credibly compete <span style="white-space: nowrap">with <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">tip</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span></span> (this necessarily exists because <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">tip</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span> does not credibly compete with itself).</li>
+</ul>
+<p>This is also motivated by the suggested change in the next section.</p>
+<p>Enforcing this is relatively straightforward if the evidence is a SNARK. It can also be enforced with aggregate signatures even for schemes that only allow aggregation of signatures over a common message: we just collect the distinct messages (corresponding to either “no competing chain” or each distinct competing chain) and aggregate them separately.</p>
+<h3 id="strengthening-the-increasing-tip-score-rule"><a class="header" href="#strengthening-the-increasing-tip-score-rule">Strengthening the Increasing Tip Score rule</a></h3>
+<p>Assume that votes include competing chain information as discussed above. We can assume that an honest proposer has read all of this information from its parent bft‑block. Therefore, we can require the tip score of its proposal to have at least the score of the best tip implied by that information:</p>
+<p>Let <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">best-tip</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span> be the tip mentioned in <span style="white-space: nowrap">bft‑block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span></span> with the highest score. <span style="white-space: nowrap">A bft‑block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span></span> “mentions” the two best tips defined in the previous section.</p>
+<p><strong>Strong Increasing Tip Score rule:</strong> Either <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0972em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord mathsf">score</span></span><span class="mopen">(</span><span class="mord text"><span class="mord textsf">best-tip</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">))</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">score</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">tip</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">))</span></span></span></span> or <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0972em;vertical-align:-0.2831em;"></span><span class="mord text"><span class="mord textsf">best-tip</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">tip</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span>.</span></p>
+<p>Note that this rule is really quite constraining for a potential adversary, especially in partitioning attacks. It means that if the adversary does not want to acknowledge the existence of a given chain, it cannot use any votes or build on any previous bft‑block that signals the existence of that chain. Essentially, a partitioning adversary with control over only the minimum one‑third of the stake would have to have ensure a perfectly complete partition; it could not get away with any information leakage between honest validators.</p>
+<h2 id="attempts-to-improve-finalization-latency"><a class="header" href="#attempts-to-improve-finalization-latency">Attempts to improve finalization latency</a></h2>
+<h3 id="broken-adjusting-the-last-snapshot-definition"><a class="header" href="#broken-adjusting-the-last-snapshot-definition">[Broken] Adjusting the last snapshot definition</a></h3>
+<p>The Crosslink 1 design imposes a finalization latency of at least <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> block times.</span> Intuitively, this is because in <span class="katex-display fleqn"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathrm">fin</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mord text"><span class="mord"> for </span></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1em;vertical-align:-0.35em;"></span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">context_bft</span></span><span class="mclose">)]</span><span class="mpunct">,</span></span></span></span></span> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">context_bft</span></span><span class="mclose">))</span></span></span></span> is at least <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> blocks</span> back <span style="white-space: nowrap">from <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0331em;vertical-align:-0.2831em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span></span> (as argued in <a href="./questions.html#Why-don%E2%80%99t-we-have-a-bc-block-validity-rule-snapshotfinal-bftH-%E2%AA%AFbc-H-">Questions about Crosslink 1</a>), and therefore <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> blocks</span> back <span style="white-space: nowrap">from <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span>.</span> So the total finalization latency is <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> block times + BFT overhead + <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span> block times + snapshot overhead.</span></p>
+<p>However, the snapshot headers contain information about the proposer’s bc‑best‑chain.</p>
+<p>Define <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1em;vertical-align:-0.35em;"></span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">context_bft</span></span><span class="mclose">)</span></span></span></span>. Although it is not guaranteed, normally <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0331em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.1em;"></span><span class="mclose">)</span></span></span></span> will be an ancestor <span style="white-space: nowrap">of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0331em;vertical-align:-0.2831em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span>.</span> What if we were to optimistically allow the last snapshot to be taken as <span class="katex-display fleqn"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord textsf">last-common-ancestor</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathsf">tip</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">))</span><span class="mpunct">,</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.1em;"></span><span class="mclose">)</span><span class="mpunct">,</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord">if it extends </span></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.1em;"></span><span class="mclose">)</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord">otherwise</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span>? After all, we know that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0331em;vertical-align:-0.2831em;"></span><span class="mord text"><span class="mord textsf">last-common-ancestor</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathsf">tip</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">))</span></span></span></span> is confirmed.</p>
+<p>Oh, this won’t work. The problem is that we want safety of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> not to depend on safety of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>. So we cannot assume (for this purpose) that nodes see the <span style="white-space: nowrap">same <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0331em;vertical-align:-0.2831em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span>.</span></p>
+<h3 id="replacing-log_bda-with-log_opt"><a class="header" href="#replacing-log_bda-with-log_opt">Replacing \LOG_{\bda}} with \LOG_{\opt}}</a></h3>
+<p>What if we instead take this to be the definition of \LOG_{\opt}}, replacing \LOG_{\bda}} (&quot;opt&quot; meaning optimistic)?</p>
+<p>As stated, a malicious proposer can try to maximize the latency of \LOG_{\opt}} (subject to the Increasing Score rule). For example, if there exists a fork of <span style="white-space: nowrap">length <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span></span></span></span>,</span> the malicious proposer can force the latency of \LOG_{\opt}} to be <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span> block times + BFT overhead.</span> However, this can be improved by applying the idea to each bft‑block in turn after the one pointed to by the best confirmed bc‑block. Then a malicious proposer cannot do anything that it could not do anyway (keeping the finalization point at its current position).</p>
+<p>Pros:</p>
+<ul>
+<li>This is always more conservative in terms of safety than the current design.</li>
+<li>The latency of \LOG_{\opt}} will typically be <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> bc‑block times,</span> rather than <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> block times.</span></li>
+</ul>
+<p>Cons:</p>
+<ul>
+<li>\LOG_{\opt}} is <em>not</em> dynamically available in any sense. It just has lower latency and different security characteristics.</li>
+<li>Even under optimistic conditions, \LOG_{\opt}} will lag slightly behind where it would be for the Crosslink 1 design, because <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0331em;vertical-align:-0.2831em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span> will necessarily be ahead of <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0331em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord mathsf">tip</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">))</span></span></span></span>.</span></li>
+</ul>
+<h3 id="broken-using-snapshots-from-the-lastseen-bftchain-when-it-is-consistent-with-the-bcbestchain"><a class="header" href="#broken-using-snapshots-from-the-lastseen-bftchain-when-it-is-consistent-with-the-bcbestchain">[Broken] Using snapshots from the last‑seen bft‑chain when it is consistent with the bc‑best‑chain</a></h3>
+<p>The following idea is broken for safety when <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has been subverted:</p>
+<div id="admonition-info" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Info</p>
+<p><a class="admonition-anchor-link" href="#admonition-info"></a></p>
+</div>
+<div>
+<p>We have two potential sources of information about blocks that could plausibly be considered finalized:</p>
+<ol>
+<li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0331em;vertical-align:-0.2831em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span></li>
+<li>the snapshots on the chain of the last <em>seen</em> <span style="white-space: nowrap">final bft‑block, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.06944em;">lsf</span></span></span></span></span>.</span></li>
+</ol>
+<p>We cannot rely only on 1. because we want assured finalization even under partition.
+We cannot rely only on 2. because if <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has been subverted, then the chain of final bft‑blocks could fork.</p>
+<p>But intuitively, if we combine these sources of information, using them over the Crosslink 1 finalization only when they are consistent, the resulting protocol should still be as safe as the safer of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> <span style="white-space: nowrap">and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</span> In particular, 2. will not roll back <em>unless</em> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has been subverted.</p>
+<p>If this idea were to pan out, it could improve the latency of finalization by <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> block times.</span></p>
+<p>This approach is essentially a hybrid of Snap‑and‑Chat and Crosslink 1:</p>
+<ul>
+<li>the Snap‑and‑Chat construction gives a finalized ledger under the assumption that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has not been subverted;</li>
+<li>the main crosslink idea is used to make sure that outputs from all finalized transactions are <em>eventually</em> spendable;</li>
+<li>safety is still only dependent on the stronger of the safety of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> <span style="white-space: nowrap">and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>,</span> because we use the additional information from snapshots in final bft‑blocks only up to the point at which they agree with the best confirmed bc‑block.</li>
+</ul>
+</div>
+</div>
+<p>To explain the safety problem with this idea: suppose that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has been subverted. In that case it is possible for a snapshot to be finalized without having being confirmed as in any honest node’s bc‑best‑chain; that is, it is possible for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> to include transactions <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span></span></span></span> from a snapshot <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span></span></span></span> in bft‑block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> such that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span></span></span></span> is not on the consensus bc‑best‑chain. And, because <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has been subverted, it is also possible that a conflicting final bft‑block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> <span style="white-space: nowrap">omits <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span></span></span></span>.</span> And so a node that has seen <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> will think that it is consistent with the best bc‑chain (so that its <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> does not include <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span></span></span></span> but does include later transactions on the consensus bc‑best‑chain), but a node that has <span style="white-space: nowrap">seen <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span></span> will compute a <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> that does <span style="white-space: nowrap">include <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span></span></span></span>.</span></p>
+<details id="admonition-more-detailed-specification-of-the-above-broken-idea" class="admonition admonish-info">
+<summary class="admonition-title">
+<p>More detailed specification of the above broken idea.</p>
+<p><a class="admonition-anchor-link" href="#admonition-more-detailed-specification-of-the-above-broken-idea"></a></p>
+</summary>
+<div>
+<p>Define <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1em;vertical-align:-0.35em;"></span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">context_bft</span></span><span class="mclose">)</span></span></span></span> as before.</p>
+<p>For simplicity assume that <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.06944em;">lsf</span></span></span></span></span> extends <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0331em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span> by only one bft‑block. (This assumption could have been removed if the idea had panned out.)</p>
+<p>Then this proposal was to consider this bc‑block as contributing the last finalized snapshot:
+<span class="katex-display fleqn"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord textsf">last-common-ancestor</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf" style="margin-right:0.06944em;">lsf</span></span><span class="mclose">))</span><span class="mpunct">,</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.1em;"></span><span class="mclose">)</span><span class="mpunct">,</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord">if it extends </span></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.1em;"></span><span class="mclose">)</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord">otherwise</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>
+<p>There is no need for a tie‑breaking rule for 2.: if we ever see two context bft‑blocks for which the last‑final blocks are conflicting, we know that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has been subverted, so we should stall or crash.</p>
+<p>Caveat: for a given node, <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0331em;vertical-align:-0.2831em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span> can</span> in theory roll back <span style="white-space: nowrap">past <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf" style="margin-right:0.06944em;">lsf</span></span><span class="mclose">)</span></span></span></span>,</span> therefore <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span></span></span> can also roll back. It is okay if we keep state here and refuse to roll back. We should set a “crisis flag”, and unset it if at any point <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0331em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> extends <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">lsf_at_crisis</span></span></span></span></span>.</span> <span style="white-space: nowrap">(If <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span> is safe and live, it will.)</p>
+<p>A similar rule that would give the same result in almost all circumstances is: <span class="katex-display fleqn"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:4.32em;vertical-align:-1.91em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.35em;"><span style="top:-2.2em;"><span class="pstrut" style="height:3.15em;"></span><span class="delimsizinginner delim-size4"><span>⎩</span></span></span><span style="top:-2.192em;"><span class="pstrut" style="height:3.15em;"></span><span style="height:0.316em;width:0.8889em;"><svg xmlns="http://www.w3.org/2000/svg" width='0.8889em' height='0.316em' style='width:0.8889em' viewBox='0 0 888.89 316' preserveAspectRatio='xMinYMin'><path d='M384 0 H504 V316 H384z M384 0 H504 V316 H384z'/></svg></span></span><span style="top:-3.15em;"><span class="pstrut" style="height:3.15em;"></span><span class="delimsizinginner delim-size4"><span>⎨</span></span></span><span style="top:-4.292em;"><span class="pstrut" style="height:3.15em;"></span><span style="height:0.316em;width:0.8889em;"><svg xmlns="http://www.w3.org/2000/svg" width='0.8889em' height='0.316em' style='width:0.8889em' viewBox='0 0 888.89 316' preserveAspectRatio='xMinYMin'><path d='M384 0 H504 V316 H384z M384 0 H504 V316 H384z'/></svg></span></span><span style="top:-4.6em;"><span class="pstrut" style="height:3.15em;"></span><span class="delimsizinginner delim-size4"><span>⎧</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.85em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.41em;"><span style="top:-4.41em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mpunct">,</span></span></span><span style="top:-2.97em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf" style="margin-right:0.06944em;">lsf</span></span><span class="mclose">)</span><span class="mpunct">,</span></span></span><span style="top:-1.53em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.1em;"></span><span class="mclose">)</span><span class="mpunct">,</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.91em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.41em;"><span style="top:-4.41em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord">if </span></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.1em;"></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf" style="margin-right:0.06944em;">lsf</span></span><span class="mclose">)</span></span></span><span style="top:-2.97em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord">if </span></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.1em;"></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf" style="margin-right:0.06944em;">lsf</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span><span style="top:-1.53em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord">otherwise</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.91em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>
+</div>
+</details>
+<h2 id="what-about-making-the-bcblockproducer-the-bftproposer"><a class="header" href="#what-about-making-the-bcblockproducer-the-bftproposer">What about making the bc‑block‑producer the bft‑proposer?</a></h2>
+<p>The answer given for this question at <a href="./construction.html#%CE%A0bft-proposal-and-block-validity">The Crosslink 2 Construction</a> is:</p>
+<blockquote>
+<p>If this were enforced, it could be an alternative way of ensuring that every bft‑proposal snapshots a new bc‑block with a higher score than previous snapshots, potentially making the <strong>Increasing Score rule</strong> redundant. However, it would require merging bc‑block‑producers and bft‑proposers, which could have concerning knock‑on effects (such as concentrating security into fewer participants).</p>
+</blockquote>
+<p>This may have been too hasty. It is not clear that merging bc‑block‑producers and bft‑proposers actually does “concentrate security into fewer participants” in a way that can have any harmful effect.</p>
+<p>Remember, the job of a bft‑proposer in Crosslink is primarily to snapshot the bc‑best‑chain (even more so if the <a href="#changing-the-increasing-score-rule-to-require-the-score-of-the-tip-rather-than-the-score-of-the-snapshot-to-increase">Increasing Tip Score rule</a> is adopted). An honest miner <em>by definition</em> is claiming to build on the best chain, and miners have a strong economic incentive to do so. Therefore, it is entirely reasonable for every newly produced block to be treated as a bft‑proposal. This arguably decentralizes the task of proposing bft‑blocks more effectively than using a leader election protocol would — especially given that in a hybrid protocol we <em>necessarily</em> still rely on there being sufficient honest miners.</p>
+<p><a href="https://arxiv.org/pdf/2010.08154.pdf">[DKT2021]</a>, for example, argues for the importance of “the complete unpredictability of who will get to propose a block next, even by the winner itself.” The main basis of this argument is that it makes subversion of the proposer significantly more difficult. A PoW protocol has that property, and most PoS protocols do not. (It is not that PoS protocols are unable to provide this property; indeed, <a href="https://arxiv.org/pdf/2010.08154.pdf">[DKT2021]</a> constructs a PoS protocol, “PoSAT”, that provides it.)</p>
+<p>So let’s explore this in more detail. A newly produced bc‑block would implicitly be a bft‑proposal with itself as the tip. The <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">bc_headers</span></span></span></span></span> field is therefore not needed. The <a href="./construction.html#%CE%A0bft-proposal-and-block-validity"><strong>Tail Confirmation rule</strong></a> goes away since its intent is automatically satisfied. This is already a significant simplification.</p>
+<p>The inner proposer signature is also not needed (since the bc‑header is self-authenticating), but the block producer would have to include a <span style="white-space: nowrap">public key <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">pubkey</span></span></span></span></span></span> that can be used to verify its outer signature. It would sign the notarized bft‑block with the corresponding private key. This change is a wash in terms of protocol complexity.</p>
+<p>Considered as a bft‑proposal, a bc‑block needs to refer to a parent bft‑block, which requires a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">parent_bft</span></span></span></span></span> field</span> in the bc‑header. With some caveats depending on the design <span style="white-space: nowrap">of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>,</span> it might be possible to merge this with the <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">context_bft</span></span></span></span></span> field,</span> but for now we will assume that it is not merged.</p>
+<details id="admonition-what-are-the-caveats" class="admonition admonish-info">
+<summary class="admonition-title">
+<p>What are the caveats?</p>
+<p><a class="admonition-anchor-link" href="#admonition-what-are-the-caveats"></a></p>
+</summary>
+<div>
+<p>If we are in an execution where <strong>Final Agreement</strong> holds <span style="white-space: nowrap">for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>,</span> then it is possible to show that merging the two fields has no negative effect, <em>provided</em> that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> has no additional rules that could disallow it in some cases.</p>
+<p>This is because, by <strong>Final Agreement</strong>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1019em;vertical-align:-0.35em;"></span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">context_bft</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:-0.5em;"></span></span><span class="base"><span class="strut" style="height:0.786em;vertical-align:-0.15em;"></span><span class="mrel"><span class="mrel">⪰</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span></span></span></span> for any potential <span style="white-space: nowrap">bft‑block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span></span> that the bc‑block‑producer of a new block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> could choose <span style="white-space: nowrap">as <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">context_bft</span></span></span></span></span>.</span> Suppose that the bc‑block‑producer freely chooses <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">parent_bft</span></span></span></span></span> according to the desired honest behaviour for a bft‑proposer <span style="white-space: nowrap">in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>,</span> and then chooses <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span> to be the same block (which is always reasonable as long as it is allowed).</p>
+<p>In the case <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1019em;vertical-align:-0.35em;"></span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">context_bft</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mclose">)</span></span></span></span>,</span> we are done, because this choice of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">context_bft</span></span></span></span></span> is allowed by the <strong>Extension rule</strong>.</p>
+<p>In the case <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1019em;vertical-align:-0.35em;"></span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">context_bft</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord">⪲</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mclose">)</span></span></span></span>,</span> we can argue that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1019em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">context_bft</span></span></span></span></span> would be a better choice than <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span> for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">parent_bft</span></span></span></span></span> as well as for <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">context_bft</span></span></span></span></span>,</span> because it has a later final ancestor. This is where the argument might fall down if <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> <span style="white-space: nowrap">(and therefore <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>)</span> has any additional rules that could disallow this choice. For now let’s suppose that situtation does not arise, but it is one of the caveats.</p>
+<p>Another potential problem is that in an execution where <strong>Final Agreement</strong> does <em>not</em> hold for <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>,</span> we can no longer infer that either <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1019em;vertical-align:-0.35em;"></span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">context_bft</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mclose">)</span></span></span></span> or <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1019em;vertical-align:-0.35em;"></span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">context_bft</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord">⪲</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mclose">)</span></span></span></span>. In particular it could be the case that the producer of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> was adversarial, and chose <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1019em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">context_bft</span></span></span></span></span> in such a way as to favour its own bft‑block that is final in that context.</p>
+<p>However, in that situation it must be possible for the bc‑block‑producer to see (and prove) that the bft‑chain has a final fork. That is, it can produce a <span style="white-space: nowrap">witness <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span></span> to the violation of <strong>Final Agreement</strong>, showing that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1019em;vertical-align:-0.35em;"></span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">context_bft</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:-0.5em;"></span></span><span class="base"><span class="strut" style="height:0.786em;vertical-align:-0.15em;"></span><span class="mrel"><span class="mrel">⪰</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span></span></span></span> does not hold, as discussed in the section <a href="#recommended-recording-more-info-about-the-bftchain-in-bcblocks">Recording more info about the bft‑chain in bc‑blocks</a> above.</p>
+<p>The second caveat is that in that situation, we still need to set <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">parent_bft</span></span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">context_bft</span></span></span></span></span> in order to be able to recover, and they typically should not be the same in order to do so.</p>
+</div>
+</details>
+<p>The <strong>Increasing Tip Score rule</strong> is still needed, but it can be simplified. A newly produced <span style="white-space: nowrap">bc‑block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span></span> is also a bft‑proposal <span style="white-space: nowrap">such that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span>.</span> This would yield the following bft‑proposal / bc‑block validity rule:</p>
+<blockquote>
+<p>[Candidate rule for discussion] Either <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">score</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">tip</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">parent_bft</span></span><span class="mclose">))</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">score</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span></span></span> <span style="white-space: nowrap">or <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">tip</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">parent_bft</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span>.</span></p>
+</blockquote>
+<p>except that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">tip</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">parent_bft</span></span><span class="mclose">)</span></span></span></span> <em>cannot</em> <span style="white-space: nowrap">be <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span>,</span> because <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> is newly produced. It turns out we can just drop that part:</p>
+<blockquote>
+<p><strong>Increasing Tip Score (producer = proposer) rule:</strong> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">score</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">tip</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">parent_bft</span></span><span class="mclose">))</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">score</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span></span></span>.</p>
+</blockquote>
+<p>This works because, if <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> does not have a higher score than the <span style="white-space: nowrap">bc‑block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">tip</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">parent_bft</span></span><span class="mclose">)</span></span></span></span>,</span> the bc‑block‑producer should instead have built on top of that bc‑block — which was necessarily known to the producer in order for it to set <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">parent_bft</span></span></span></span></span> in the header of the new block.</p>
+<p>The voting would be the same, performed by the same parties. Therefore, there is no concentration of voting into fewer parties. There is no change in the producer/proposer’s incentive to make the bft‑notarization‑proof or its soundness properties. Everything else is roughly the same, including the use of the <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">context_bft</span></span></span></span></span> field</span> of a bc‑block and the validity rules related to it. As far as I can see, all of the security analysis goes through essentially unchanged.</p>
+<p>There may be some complication due to the fact that BFT protocols are typically designed to use epochs with a fixed period, whereas bc‑blocks are found at less predictable intervals. However, as long as BFT messages are labelled with the bc‑block they apply to, it seems like most BFT protocols would be tolerant to this change. In fact the adaptations of Snap‑and‑Chat to Hotstuff and PBFT in <a href="https://eprint.iacr.org/2020/1091.pdf">[NTT2020]</a> already assume that BFT messages can be queued and processed at a later time, and rely on those BFT protocols' tolerance to this.</p>
+<div id="admonition-info-1" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Info</p>
+<p><a class="admonition-anchor-link" href="#admonition-info-1"></a></p>
+</div>
+<div>
+<p>In most PoS protocols, the requirement to have a minimum amount of stake in order to make a proposal acts as a gatekeeping filter on <em>proposals</em>, and potentially allows parties that make invalid <em>proposals</em> to be slashed.</p>
+<p>Strictly speaking, whether there is a stake requirement to make a proposal is independent of whether bc‑block‑producers (e.g. miners) are merged with bft‑proposers. It could be, for example, that a miner is still able to produce bc‑blocks, but is not able to make them into a proposal unless they satisfy a stake requirement. (This would have significant effects on the economics of mining that would need to be analyzed, and that might have governance consequences.)</p>
+<p>In a system that uses PoS, validators by definition need to have stake in order to control the ability to vote. This also allows validators to be slashed.</p>
+<p>On the other hand, there is no <em>technical</em> reason why the ability to make a bft‑proposal has to be gatekept by a stake requirement — given the situation of Zcash in which we already have a mining infrastructure, and that in a Snap‑and‑Chat or Crosslink‑style hybrid protocol we necessarily still rely on miners not to censor transactions. The potential to make proposals that are expensive to validate as a denial of service is made sufficiently difficult by proof‑of‑work. This option has probably been underexplored by previous PoS protocols because they cannot assume an existing mining infrastructure.</p>
+</div>
+</div>
+<p>It could be argued that this approach goes less far toward a pure PoS‑based block‑chain protocol, leaving more to be done in the second stage. However, there is a clear route to that second stage, by replacing PoW with a protocol like PoSAT that has full unpredictabilty and dynamic availability. PoSAT does this using a VDF, and as it happens, Halo 2 is a strong candidate to be used to construct such a VDF.</p>
+<p>If the arguments in <a href="https://arxiv.org/pdf/2010.08154.pdf">[DKT2021]</a> about the need for proposer unpredictability are persuasive, then this approach defers the complexity of requiring a VDF without losing any security, since Zcash’s PoW is already unpredictable.</p>
+<h3 id="do-we-need-an-explicit-bftchain-at-all"><a class="header" href="#do-we-need-an-explicit-bftchain-at-all">Do we need an explicit bft‑chain at all?</a></h3>
+<p>Building on the previous idea, we can try to eliminate the explicit bft‑chain by piggybacking the information it would hold onto a bc‑block (in the header and/or the block contents). In the previous section we merged the concepts of a bft‑proposal and a bc‑block; the <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">parent</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.242em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mclose">)</span></span></span></span> and <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">context_bft</span></span></span></span></span> fields</span> of a bft‑proposal were moved into the <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">parent_bft</span></span></span></span></span> and <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">context_bft</span></span></span></span></span> fields</span> of a bc‑header respectively. <span style="white-space: nowrap">A field <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">pubkey</span></span></span></span></span></span> was also added to hold the producer’s public key, so that the producer can sign the bft‑block constructed from it using the corresponding private key.</p>
+<p>This left the concept of a bft‑block intact. Recall that in Crosslink 1, a <span style="white-space: nowrap">bft‑block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span></span> consists of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord mathsf" style="margin-right:0.06944em;">proof</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2342em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">P</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> signed by the proposer. So in “Crosslink with proposer = producer”, a bft‑block consists of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord mathsf" style="margin-right:0.06944em;">proof</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2342em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.08125em;">H</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> signed by the producer.</p>
+<p>What if a <span style="white-space: nowrap">bc‑block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span></span> were to “inline” its parent and context bft‑blocks rather than referring to them? <span style="white-space: nowrap">I.e. a bc‑block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span></span> with <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">parent_bft</span></span></span></span></span> <em>referring to</em> <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord mathsf" style="margin-right:0.06944em;">proof</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2342em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6828em;"><span style="top:-2.786em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> signed for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9463em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">pubkey</span></span></span></span></span>,</span> would instead <span style="white-space: nowrap">
+<em>literally include</em></span> (either in the header or the coinbase transaction) <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord mathsf" style="margin-right:0.06944em;">proof</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2342em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6828em;"><span style="top:-2.786em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> signed <span style="white-space: nowrap">for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9463em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">pubkey</span></span></span></span></span> —</span> and similarly <span style="white-space: nowrap">for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">context_bft</span></span></span></span></span>.</span></p>
+<p>It would still be necessary to have the message type that the proposer/producer previously used to submit a notarized bft‑block. (It cannot be merged with a bc‑block announcement: the producer of a new block is not in general the producer of its parent, and their incentives may differ; also we cannot wait until a new block is found before publishing the previous notarization.) It would also still be necessary for Crosslink nodes to keep track of notarizations that have not made it into any bc‑block. Nevertheless, this is a potential simplification.</p>
+<p>Note that unless notarization proofs are particularly short and constant-length, it would not be appropriate to include them in the bc‑block headers, and so they would have to go into the coinbase transaction or another similar variable-length data structure. In that case we would still have an indirection to obtain the bft‑block information; it would just be merged with the indirection to obtain a coinbase transaction (or similar) — which is already needed in order to check validity of the bc‑block.</p>
+<p>As discussed under <a href="#recommended-recording-more-info-about-the-bftchain-in-bcblocks">Recording more info about the bft‑chain in bc‑blocks</a> above, we might in any case want to record information about other proposed and notarized bft‑blocks, and the data structure needed for this would necessarily be variable-length. The complexity burden of doing so would be shared between these two changes.</p>
+<p>It would be possible to save some space in headers (while keeping them fixed-length), by inlining only one of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">parent_bft</span></span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">context_bft</span></span></span></span></span> in the header and keeping the other as a hash. As discussed under “What are the caveats?” above, the only reason for the two bft‑blocks referred to by these fields to be different, is that the bc‑block producer has observed a violation of <strong>Final Agreement</strong> <span style="white-space: nowrap">in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</span> In that case, we can include an inlining of the other <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathsf">.</span><span class="mord"><span class="mord"><span class="mord">∗</span></span><span class="mord mathsf">_bft</span></span></span></span></span> block,</span> and any other information needed to prove that a violation of <strong>Final Agreement</strong> has occurred, in a variable-length overflow structure.</p>
+<p>Pros:</p>
+<ul>
+<li>No additional mechanism or messages are needed to obtain bft‑blocks given their hashes.</li>
+<li>It could be a performance/latency improvement to not have to separately fetch bft‑blocks.</li>
+</ul>
+<p>Con:</p>
+<ul>
+<li>Additional complexity of the variable-length overflow mechanism suggested above, if it is used.</li>
+<li>Assumes that notarization proofs are not too large.</li>
+</ul>
+<h2 id="linearity-and-last-final-snapshot-rules"><a class="header" href="#linearity-and-last-final-snapshot-rules">Linearity and Last Final Snapshot rules</a></h2>
+<p>A potential simplification can be obtained by combining the following two ideas:</p>
+<ul>
+<li>Str4d suggested that the snapshot of each bft‑block should descend from the snapshot of its parent bft‑block.</li>
+<li>Nate suggested that each bc‑block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> should descend from <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.1em;"></span><span class="mclose">)</span></span></span></span>.</li>
+</ul>
+<p>Str4d’s suggestion can be written as:</p>
+<p><strong>Linearity rule:</strong> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0972em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span>.</p>
+<p>Notice that this implies the existing <strong>Increasing Score rule</strong> in Crosslink 1, because score necessarily increases within a bc‑valid‑chain. Therefore it would in practice be a replacement for the <strong>Increasing Score rule</strong>. It does not imply the <strong>Increasing Tip Score rule</strong> discussed <a href="#changing-the-increasing-score-rule-to-require-the-score-of-the-tip-rather-than-the-score-of-the-snapshot-to-increase">above</a>, and in fact it could make sense to enforce both the <a href="#linearity-rule"><strong>Linearity rule</strong></a> and the <strong>Increasing Tip Score rule</strong>.</p>
+<p>The <a href="#linearity-rule"><strong>Linearity rule</strong></a> implies that it is no longer possible for a bft‑valid‑chain to snapshot a bc‑chain that rolls back relative to the previous snapshot. This makes it unnecessary to sanitize <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathsf">LO</span><span class="mord"><span class="mord mathsf">G</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathsf mtight">fin</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span>: the sequence of snapshots considered by <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5714em;"></span><span class="mord text"><span class="mord textsf">san-ctx</span></span></span></span></span> is linear, and so the “sanitization” would just return the transactions in the last snapshot.</p>
+<p>To remove the need to sanitize <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathsf">LO</span><span class="mord"><span class="mord mathsf">G</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathsf mtight">bda</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span> as well, we need a further modification to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>. Recall that in Crosslink 1 we define: <span class="katex-display fleqn"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">fin</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord textsf">bda-ctx</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">μ</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mrel">:=</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mrel">:=</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">[</span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mord text"><span class="mord"> for </span></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)]</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord textsf">san-ctx</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathrm">fin</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">∣∣</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em;"><span style="top:-2.3987em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.1809em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3013em;"><span></span></span></span></span></span></span><span class="mclose">])</span><span class="mord">.</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span></span></span></span></span>
+The <a href="#linearity-rule"><strong>Linearity rule</strong></a> ensures that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathrm">fin</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span></span></span> is a linear sequence of snapshots, but for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0836em;vertical-align:-0.3013em;"></span><span class="mord"><span class="mord mathrm">fin</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">∣∣</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em;"><span style="top:-2.3987em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.1809em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3013em;"><span></span></span></span></span></span></span><span class="mclose">]</span></span></span></span> to be linear, we also need <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">last</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathrm">fin</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">))</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0331em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.1em;"></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0836em;vertical-align:-0.3013em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em;"><span style="top:-2.3987em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.1809em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3013em;"><span></span></span></span></span></span></span></span></span></span>. In order for this to hold for any choice of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span></span></span></span> with <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7804em;vertical-align:-0.136em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8304em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>, we require the strongest version of this condition with <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>, i.e. <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0331em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.1em;"></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0331em;vertical-align:-0.2831em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span>.</p>
+<p>Since we can only enforce that this holds for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0331em;vertical-align:-0.2831em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span> by enforcing that it holds for an arbitrary bc‑valid‑block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span>, the rule becomes:</p>
+<p><strong>Last Final Snapshot rule:</strong> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.1em;"></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span>.</p>
+<p>This is exactly Nate’s suggestion discussed in <a href="questions.html">Questions about Crosslink</a>. In that document we argued against this rule, but that argument was made in the context of a protocol without the <a href="./construction.html##linearity-rule"><strong>Linearity rule</strong></a> (and originally, even without the <strong>Increasing Score rule</strong>).</p>
+<p>Combining the <a href="./construction.html#linearity-rule"><strong>Linearity rule</strong></a> and <a href="./construction.html#last-final-snapshot-rule"><strong>Last Final Snapshot rule</strong></a>, on the other hand, completely eliminates the need for sanitization. This could be a huge simplification — and potentially safer, since it would avoid breaking assumptions that may be made by existing Zcash node implementations and other consumers of the Zcash block chain.</p>
+<p>To spell out the resulting simplifications to the definitions of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span> and \LOG^t_{\bda,\mu,i}, we would just have: 
+\begin{array}{rcl}
+\LOG_{\fin,i}^t &amp;:=&amp; \snapshotlf{\ch_i^t \trunc_{\bc}^\sigma} \
+\LOG_{\bda,\mu,i}^t &amp;:=&amp; \ch_i^t \trunc_{\bc}^\mu
+\end{array}</p>
+<p>Here it is no longer necessary to define <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span> and \LOG^t_{\bda,\mu,i} as sequences of transactions, since the final and bounded-available chains are both just bc‑valid‑chains.</p>
+<p>The definition of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord textsf">finality-depth</span></span></span></span></span> in the <a href="./construction.html#finality-depth-rule"><strong>Finality depth rule</strong></a> becomes much simpler: <span class="katex-display fleqn"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">finality-depth</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">height</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">height</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.1em;"></span><span class="mclose">))</span></span></span></span></span> As before, either <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">finality-depth</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span> or <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">is-stalled-block</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span></span></span>.</span></p>
+<p>Avoiding sanitization also means that the bug we described in Snap‑and‑Chat, that could prevent spending outputs from a snapshotted chain after a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>-block</span> rollback, cannot occur by construction. That is, the changes in <a href="./construction.html#%CE%A0bc-contextual-validity"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> contexual validity</a> relative to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> are not needed any more.</p>
+<p>This almost seems too simple, and indeed we should be skeptical, because the security analysis essentially has to be redone. The reason why Snap‑and‑Chat didn’t take this approach is that it requires a more complicated argument to show that it is reasonable to believe in the safety assumptions of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> whenever it is reasonable to believe in the corresponding assumptions for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>. This is because the ... We will need to do some work to show that the changes are benign.</p>
+<h3 id="security-analysis"><a class="header" href="#security-analysis">Security Analysis</a></h3>
+<p>The key observation needed for this analysis is that neither the <a href="#linearity-rule"><strong>Linearity rule</strong></a> nor the <strong>Last Final Snapshot rule</strong> affect the evolution of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> <em>unless</em> we are in a situation where its <strong>Prefix Consistency</strong> or <strong>Prefix Agreement</strong> properties would be violated.</p>
+<p>This implies that any safety property that we can prove given <strong>Prefix Consistency</strong> plus ****.</p>
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+                <ol class="chapter"><li class="chapter-item expanded "><a href="../../introduction.html"><strong aria-hidden="true">1.</strong> Introduction</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../introduction/a-path-to-pos-zcash.html"><strong aria-hidden="true">1.1.</strong> A Path to Proof-of-Stake Zcash</a></li><li class="chapter-item expanded "><a href="../../introduction/trailing-finality-layer-in-a-nutshell.html"><strong aria-hidden="true">1.2.</strong> Trailing Finality Layer in a Nutshell</a></li><li class="chapter-item expanded "><a href="../../introduction/status-and-next-steps.html"><strong aria-hidden="true">1.3.</strong> Status and Next Steps</a></li><li class="chapter-item expanded "><a href="../../introduction/motivating-finality.html"><strong aria-hidden="true">1.4.</strong> Motivating Finality</a></li><li class="chapter-item expanded "><a href="../../introduction/get-involved.html"><strong aria-hidden="true">1.5.</strong> Get Involved</a></li></ol></li><li class="chapter-item expanded "><a href="../../terminology.html"><strong aria-hidden="true">2.</strong> Terminology</a></li><li class="chapter-item expanded "><a href="../../design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../design/design-at-a-glance.html"><strong aria-hidden="true">3.1.</strong> Design at a Glance</a></li><li class="chapter-item expanded "><a href="../../design/goals.html"><strong aria-hidden="true">3.2.</strong> Design Goals</a></li><li class="chapter-item expanded "><a href="../../design/crosslink.html"><strong aria-hidden="true">3.3.</strong> Crosslink</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../design/crosslink/the-arguments-for-bounded-availability-and-finality-overrides.html"><strong aria-hidden="true">3.3.1.</strong> The Arguments for Bounded Availability and Finality Overrides</a></li><li class="chapter-item expanded "><a href="../../design/crosslink/notes-on-snap-and-chat.html"><strong aria-hidden="true">3.3.2.</strong> Notes On Snap‑and‑Chat</a></li><li class="chapter-item expanded "><a href="../../design/crosslink/construction.html"><strong aria-hidden="true">3.3.3.</strong> The Crosslink 2 Construction</a></li><li class="chapter-item expanded "><a href="../../design/crosslink/security-analysis.html"><strong aria-hidden="true">3.3.4.</strong> Security Analysis of Crosslink 2</a></li><li class="chapter-item expanded "><a href="../../design/crosslink/questions.html" class="active"><strong aria-hidden="true">3.3.5.</strong> Questions About Crosslink</a></li><li class="chapter-item expanded "><a href="../../design/crosslink/potential-changes.html"><strong aria-hidden="true">3.3.6.</strong> Potential Changes to Crosslink</a></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">3.4.</strong> Model Code Architecture</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">3.5.</strong> Network Architecture</div></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">4.</strong> Protocol Specification</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">4.1.</strong> Abstract Protocol</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">4.1.1.</strong> Token Dynamics</div></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">4.2.</strong> Concrete Protocol</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">4.2.1.</strong> Token Dynamics</div></li></ol></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.</strong> Security</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.</strong> Abstract Analysis</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.1.</strong> Ebb-and-Flow analysis</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.2.</strong> Subprotocol Compromise Analysis</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.3.</strong> Model Code Architecture Analysis</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.4.</strong> Network Architecture Analysis</div></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.2.</strong> Concrete Analysis</div></li></ol></li><li class="chapter-item expanded "><a href="../../references.html"><strong aria-hidden="true">6.</strong> References</a></li><li class="chapter-item expanded "><a href="../../version-history.html"><strong aria-hidden="true">7.</strong> Version History</a></li></ol>
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+<h1 id="questions-about-crosslink"><a class="header" href="#questions-about-crosslink">Questions about Crosslink</a></h1>
+<h3 id="why-dont-we-have-a-bcblockvalidity-rule-snapshotlfh-bc-h-"><a class="header" href="#why-dont-we-have-a-bcblockvalidity-rule-snapshotlfh-bc-h-">Why don’t we have a bc‑block‑validity rule snapshot(LF(H)) ⪯<sub>bc</sub> H ?</a></h3>
+<p>Rationale: Can we outright prevent rollbacks &gt; <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> from ever appearing in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">Ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span> ?</p>
+<div id="admonition-info" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Info</p>
+<p><a class="admonition-anchor-link" href="#admonition-info"></a></p>
+</div>
+<div>
+<p>This document analyzes the effect of this rule <em>on its own</em>. For the effect in combination with an additional <a href="#linearity-rule"><strong>Linearity rule</strong></a>, see the <a href="potential-changes.html#linearity-and-last-final-snapshot-rules">Linearity and Last Final Snapshot rules</a> section of <a href="potential-changes.html">Potential changes for Crosslink 2</a>.</p>
+</div>
+</div>
+<p>Daira-Emma: In a variant of Crosslink with this rule, an adversary’s strategy would be to keep the <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">context_bft</span></span></span></span></span> fields in its blocks as <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span> such that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mclose">)</span></span></span></span> when the attack starts, and then fork from the bc‑best‑chain that extends <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span>. If its private chain falls behind the public bc‑best‑chain, it resets, just like in a conventional private mining attack.</p>
+<p>Note that the proposed rule does <em>not</em> prevent the adversary’s private chain from just staying at the same <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">context_bft</span></span></span></span></span> block. The reason is that Crosslink does not change the fork-choice rule of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>. That is, even if the adversary’s chain has a <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">context_bft</span></span></span></span></span> that is far behind the current bft‑block, it is still allowed to become the bc‑best‑chain.</p>
+<p>(Eventually the adversary’s chain using this strategy will hit the finality gap bound of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span> blocks. But that must be significantly greater than <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>, to avoid availability problems. So it does not prevent the adversary from performing a rollback longer than <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> blocks before they hit the bound. Also, going into Stalled Mode for new blocks does not prevent the attacker’s chain from having included harmful transactions before that point.)</p>
+<p>It is possible to change the fork-choice rule, for example so that the bc‑best‑chain for a node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> is required to extend <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span> where <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> is the last final block for any bft-chain in node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>’s view.</p>
+<p>This would break the current safety and liveness arguments for Crosslink. But for the sake of argument, suppose we did it anyway.</p>
+<p>The adversary’s strategy would change slightly: it resets if <em>either</em> its private chain falls behind the public bc‑best‑chain, or its private chain is invalidated because it forks before <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span> for some new last final block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> of a bft-chain. During the attack, it also attempts to impede progress of the BFT protocol as far as possible.</p>
+<p>In that case, the proposed rule still does not preclude a rollback of more than <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> blocks, for several reasons:</p>
+<ul>
+<li>
+<p>In general we can’t say anything about how many bc‑blocks are mined in any given interval, so it could be the case that more than <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> blocks are mined on both the honest chain and the adversary’s chain before it would be realistically possible to go through even a single round of the BFT protocol.</p>
+</li>
+<li>
+<p>Nor can we say anything about how quickly those blocks are finalized, unless we enforce it, which we don’t. (In Crosslink we do enforce a finalization gap bound <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span>, but as explained above <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span> must be significantly greater than <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>, so that doesn’t really help.)</p>
+<ul>
+<li>In particular, the adversary could be suppressing publication of final bft‑blocks, or attacking the <em>liveness</em> of the BFT protocol in other ways. An attack against BFT liveness is potentially easier than an attack against BFT safety, and it would be difficult to characterize exactly how much this rule gains you in terms of security given that (at best) it’s dependent on that.</li>
+</ul>
+</li>
+<li>
+<p><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.1em;"></span><span class="mclose">)</span></span></span></span> will typically be at least <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> blocks back from <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span>.
+The argument for that goes:</p>
+<blockquote>
+<p>None of the block hashes in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">headers_bc</span></span></span></span></span> can point to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> because that would be a hash cycle. In a typical case where no block withholding and no <em>other</em> rollback (not caused by the adversary) occurs on the honestly mined chain, the proposer of the last final block before a context bft‑block that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> can point to will have, at the latest, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0972em;vertical-align:-0.2831em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span> as <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">tip</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span>. Under these conditions, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.1em;"></span><span class="mclose">)</span></span></span></span> will point, at the latest, to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> blocks before <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0972em;vertical-align:-0.2831em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span>, i.e. <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> blocks before <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span>.</p>
+</blockquote>
+<p>This means that by the time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.1em;"></span><span class="mclose">)</span></span></span></span> could catch up to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span>, on average <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> block times will have occurred. So, roughly speaking, the rule that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.1em;"></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> does not usefully constrain the adversary until <span style="white-space: nowrap">after <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> block times.</span> <span style="white-space: nowrap">Wlog let’s assume <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span> <span style="white-space: nowrap">uses PoW: if <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span></span> is chosen reasonably, then being able to do <span style="white-space: nowrap">a <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>-block rollback</span> at all probably requires having somewhere close to 50% of network hash rate. And so in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> block times the adversary has a significant chance of being able to do the required rollback before the suggested rule “kicks in”. (It also has however much additional latency is added by the BFT protocol, which is simultaneously under attack to maximize this latency.)</p>
+</li>
+</ul>
+<p>All that said, does the suggested rule help? First we have to ask whether it introduces any weaknesses.</p>
+<ul>
+<li>One potential issue is that the rule cuts both ways: if an adversarial rollback of more than <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> blocks <em>does</em> occur, then the adversary can make a proposal that will “lock in” its success. But it can be argued that this is intended behaviour anyway: the adversary has a confirmed chain, and is entitled to propose to finalize it.</li>
+<li>What happens if <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> is broken?
+<ul>
+<li><strong>Edit: the answer below refers to Crosslink before the Increasing Score rule was added.</strong></li>
+<li>This could definitely be a problem. If either an adversary has a two‑thirds supermajority of validators, <em>or</em> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> is completely broken (e.g. by a bug in the implementation of the validation signature scheme), then they can add a final block to the bft-chain with a <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">headers_bc</span></span></span></span></span> field that does not satisfy the honest voting condition. They do not need this to be bft-valid (although greater vulnerability to bugs in the implementation of bft-validity are also an issue). Then they can use the proposed rule to prevent an arbitrary bc-valid-chain from being extended (e.g. by mining <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> blocks from <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> at the genesis difficulty, then getting that chain into a final bft‑block). <strong>[Edit: they would no longer be able to do this using a chain mined at a much lower difficulty, because of the Increasing Score rule.]</strong>
+<ul>
+<li>In Crosslink, this is very carefully avoided. The only similar rule that depends on <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.1em;"></span><span class="mclose">)</span></span></span></span> and that could potentially affect liveness of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> is the <a href="./construction.html#%CE%A0bc%E2%80%91block-validity"><strong>Finality depth</strong></a> rule. But that rule <em>always</em> allows the alternative of producing a Stalled Mode block on the current bc‑best‑chain — and honest block producers will do so. Therefore, the effect of trying to exploit even a catastrophic break in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> in order to cause a rollback of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>, as long as <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has not also been broken, is to go into Stalled Mode.</li>
+<li>This does not mean that a break of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> is not a problem for Crosslink. In particular, an adversary that can violate safety of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> can violate safety of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">ba</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> (and of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> if there is also <span style="white-space: nowrap">a <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>‑block rollback</span> in some node’s bc‑best‑chain).</li>
+<li>The difference is that a safety violation of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> can be directly observed by nodes <em>without any chance of false positives</em>, which is not necessarily the case for all possible attacks against <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>. (The attack described above does <em>not</em> violate safety of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>; it just adds a final bft‑block with a suspiciously long <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> rollback. It could alternatively have added a block with a less suspiciously long rollback, say exactly <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> blocks. That is, in pursuit of preventing an attack against <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>, we have enabled attacks against <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> to achieve the same effect — precisely what Crosslink is designed to prevent.)</li>
+<li>This raises an interesting idea: if any node sees a rollback in the chain of final bft‑blocks, it could provide objective evidence of that rollback in the form of a “bft-final-vee”: two final bft‑blocks with the same parent. Similarly, if any node sees more than one third of stake vote for conflicting blocks in a given epoch, then the assumption bounding the adversary’s stake must have been violated. This evidence can be posted in a transaction to the bc-chain. In that case, any node that is synced to the bc-chain can see that the bft-chain suffered a violation of safety or of a safety assumption, without needing to have seen that violation itself. This can be generalized to allow other proofs of flaws in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>. Optionally, a bc-chain that has posted such a proof could be latched into Stalled Mode until manual intervention can occur. (Obviously we need to make sure that this cannot be abused for denial-of-service.)</li>
+<li>This is now described in <a href="./potential-changes.html#recommended-recording-more-info-about-the-bftchain-in-bcblocks">Potential changes to Crosslink</a>.</li>
+</ul>
+</li>
+</ul>
+</li>
+</ul>
+<p><strong>Okay, but is it a good idea to make that change to the fork-choice rule anyway?</strong></p>
+<p>Probably not. I don’t know how to repair the safety and liveness arguments.</p>
+<p>The change was that the bc‑best‑chain for a node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> would be required to extend <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span> where <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> is the last final bft‑block in node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>’s view.</p>
+<p>From the point of view of any modular analysis that treats <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathsf mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> as potentially subverted, we cannot say anything useful about <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span>. It seems as though any repair would have to assume much more about the BFT protocol than is desirable.</p>
+<p>In general, changes to fork‑choice rules are tricky; it was a fork-choice rule problem that allowed the liveness attack against Casper FFG described in [<a href="https://eprint.iacr.org/2020/1091.pdf">NTT2020</a>, Appendix E].</p>
+<p><strong>What if validators who see that a long rollback occurred, refuse to vote for it?</strong></p>
+<p>Yep that is allowed. The rule is “An honest validator will only vote for a proposal <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span> if ...” (not if‑and‑only‑if). If an honest validator sees a “good” reason not to vote for a proposal, including reasons based on out‑of‑band information, they should not. The <a href="./the-arguments-for-bounded-availability-and-finality-overrides.html#complementarity">Complementarity argument</a> made in <em>The Argument for Bounded Availability and Finality Overrides</em> actually depends on this. Obviously, it may affect BFT liveness (and that’s okay).</p>
+<p>The only reason why we don’t make this part of the voting condition is that it’s a stateful rule. A new validator could come along and wouldn’t have the state needed to enforce it. Perhaps that could be fixed.</p>
+
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+                <ol class="chapter"><li class="chapter-item expanded "><a href="../../introduction.html"><strong aria-hidden="true">1.</strong> Introduction</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../introduction/a-path-to-pos-zcash.html"><strong aria-hidden="true">1.1.</strong> A Path to Proof-of-Stake Zcash</a></li><li class="chapter-item expanded "><a href="../../introduction/trailing-finality-layer-in-a-nutshell.html"><strong aria-hidden="true">1.2.</strong> Trailing Finality Layer in a Nutshell</a></li><li class="chapter-item expanded "><a href="../../introduction/status-and-next-steps.html"><strong aria-hidden="true">1.3.</strong> Status and Next Steps</a></li><li class="chapter-item expanded "><a href="../../introduction/motivating-finality.html"><strong aria-hidden="true">1.4.</strong> Motivating Finality</a></li><li class="chapter-item expanded "><a href="../../introduction/get-involved.html"><strong aria-hidden="true">1.5.</strong> Get Involved</a></li></ol></li><li class="chapter-item expanded "><a href="../../terminology.html"><strong aria-hidden="true">2.</strong> Terminology</a></li><li class="chapter-item expanded "><a href="../../design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../design/design-at-a-glance.html"><strong aria-hidden="true">3.1.</strong> Design at a Glance</a></li><li class="chapter-item expanded "><a href="../../design/goals.html"><strong aria-hidden="true">3.2.</strong> Design Goals</a></li><li class="chapter-item expanded "><a href="../../design/crosslink.html"><strong aria-hidden="true">3.3.</strong> Crosslink</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../design/crosslink/the-arguments-for-bounded-availability-and-finality-overrides.html"><strong aria-hidden="true">3.3.1.</strong> The Arguments for Bounded Availability and Finality Overrides</a></li><li class="chapter-item expanded "><a href="../../design/crosslink/notes-on-snap-and-chat.html"><strong aria-hidden="true">3.3.2.</strong> Notes On Snap‑and‑Chat</a></li><li class="chapter-item expanded "><a href="../../design/crosslink/construction.html"><strong aria-hidden="true">3.3.3.</strong> The Crosslink 2 Construction</a></li><li class="chapter-item expanded "><a href="../../design/crosslink/security-analysis.html" class="active"><strong aria-hidden="true">3.3.4.</strong> Security Analysis of Crosslink 2</a></li><li class="chapter-item expanded "><a href="../../design/crosslink/questions.html"><strong aria-hidden="true">3.3.5.</strong> Questions About Crosslink</a></li><li class="chapter-item expanded "><a href="../../design/crosslink/potential-changes.html"><strong aria-hidden="true">3.3.6.</strong> Potential Changes to Crosslink</a></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">3.4.</strong> Model Code Architecture</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">3.5.</strong> Network Architecture</div></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">4.</strong> Protocol Specification</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">4.1.</strong> Abstract Protocol</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">4.1.1.</strong> Token Dynamics</div></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">4.2.</strong> Concrete Protocol</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">4.2.1.</strong> Token Dynamics</div></li></ol></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.</strong> Security</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.</strong> Abstract Analysis</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.1.</strong> Ebb-and-Flow analysis</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.2.</strong> Subprotocol Compromise Analysis</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.3.</strong> Model Code Architecture Analysis</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.4.</strong> Network Architecture Analysis</div></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.2.</strong> Concrete Analysis</div></li></ol></li><li class="chapter-item expanded "><a href="../../references.html"><strong aria-hidden="true">6.</strong> References</a></li><li class="chapter-item expanded "><a href="../../version-history.html"><strong aria-hidden="true">7.</strong> Version History</a></li></ol>
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+<h1 id="security-analysis-of-crosslink-2"><a class="header" href="#security-analysis-of-crosslink-2">Security Analysis of Crosslink 2</a></h1>
+<p>This document analyzes the security of Crosslink 2 in terms of liveness and safety. It assumes that you have read <a href="./construction.html">The Crosslink 2 Construction</a> which defines the protocol.</p>
+<h2 id="liveness-argument"><a class="header" href="#liveness-argument">Liveness argument</a></h2>
+<p>First note that Crosslink 2 <em>intentionally</em> sacrifices <strong>availability</strong> if there is a long finalization stall.</p>
+<div id="admonition-info" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Info</p>
+<p><a class="admonition-anchor-link" href="#admonition-info"></a></p>
+</div>
+<div>
+<p>This is technically independent of the other changes; you can omit the <a href="./construction.html#finality-depth-rule"><strong>Finality depth rule</strong></a> and the protocol would still have security advantages over Snap‑and‑Chat, as well as being much simpler and solving its “spending from finalized outputs” issue. In that case the incentives to “pull” the finalization point forward to include new final bft‑blocks would be weaker, but honest bc‑block‑producers would still do it.</p>
+</div>
+</div>
+<p>It would still be a bug if there were any situation in which <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> failed to be <strong>live</strong>, though, because that would allow <a href="./the-arguments-for-bounded-availability-and-finality-overrides.html#tail-thrashing-attacks">tail‑thrashing attacks</a>.</p>
+<p>Crosslink 2 involves a bidirectional dependence between <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathsf mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathsf mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>. The Ebb‑and‑Flow paper <a href="https://eprint.iacr.org/2020/1091.pdf">[NTT2020]</a> argues in Appendix E (&quot;Bouncing Attack on Casper FFG&quot;) that it can be more difficult to reason about liveness given a bidirectional dependence between protocols:</p>
+<blockquote>
+<p>To ensure consistency among the two tiers [of Casper FFG], the fork choice rule of the blockchain is modified to always respect ‘the justified checkpoint of the greatest <span style="white-space: nowrap">height [*]’ <a href="https://arxiv.org/abs/1710.09437">[22]</a>.</span> There is thus a bidirectional interaction between the block proposal and the finalization layer: blocks proposed by the blockchain are input to finalization, while justified checkpoints constrain future block proposals. This bidirectional interaction is intricate to reason about and a gateway for liveness attacks.</p>
+</blockquote>
+<div id="admonition-info-1" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Info</p>
+<p><a class="admonition-anchor-link" href="#admonition-info-1"></a></p>
+</div>
+<div>
+<p>[*] The quotation changes this to “[depth]”, but our terminology is consistent with Ethereum here and not with [NTT2020]’s idiosyncratic use of “depth” to mean “block height”.</p>
+</div>
+</div>
+<p>The argument is correct as far as it goes. The main reason why this does not present any great difficulty to proving liveness of Crosslink, is due to a fundamental difference from Casper FFG: in Crosslink 2 the fork‑choice rule of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> is not modified.</p>
+<p>Let <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathcal" style="margin-right:0.075em;">S</span></span></span></span> be the subset of bc‑blocks <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">is-coinbase-only-block</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mord text"><span class="mord"> and </span></span><span class="mord text"><span class="mord textsf">is-stalled-block</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)}</span></span></span></span>. Assume that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathcal" style="margin-right:0.075em;">S</span></span></span></span> is such that a bc‑block‑producer can always produce a block in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathcal" style="margin-right:0.075em;">S</span></span></span></span>.</p>
+<p>In that case it is straightforward to convince ourselves that the additional bc‑block‑validity rules are never an obstacle to producing a new bc‑block in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathcal" style="margin-right:0.075em;">S</span></span></span></span>:</p>
+<ul>
+<li>The changes to the definition of contextual validity do not interfere with liveness, since they do not affect coinbase transactions, and therefore do not affect blocks in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathcal" style="margin-right:0.075em;">S</span></span></span></span>. That is, the bc‑block‑producer can always just omit non‑coinbase transactions that are contextually invalid.</li>
+<li>The <a href="./construction.html#genesis-bc-block-rule"><strong>Genesis bc‑block rule</strong></a> doesn’t apply to new bc‑blocks.</li>
+<li>The <a href="./construction.html#valid-context-rule"><strong>Valid context rule</strong></a>, <a href="./construction.html#extension-rule"><strong>Extension rule</strong></a>, and <a href="./construction.html#last-final-snapshot-rule"><strong>Last Final Snapshot rule</strong></a> are always satisfiable by referencing the same context bft‑block as the parent bc‑block.</li>
+<li>The <a href="./construction.html#finality-depth-rule"><strong>Finality depth rule</strong></a> always allows the option of producing a stalled block, and therefore does not affect blocks in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathcal" style="margin-right:0.075em;">S</span></span></span></span>.</li>
+</ul>
+<p>The instructions to an honest bc‑block‑producer allow it to produce a block in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathcal" style="margin-right:0.075em;">S</span></span></span></span>. Therefore, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> remains live under the same conditions as <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>.</p>
+<p>The additional bft‑proposal‑validity, bft‑block‑validity, bft‑finality, and honest voting rules are also not an obstacle to making, voting for, or finalizing bft‑proposals:</p>
+<ul>
+<li>Because <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> is live, there will always be some point in time at which a fresh valid bc‑header chain that satisfies both the <a href="./construction.html#linearity-rule"><strong>Linearity rule</strong></a> and the <a href="./construction.html#tail-confirmation-rule"><strong>Tail confirmation rule</strong></a> exists for use in the <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord">.</span><span class="mord"><span class="mord mathsf">headers_bc</span></span></span></span></span> field.</li>
+<li>If no fresh valid bc‑header chain is available, the <a href="./construction.html#linearity-rule"><strong>Linearity rule</strong></a> and <a href="./construction.html#tail-confirmation-rule"><strong>Tail confirmation rule</strong></a> allow an honest bft‑proposer to choose <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">headers_bc</span></span></span></span></span> to be the same as in the previous bft‑block. So, if liveness of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> depends on an honest proposer always being able to make a proposal (as it does in adapted‑Streamlet for example), then this requirement will not be violated.</li>
+<li>The changes to voting are only requiring a vote to be for a proposal that could have been honestly proposed.</li>
+<li>The bft‑finality rules are unchanged from origbft‑finality.</li>
+</ul>
+<p>Therefore, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> remains live under the same conditions as <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>.</p>
+<p>The only other possibility for a liveness issue in Crosslink 2 would be if the change to the constructions of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> or <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ba</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> could cause either of them to stall, even when <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> are both still live.</p>
+<p>However, liveness of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> and the <a href="./construction.html#linearity-rule"><strong>Linearity rule</strong></a> together imply that at each point in time, provided there are sufficient honest bft‑proposers/validators, eventually a new bft‑block with a higher-scoring snapshot will become final in the context of the longest bft‑valid‑chain. <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="enclosing" style="background-color:yellow;color:black;"><span class="mord text"><span class="mord textsf">TODO:</span></span></span></span></span></span> make that more precise.</p>
+<p>Because of the <a href="./construction.html#extension-rule"><strong>Extension rule</strong></a>, this new bft‑block must be a descendent of the previous final bft‑block in the context visible to bc‑block‑producers. Therefore, the new finalized chain will extend the old finalized chain.</p>
+<p>Finally, we need to show that Stalled Mode is only triggered when it should be; that is, when the assumptions needed for liveness of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> are violated. Informally, that is the case because, as long as there are sufficient honest bc‑block‑producers <em>and</em> sufficient honest bft‑proposers/validators, the finalization point implied by the <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">context_bft</span></span></span></span></span> field at the tip of the bc‑best chain in any node’s view will advance fast enough for the finalization gap bound <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span> not to be hit. This depends on the value of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span> relative to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>, the network delay, the hash rate of honest bc‑block‑producers, the number of honest bft‑proposers and the proportion of voting units they hold, and other details of the BFT protocol. <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="enclosing" style="background-color:yellow;color:black;"><span class="mord text"><span class="mord textsf">TODO:</span></span></span></span></span></span> more detailed argument needed, especially for the dependence on <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span>.</p>
+<h2 id="safety-argument"><a class="header" href="#safety-argument">Safety argument</a></h2>
+<p><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="enclosing" style="background-color:yellow;color:black;"><span class="mord text"><span class="mord textsf">TODO:</span></span></span></span></span></span> Not updated for Crosslink 2 below this point.</p>
+<h3 id="discussion"><a class="header" href="#discussion">Discussion</a></h3>
+<p>Recall the definition of <a href="./construction.html#assured-finality"><strong>Assured Finality</strong></a>.</p>
+<p><span id="assured-finality"></span></p>
+<div id="admonition-definition-assured-finality" class="admonition admonish-success">
+<div class="admonition-title">
+<p>Definition: Assured Finality</p>
+<p><a class="admonition-anchor-link" href="#admonition-definition-assured-finality"></a></p>
+</div>
+<div>
+<p>An execution of Crosslink 2 has <strong>Assured Finality</strong> iff <span style="white-space: nowrap">for all times <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span></span> and <span style="white-space: nowrap">all nodes <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span></span> (potentially the same) such that <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span></span> and <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span>,</span> we have <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:-0.5em;"></span></span><span class="base"><span class="strut" style="height:0.786em;vertical-align:-0.15em;"></span><span class="mrel"><span class="mrel">⪰</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0892em;vertical-align:-0.3948em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3948em;"><span></span></span></span></span></span></span></span></span></span>.</span></p>
+</div>
+</div>
+<p>First we prove that <strong>Assured Finality</strong> is implied by <a href="./construction.html#prefix-agreement"><strong>Prefix Agreement</strong></a> of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</p>
+<p><span id="prefix-agreement"></span></p>
+<div id="admonition-definition-prefix-agreement" class="admonition admonish-success">
+<div class="admonition-title">
+<p>Definition: Prefix Agreement</p>
+<p><a class="admonition-anchor-link" href="#admonition-definition-prefix-agreement"></a></p>
+</div>
+<div>
+<p>An execution of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has <strong>Prefix Agreement</strong> at confirmation <span style="white-space: nowrap">depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>,</span> iff <span style="white-space: nowrap">for all times <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span></span> and <span style="white-space: nowrap">all nodes <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span></span> (potentially the same) such that <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span></span> and <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span>,</span> we have <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.161em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mspace mtight" style="margin-right:-0.1952em;"></span><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:-0.5em;"></span></span><span class="base"><span class="strut" style="height:0.786em;vertical-align:-0.15em;"></span><span class="mrel"><span class="mrel">⪰</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1331em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mspace mtight" style="margin-right:-0.1952em;"></span><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span>.</span></p>
+</div>
+</div>
+<p><span id="prefix-agreement-implies-assured-finality"></span></p>
+<div id="admonition-safety-theorem-prefix-agreement-of-Πbc-implies-assured-finality" class="admonition admonish-success">
+<div class="admonition-title">
+<p>Safety Theorem: Prefix Agreement of Π<sub>bc</sub> implies Assured Finality</p>
+<p><a class="admonition-anchor-link" href="#admonition-safety-theorem-prefix-agreement-of-Πbc-implies-assured-finality"></a></p>
+</div>
+<div>
+<p>In an execution of Crosslink 2 for which the <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> subprotocol has <strong>Prefix Agreement</strong> at confirmation <span style="white-space: nowrap">depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>,</span> that execution has <strong>Assured Finality</strong>.</p>
+<p>Proof: Suppose that we have times <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span></span> and <span style="white-space: nowrap">nodes <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span></span> (potentially the same) such that <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span></span> and <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span>.</span> Then, by the <a href="./construction.html#local-fin-depth"><strong>Local fin-depth lemma</strong></a> applied to each of <span style="white-space: nowrap">node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span></span> and <span style="white-space: nowrap">node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span>,</span> there exist <span style="white-space: nowrap">times <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span></span></span></span></span> at which <span style="white-space: nowrap">node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span></span> is honest and <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span> at which <span style="white-space: nowrap">node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span></span> is honest, such that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0331em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span> and <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0892em;vertical-align:-0.3948em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3948em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.4099em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0268em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8278em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span>.</span> By <a href="./construction.html#prefix-agreement"><strong>Prefix Agreement at confirmation depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">σ</span></span></span></span></span></span></strong></a>, we have <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0331em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:-0.5em;"></span></span><span class="base"><span class="strut" style="height:0.786em;vertical-align:-0.15em;"></span><span class="mrel"><span class="mrel">⪰</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.4099em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0268em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8278em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span>.</span> Wlog due to symmetry, suppose <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0331em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.4099em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0268em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8278em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span>.</span> Then <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.4099em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0268em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8278em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span></span> (by transitivity <span style="white-space: nowrap">of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.786em;vertical-align:-0.15em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>)</span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0892em;vertical-align:-0.3948em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3948em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.4099em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0268em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8278em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span> (as above),</span> so <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:-0.5em;"></span></span><span class="base"><span class="strut" style="height:0.786em;vertical-align:-0.15em;"></span><span class="mrel"><span class="mrel">⪰</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0892em;vertical-align:-0.3948em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3948em;"><span></span></span></span></span></span></span></span></span></span></span> by the <a href="./construction.html#linear-prefix-lemma"><strong>Linear prefix lemma</strong></a>.</p>
+</div>
+</div>
+<p>Then we prove that <strong>Assured Finality</strong> is also implied by <a href="./construction.html#final-agreement"><strong>Final Agreement</strong></a> of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</p>
+<p><span id="final-agreement"></span></p>
+<div id="admonition-definition-final-agreement" class="admonition admonish-success">
+<div class="admonition-title">
+<p>Definition: Final Agreement</p>
+<p><a class="admonition-anchor-link" href="#admonition-definition-final-agreement"></a></p>
+</div>
+<div>
+<p>An execution of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has <strong>Final Agreement</strong> iff for all <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑valid blocks <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span></span> in honest view at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span></span> and <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> in honest view</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span>,</span> we have <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∗</span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:-0.5em;"></span></span><span class="base"><span class="strut" style="height:0.786em;vertical-align:-0.15em;"></span><span class="mrel"><span class="mrel">⪰</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord">∗</span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>.</span></p>
+</div>
+</div>
+<p><span id="final-agreement-implies-assured-finality"></span></p>
+<div id="admonition-safety-theorem-final-agreement-of-Πbft-implies-assured-finality" class="admonition admonish-success">
+<div class="admonition-title">
+<p>Safety Theorem: Final Agreement of Π<sub>bft</sub> implies Assured Finality</p>
+<p><a class="admonition-anchor-link" href="#admonition-safety-theorem-final-agreement-of-Πbft-implies-assured-finality"></a></p>
+</div>
+<div>
+<p>In an execution of Crosslink 2 for which the <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> subprotocol has <strong>Final Agreement</strong>, that execution has <strong>Assured Finality</strong>.</p>
+<p>Proof: Suppose that we have times <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span></span> and <span style="white-space: nowrap">nodes <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span></span> (potentially the same) such that <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span></span> and <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span>.</span> Then, by the <a href="./construction.html#local-fin-depth"><strong>Local fin-depth lemma</strong></a> applied to each of <span style="white-space: nowrap">node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span></span> and <span style="white-space: nowrap">node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span>,</span> there exist <span style="white-space: nowrap">times <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span></span></span></span></span> at which <span style="white-space: nowrap">node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span></span> is honest and <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span> at which <span style="white-space: nowrap">node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span></span> is honest, such that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0331em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span> and <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0892em;vertical-align:-0.3948em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3948em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.4099em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0268em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8278em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span>.</span> By <a href="./construction.html#prefix-agreement"><strong>Prefix Agreement at confirmation depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">σ</span></span></span></span></span></span></strong></a>, we have <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0331em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:-0.5em;"></span></span><span class="base"><span class="strut" style="height:0.786em;vertical-align:-0.15em;"></span><span class="mrel"><span class="mrel">⪰</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.4099em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0268em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8278em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span>.</span> Wlog due to symmetry, suppose <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0331em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.4099em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0268em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8278em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span>.</span> Then <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.4099em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0268em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8278em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span></span> (by transitivity <span style="white-space: nowrap">of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.786em;vertical-align:-0.15em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>)</span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0892em;vertical-align:-0.3948em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3948em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.4099em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0268em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8278em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span> (as above),</span> so <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:-0.5em;"></span></span><span class="base"><span class="strut" style="height:0.786em;vertical-align:-0.15em;"></span><span class="mrel"><span class="mrel">⪰</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0892em;vertical-align:-0.3948em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3948em;"><span></span></span></span></span></span></span></span></span></span></span> by the <a href="./construction.html#linear-prefix-lemma"><strong>Linear prefix lemma</strong></a>.</p>
+</div>
+</div>
+<p><span id="prefix-consistency"></span></p>
+<div id="admonition-safety-theorem-prefix-consistency-of-Πbc-implies-prefix-consistency-of-ba" class="admonition admonish-success">
+<div class="admonition-title">
+<p>Safety Theorem: Prefix Consistency of Π<sub>bc</sub> implies Prefix Consistency of ba</p>
+<p><a class="admonition-anchor-link" href="#admonition-safety-theorem-prefix-consistency-of-Πbc-implies-prefix-consistency-of-ba"></a></p>
+</div>
+<div>
+<p>By the <a href="./construction.html#local-ba-depth"><strong>Local ba-depth lemma</strong></a>, we have:</p>
+<blockquote>
+<p>In any execution of Crosslink 2, for any <span style="white-space: nowrap">confirmation depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8304em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span></span> and any <span style="white-space: nowrap">node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span></span> that is honest at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>,</span> there exists a <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span></span> <span style="white-space: nowrap">such that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0797em;vertical-align:-0.2861em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ba</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0836em;vertical-align:-0.3013em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em;"><span style="top:-2.3987em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.1809em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3013em;"><span></span></span></span></span></span></span></span></span></span>.</span></p>
+</blockquote>
+<p>Renaming <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span> to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span></span></span></span> in the definition of <a href="./construction.html#prefix-consistency"><strong>Prefix Consistency</strong></a> gives:</p>
+<blockquote>
+<p>An execution of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has <strong>Prefix Consistency</strong> at confirmation depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span></span></span></span>, iff <span style="white-space: nowrap">for all times <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span></span> and <span style="white-space: nowrap">all nodes <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span></span> (potentially the same) such that <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span></span></span></span></span> and <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span>,</span> we have that <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0836em;vertical-align:-0.3013em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em;"><span style="top:-2.3987em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.1809em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3013em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1318em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span>.</span></p>
+</blockquote>
+<p>Since any node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> that is honest at time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span> is also honest at time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>, and by transitivity of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.786em;vertical-align:-0.15em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>, we therefore have:</p>
+<blockquote>
+<p>In any execution of Crosslink 2 that has <strong>Prefix Consistency</strong> at confirmation depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span></span></span></span>, <span style="white-space: nowrap">for all times <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span></span> and <span style="white-space: nowrap">all nodes <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span></span> (potentially the same) such that <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span></span> and <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span>,</span> we have that <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0797em;vertical-align:-0.2861em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ba</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1318em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span>.</span></p>
+</blockquote>
+</div>
+</div>
+----
+<p>The <strong>Extension rule</strong> ensures that, informally, if a given node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>’s view of its bc‑best‑chain at a depth of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> blocks does not roll back, then neither does its view of the bft‑final block referenced by its bc‑best‑chain, and therefore neither does its view of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span>.</p>
+<p>This does <em>not</em> by itself imply that all nodes are seeing the “same” confirmed bc‑best‑chain (up to propagation timing), or the same <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span>. If the network is partitioned <em>and</em> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> is subverted, it could be that the nodes on each side of the partition follow a different fork, and the adversary arranges for each node’s view of the last final bft‑block to be consistent with the fork it is on. It can potentially do this if it has more than one third of validators, because if the validators are partitioned in the same way as other nodes, it can vote with an additional one third of them on each side of the fork.</p>
+<p>This is, if you think about it, unavoidable. <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> doesn’t include the mechanisms needed to maintain finality under partition; it takes a different position on the CAP trilemma. In order to maintain finality under partition, we need <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> not to be subverted (and to actually work!)</p>
+<p>So what is the strongest security property we can realistically get? It is stronger than what Snap‑and‑Chat provides. Snap‑and‑Chat is unsafe <em>even without a partition</em> if <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> is subverted. Ideally we would have a protocol with safety that is <em>only</em> limited by attacks “like” the unavoidable attack described above — which also applies to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> used on its own.</p>
+<h3 id="proof-of-safety-for-logfin"><a class="header" href="#proof-of-safety-for-logfin">Proof of safety for LOG<sub>fin</sub></a></h3>
+<p>In order to capture the intuition that it is hard in practice to cause a consistent partition of the kind described in the previous section, we will need to assume that the <a href="./construction.html#prefix-agreement"><strong>Prefix Agreement</strong></a> safety property holds for the relevant executions of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>. The structural and consensus modifications to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> relative to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> seem unlikely to have any significant effect on this property, given that we proved above that they do not affect liveness. ==TODO: that is a handwave; we should be able to do better, as we do for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> below.== So, to the extent that it is reasonable to assume that <a href="./construction.html#prefix-agreement"><strong>Prefix Agreement</strong></a> holds for executions of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> under some conditions, it should also be reasonable to assume it holds for executions of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> under the same conditions.</p>
+<p>Recall that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1em;vertical-align:-0.35em;"></span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">context_bft</span></span><span class="mclose">)</span></span></span></span>.</p>
+<p><span id="prefix-lemma"></span></p>
+<div id="admonition-prefix-lemma" class="admonition admonish-success">
+<div class="admonition-title">
+<p>Prefix Lemma</p>
+<p><a class="admonition-anchor-link" href="#admonition-prefix-lemma"></a></p>
+</div>
+<div>
+<p>If <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> are bc‑valid blocks with <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>, then <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>.</p>
+<p>Proof: Using the <strong>Extension rule</strong>, by induction on the distance between <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</p>
+</div>
+</div>
+<p>Using the <strong>Prefix Lemma</strong> once for each direction, we can transfer the <a href="./construction.html#prefix-agreement"><strong>Prefix Agreement</strong></a> property to the referenced bft‑blocks:</p>
+<p><span id="prefix-agreement-lemma"></span></p>
+<div id="admonition-prefix-agreement-lemma" class="admonition admonish-success">
+<div class="admonition-title">
+<p>Prefix Agreement Lemma</p>
+<p><a class="admonition-anchor-link" href="#admonition-prefix-agreement-lemma"></a></p>
+</div>
+<div>
+<p>In an execution of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> that has <a href="./construction.html#prefix-agreement"><strong>Prefix Agreement</strong></a> at confirmation <span style="white-space: nowrap">depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>,</span> <span style="white-space: nowrap">for all times <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span></span> and <span style="white-space: nowrap">all nodes <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span></span> (potentially the same) such that <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span></span> and <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span>,</span> we have <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.161em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:-0.5em;"></span></span><span class="base"><span class="strut" style="height:0.786em;vertical-align:-0.15em;"></span><span class="mrel"><span class="mrel">⪰</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1331em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>.</span></p>
+</div>
+</div>
+<p>Let <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">chain-txns</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">ch</span></span><span class="mclose">)</span></span></span></span> be the sequence of transactions in the given <span style="white-space: nowrap">chain <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathsf">ch</span></span></span></span></span>,</span> starting from genesis.</p>
+<p>Recall that <span class="katex-display fleqn"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:9.722em;vertical-align:-4.611em;"></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:5.111em;"><span style="top:-7.271em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord textsf">san-ctx</span></span></span></span><span style="top:-6.071em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord textsf">san-ctx</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mclose">)</span></span></span><span style="top:-4.871em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord textsf">fin</span></span></span></span><span style="top:-3.671em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">fin</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span></span><span style="top:-2.471em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord textsf">fin-ctx</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span></span><span style="top:-1.271em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord textsf">ba-ctx</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">μ</span><span class="mclose">)</span></span></span><span style="top:-0.0331em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span><span style="top:1.2279em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">ba</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">μ</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:4.611em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:5.111em;"><span style="top:-7.271em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mrel"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.1293em;"><span style="top:-2.6293em;"><span class="pstrut" style="height:2.5em;"></span><span class="mord"><span class="mord sizing reset-size6 size1">⦂</span></span></span></span></span></span></span></span></span><span style="top:-6.071em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mrel">:=</span></span></span><span style="top:-4.871em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mrel"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.1293em;"><span style="top:-2.6293em;"><span class="pstrut" style="height:2.5em;"></span><span class="mord"><span class="mord sizing reset-size6 size1">⦂</span></span></span></span></span></span></span></span></span><span style="top:-3.671em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mrel">:=</span></span></span><span style="top:-2.471em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mrel">:=</span></span></span><span style="top:-1.271em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mrel">:=</span></span></span><span style="top:-0.0331em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mrel">:=</span></span></span><span style="top:1.2279em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mrel">:=</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:4.611em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:5.111em;"><span style="top:-7.271em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">[</span><span class="mord text"><span class="mord textsf">bc-chain</span></span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord text"><span class="mord textsf">bc-context</span></span></span></span><span style="top:-6.071em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">sanitize</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">concat</span></span><span class="mopen">([</span><span class="mord text"><span class="mord textsf">chain-txns</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mclose">)</span><span class="mord text"><span class="mord"> for </span></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord text"><span class="mord"> in </span></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mclose">]))</span></span></span><span style="top:-4.871em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord textsf">bc-block</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">[</span><span class="mord text"><span class="mord textsf">bc-chain</span></span><span class="mclose">]</span></span></span><span style="top:-3.671em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">[</span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mord text"><span class="mord"> for </span></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)]</span></span></span><span style="top:-2.471em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord textsf">san-ctx</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathrm">fin</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">))</span></span></span><span style="top:-1.271em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord textsf">san-ctx</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathrm">fin</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">∣∣</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em;"><span style="top:-2.3987em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.1809em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3013em;"><span></span></span></span></span></span></span><span class="mclose">])</span></span></span><span style="top:-0.0331em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord textsf">context-txns</span></span><span class="mopen">(</span><span class="mord text"><span class="mord textsf">fin-ctx</span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose">))</span></span></span><span style="top:1.2279em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord textsf">context-txns</span></span><span class="mopen">(</span><span class="mord text"><span class="mord textsf">ba-ctx</span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">μ</span><span class="mclose">))</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:4.611em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span></span></span></span></span></p>
+<p>Because <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathrm">fin</span></span></span></span></span> takes the form <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathrm">fin</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span><span class="mord text"><span class="mord"> for </span></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)]</span></span></span></span>,</span> we have that <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟹</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathrm">fin</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathrm">fin</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>.</span> (This would be true for any well‑typed <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span></span></span></span>.)</span></p>
+<p>By this observation and the <strong>Prefix Agreement Lemma</strong>, we also have that, in an execution of Crosslink 2 where <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has the <a href="./construction.html#prefix-agreement"><strong>Prefix Agreement</strong></a> safety property at confirmation depth <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>,</span> <span style="white-space: nowrap">for all times <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span></span> and <span style="white-space: nowrap">all nodes <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span></span> (potentially the same) such that <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span></span> and <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span>,</span> <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1279em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathrm">fin</span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:-0.5em;"></span></span><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">⪰</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1331em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord mathrm">fin</span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>.</span></p>
+<p>Because <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6794em;"></span><span class="mord"><span class="mord mathsf">sanitize</span></span></span></span></span> only considers previous state, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5714em;"></span><span class="mord text"><span class="mord textsf">context-txns</span></span></span></span></span> ∘ <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5714em;"></span><span class="mord text"><span class="mord textsf">san-ctx</span></span></span></span></span> must be a prefix-preserving map; that is, <span style="white-space: nowrap">if <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> then <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">context-txns</span></span><span class="mopen">(</span><span class="mord text"><span class="mord textsf">san-ctx</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">))</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">context-txns</span></span><span class="mopen">(</span><span class="mord text"><span class="mord textsf">san-ctx</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">))</span></span></span></span>.</span> Therefore:</p>
+<div id="admonition-theorem-logfin-safety-from-prefix-agreement-of-Πbc" class="admonition admonish-success">
+<div class="admonition-title">
+<p>Theorem: LOG<sub>fin</sub> Safety (from Prefix Agreement of Π<sub>bc</sub>)</p>
+<p><a class="admonition-anchor-link" href="#admonition-theorem-logfin-safety-from-prefix-agreement-of-Πbc"></a></p>
+</div>
+<div>
+<p>In an execution of Crosslink 2 where <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has <a href="./construction.html#prefix-agreement"><strong>Prefix Agreement</strong></a> at confirmation depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>, <span style="white-space: nowrap">for all times <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span></span> and <span style="white-space: nowrap">all nodes <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span></span> (potentially the same) such that <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span></span> and <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span>,</span> <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:-0.5em;"></span></span><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">⪰</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1318em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span></span></span></span>.</span></p>
+</div>
+</div>
+<p>Notice that this does not depend on any safety property of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>, and is an elementary proof. ([<a href="https://eprint.iacr.org/2020/1091.pdf">NTT2020</a>, Theorem 2] is a much more complicated proof that takes nearly 3 pages, not counting the reliance on results from <a href="https://eprint.iacr.org/2016/918.pdf">[PS2017]</a>.)</p>
+<p><em>In addition,</em> just as in Snap‑and‑Chat, safety of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> can be inferred from safety of <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>,</span> which follows from safety of <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>.</span> We prove this based on the <a href="./construction.html#final-agreement"><strong>Final Agreement</strong></a> property for executions of <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>:</span></p>
+<div id="admonition-definition-final-agreement-1" class="admonition admonish-success">
+<div class="admonition-title">
+<p>Definition: Final Agreement</p>
+<p><a class="admonition-anchor-link" href="#admonition-definition-final-agreement-1"></a></p>
+</div>
+<div>
+<p>An execution of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> has the <a href="./construction.html#final-agreement"><strong>Final Agreement</strong></a> safety property iff for all origbft‑valid <span style="white-space: nowrap">blocks <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span></span> <a href="#in-honest-view">in honest view</a> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> in honest view at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span>,</span> we have <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">origbft-last-final</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:-0.5em;"></span></span><span class="base"><span class="strut" style="height:0.9221em;vertical-align:-0.2861em;"></span><span class="mrel"><span class="mrel">⪰</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">origbft-last-final</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span></span></span></span>.</span></p>
+</div>
+</div>
+<p>The changes in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> relative to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> only add structural components and tighten bft‑block‑validity and bft‑proposal‑validity rules. So for any legal execution of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> there is a corresponding legal execution of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>, with the structural additions erased and with the same nodes honest at any given time. A safety property, by definition, only asserts that executions not satisfying the property do not occur. Safety properties of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> necessarily do not refer to the added structural components in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>. Therefore, for any safety property of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>, including <a href="./construction.html#final-agreement"><strong>Final Agreement</strong></a>, the corresponding safety property holds for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</p>
+<p>By the definition of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathrm">fin</span></span></span></span></span> as above, in an execution of Crosslink 2 where <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has <a href="./construction.html#final-agreement"><strong>Final Agreement</strong></a>, <span style="white-space: nowrap">for all times <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span></span> and <span style="white-space: nowrap">all nodes <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span></span> (potentially the same) such that <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span></span> and <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span>,</span> <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1279em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathrm">fin</span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:-0.5em;"></span></span><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">⪰</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1331em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord mathrm">fin</span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span></span></span></span>.</span> Therefore, by an argument similar to the one above using the fact that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5714em;"></span><span class="mord text"><span class="mord textsf">context-txns</span></span></span></span></span> ∘ <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5714em;"></span><span class="mord text"><span class="mord textsf">san-ctx</span></span></span></span></span> is a prefix-preserving map:</p>
+<div id="admonition-theorem-logfin-safety-from-final-agreement-of-Πbft-or-Πorigbft" class="admonition admonish-success">
+<div class="admonition-title">
+<p>Theorem: LOG<sub>fin</sub> Safety (from Final Agreement of Π<sub>bft</sub> or Π<sub>origbft</sub>)</p>
+<p><a class="admonition-anchor-link" href="#admonition-theorem-logfin-safety-from-final-agreement-of-Πbft-or-Πorigbft"></a></p>
+</div>
+<div>
+<p>In an execution of Crosslink 2 where <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has <a href="./construction.html#final-agreement"><strong>Final Agreement</strong></a>, <span style="white-space: nowrap">for all times <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span></span> and <span style="white-space: nowrap">all nodes <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span></span> (potentially the same) such that <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span></span> and <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span>,</span> <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:-0.5em;"></span></span><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">⪰</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1318em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span></span></span></span>.</span></p>
+</div>
+</div>
+<h3 id="proof-of-safety-for-logba"><a class="header" href="#proof-of-safety-for-logba">Proof of safety for LOG<sub>ba</sub></a></h3>
+<p>From the two <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> Safety theorems and the <a href="./construction.html#admonition-theorem-ledger-prefix-property"><strong>Ledger prefix property</strong></a>, we immediately have:</p>
+<div id="admonition-theorem-logba-does-not-roll-back-past-the-agreed-logfin" class="admonition admonish-success">
+<div class="admonition-title">
+<p>Theorem: LOG<sub>ba</sub> does not roll back past the agreed LOG<sub>fin</sub></p>
+<p><a class="admonition-anchor-link" href="#admonition-theorem-logba-does-not-roll-back-past-the-agreed-logfin"></a></p>
+</div>
+<div>
+<p>Let <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">μ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> be an arbitrary choice of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">ba</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> confirmation depth for each node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>. Consider an execution of Crosslink 2 where either <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has <a href="./construction.html#prefix-agreement"><strong>Prefix Agreement</strong></a> at confirmation depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> or <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has <a href="./construction.html#final-agreement"><strong>Final Agreement</strong></a>.</p>
+<p>In such an execution, <span style="white-space: nowrap">for all times <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span></span> and <span style="white-space: nowrap">all nodes <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span></span> (potentially the same) such that <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span></span> and <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span>,</span> either <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1318em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1931em;vertical-align:-0.4443em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">ba</span></span><span class="mpunct mtight">,</span><span class="mord mtight"><span class="mord mathnormal mtight">μ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2819em;"><span></span></span></span></span></span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4443em;"><span></span></span></span></span></span></span></span></span></span> or <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1318em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">ba</span></span><span class="mpunct mtight">,</span><span class="mord mtight"><span class="mord mathnormal mtight">μ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span>.</p>
+<p><strong>Corollary:</strong> Under the same conditions, if wlog <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1318em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span>, then <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.3222em;vertical-align:-0.4443em;"></span><span class="mopen">{</span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">ba</span></span><span class="mpunct mtight">,</span><span class="mord mtight"><span class="mord mathnormal mtight">μ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">ba</span></span><span class="mpunct mtight">,</span><span class="mord mtight"><span class="mord mathnormal mtight">μ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2819em;"><span></span></span></span></span></span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4443em;"><span></span></span></span></span></span></span><span class="mclose">}</span></span></span></span>.</p>
+</div>
+</div>
+<p>The above property is not as strong as we would like for practical uses of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">ba</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>, because it does not say anything about rollbacks up to the finalization point, and such rollbacks may be of unbounded length. (Loosely speaking, the number of non‑Stalled Mode bc‑blocks after the consensus finalization point is bounded by <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span>,</span> but we have also not proven that so far.)</p>
+<p>As documented in the <a href="./construction.html#model-for-bft-protocols-%CE%A0origbftbft">Model for BFT protocols</a> section of <a href="./construction.html">The Crosslink 2 Construction</a>):</p>
+<blockquote>
+<p>For each epoch, there is a fixed number of voting units distributed between the players, which they use to vote for a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑proposal.</span> We say that a voting unit has been cast for a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑proposal <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span></span> at a given time in a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑execution,</span> <span style="white-space: nowrap">if and only if</span> <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span> is <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑proposal‑valid</span> and a ballot <span style="white-space: nowrap">for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span></span> authenticated by the holder of the voting unit exists at that time.</p>
+<p>Using knowledge of ballots cast for a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑proposal <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span></span> that collectively satisfy a notarization rule at a given time in a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑execution,</span> and only with such knowledge, it is possible to obtain a valid <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑notarization‑proof <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9386em;vertical-align:-0.2441em;"></span><span class="mord"><span class="mord"><span class="mord mathsf" style="margin-right:0.06944em;">proof</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2342em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">P</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span></span></span></span>.</span> The notarization rule must require at least a two‑thirds absolute supermajority of voting units <span style="white-space: nowrap">in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span>’s epoch</span> to have been cast <span style="white-space: nowrap">for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span>.</span> It may also require other conditions.</p>
+<p>A voting unit is cast non‑honestly for an epoch’s proposal iff:</p>
+<ul>
+<li>it is cast other than by the holder of the unit (due to key compromise or any flaw in the voting protocol, for example); or</li>
+<li>it is double‑cast (i.e. there are two ballots casting it for distinct proposals); or</li>
+<li>the holder of the unit following the conditions for honest voting <span style="white-space: nowrap">in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>,</span> according to its view, should not have cast that vote.</li>
+</ul>
+</blockquote>
+<p><span id="one-third-bound"></span></p>
+<div id="admonition-definition-onethird-bound-on-nonhonest-voting" class="admonition admonish-success">
+<div class="admonition-title">
+<p>Definition: One‑third bound on non‑honest voting</p>
+<p><a class="admonition-anchor-link" href="#admonition-definition-onethird-bound-on-nonhonest-voting"></a></p>
+</div>
+<div>
+<p>An execution of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has the <strong>one‑third bound on non‑honest voting</strong> property iff for every epoch, <em>strictly</em> fewer than one third of the total voting units for that epoch are ever cast non‑honestly.</p>
+</div>
+</div>
+<p><span id="theorem-on-bft-valid-blocks"></span></p>
+<div id="admonition-theorem-on-bftvalid-blocks-for-a-given-epoch-in-honest-view" class="admonition admonish-success">
+<div class="admonition-title">
+<p>Theorem: On bft‑valid blocks for a given epoch in honest view</p>
+<p><a class="admonition-anchor-link" href="#admonition-theorem-on-bftvalid-blocks-for-a-given-epoch-in-honest-view"></a></p>
+</div>
+<div>
+<p>By a well known argument often used to prove safety of BFT protocols, in an execution of Crosslink 2 where <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has the <strong>one‑third bound on non‑honest voting</strong> property (and assuming soundness of notarization proofs), any bft‑valid block for a given epoch <a href="#in-honest-view">in honest view</a> must commit to the same proposal.</p>
+<p>Proof (adapted from [<a href="https://eprint.iacr.org/2020/088.pdf">CS2020</a>, Lemma 1]): Suppose there are two bft‑proposals <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span>, both for <span style="white-space: nowrap">epoch <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">e</span></span></span></span>,</span> such that <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span> is</span> committed to by some bft‑block‑valid <span style="white-space: nowrap">block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span>,</span> and <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> is</span> committed to by some bft‑block‑valid <span style="white-space: nowrap">block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span>.</span> This implies that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> have valid notarization proofs. Let the number of voting units for <span style="white-space: nowrap">epoch <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">e</span></span></span></span></span> <span style="white-space: nowrap">be <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</span> Assuming soundness of the notarization proofs, it must be that at least <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">/3</span></span></span></span> voting units for <span style="white-space: nowrap">epoch <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">e</span></span></span></span>,</span> denoted as the <span style="white-space: nowrap">set <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span></span></span></span>,</span> were cast <span style="white-space: nowrap">for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span>,</span> and at least <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">/3</span></span></span></span> voting units for <span style="white-space: nowrap">epoch <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">e</span></span></span></span>,</span> denoted as the <span style="white-space: nowrap">set <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span>,</span> were cast <span style="white-space: nowrap">for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span>.</span> Since there are <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> voting units for <span style="white-space: nowrap">epoch <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">e</span></span></span></span>,</span> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∩</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> must have size at least <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">/3</span></span></span></span>.</span> In an execution of Crosslink 2 where <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has the <strong>one‑third bound on non‑honest voting</strong> property, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∩</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> must therefore include at least one voting unit that was cast honestly. Since a voting unit for epoch <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">e</span></span></span></span> that is cast honestly is not double-cast, it must be <span style="white-space: nowrap">that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span>.</span></p>
+</div>
+</div>
+<div id="admonition-info-2" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Info</p>
+<p><a class="admonition-anchor-link" href="#admonition-info-2"></a></p>
+</div>
+<div>
+<p>In the case of a network partition, votes may not be seen on both/all sides of the partition. Therefore, it is not necessarily the case that honest nodes can directly see double‑voting. The above argument does not depend on being able to do so.</p>
+</div>
+</div>
+<p>Therefore, in an execution of Crosslink 2 for which <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has the <strong>one‑third bound on non‑honest voting</strong> property, for each <span style="white-space: nowrap">epoch <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">e</span></span></span></span></span> there will be at most one bft‑proposal‑valid <span style="white-space: nowrap">proposal <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>,</span> and at least one third of honestly cast voting units must have been cast for it. Let <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.07382em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0738em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> be the (necessarily nonempty) set of nodes that cast these honest votes; then, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2558em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9727em;"><span style="top:-2.4231em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.2421em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">e</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2819em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2769em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span> for <span style="white-space: nowrap">all <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6986em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.07382em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0738em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span> at the <span style="white-space: nowrap">times <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9012em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">e</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span></span> of their votes in epoch <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">e</span></span></span></span>. <span style="white-space: nowrap">(For simplicity,</span> we assume that for each honest <span style="white-space: nowrap">node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span></span> there is only one <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9012em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">e</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span></span> at which it obtains a successful check for the voting condition in <span style="white-space: nowrap">epoch <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">e</span></span></span></span>,</span> which it uses for any votes that it casts in that epoch.)</p>
+<p>Let <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> be any bft‑block for <span style="white-space: nowrap">epoch <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">e</span></span></span></span></span> such that <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mclose">)</span></span></span></span>,</span> where <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span> is some bft‑block‑valid block. <span style="white-space: nowrap">Since <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span>,</span> <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> is bft‑block‑valid.</span> So by the argument above, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> commits to the only bft‑proposal‑valid <span style="white-space: nowrap">proposal <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span> for <span style="white-space: nowrap">epoch <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">e</span></span></span></span>,</span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> was voted for in that epoch by a nonempty subset of honest <span style="white-space: nowrap">nodes <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.07382em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0738em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</span></p>
+<p>Let <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> be any bc‑valid block. We have by definition: <span class="katex-display fleqn"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">fin</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mspace" style="margin-right:-0.1667em;"></span><span class="mspace" style="margin-right:-0.1667em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:-0.1667em;"></span><span class="mspace" style="margin-right:-0.1667em;"></span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mspace" style="margin-right:-0.1667em;"></span><span class="mspace" style="margin-right:-0.1667em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:-0.1667em;"></span><span class="mspace" style="margin-right:-0.1667em;"></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">[</span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mord text"><span class="mord"> for </span></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)]</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">[</span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mord text"><span class="mord"> for </span></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">context_bft</span></span><span class="mclose">)]</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span></span></span></span></span> So, taking <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">context_bft</span></span></span></span></span>, each <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span> for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> of epoch <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">e</span></span></span></span> in the result of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathrm">fin</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span></span></span> satisfies <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2558em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9727em;"><span style="top:-2.4231em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.2421em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">e</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2819em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2769em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span> for all <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> in some nonempty honest set of nodes <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.07382em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0738em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</p>
+<p>For an execution of Crosslink 2 in which <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has the <a href="./construction.html#prefix-consistency"><strong>Prefix Consistency</strong></a> property at confirmation depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>, for every epoch <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">e</span></span></span></span>, for every such <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mopen">(</span><span class="mord mathnormal">i</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">e</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>, for every node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> that is honest at any time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal">u</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9012em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">e</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>, we have <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2558em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9727em;"><span style="top:-2.4231em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.2421em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">e</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2819em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2769em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1318em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span>. Let <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7651em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mop">min</span><span class="mopen">{</span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">e</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6986em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.07382em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0738em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">}</span></span></span></span>. Then, by transitivity of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.786em;vertical-align:-0.15em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>:</p>
+<div id="admonition-theorem-on-snapshots-in-logfin" class="admonition admonish-success">
+<div class="admonition-title">
+<p>Theorem: On snapshots in LOG<sub>fin</sub></p>
+<p><a class="admonition-anchor-link" href="#admonition-theorem-on-snapshots-in-logfin"></a></p>
+</div>
+<div>
+<p>In an execution of Crosslink 2 where <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has the <strong>one‑third bound on non‑honest voting</strong> property and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has the <a href="./construction.html#prefix-consistency"><strong>Prefix Consistency</strong></a> property at confirmation <span style="white-space: nowrap">depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>,</span> every bc‑chain <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span> in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1279em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathrm">fin</span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> (and therefore every snapshot that contributes to <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span>)</span> is, <span style="white-space: nowrap">at any time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal">u</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7651em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span>,</span> in the bc‑best‑chain of <span style="white-space: nowrap">every node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span></span> that is honest <span style="white-space: nowrap">at time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span></span> (where <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> commits to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> at epoch <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">e</span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7651em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> is the time of the first honest vote <span style="white-space: nowrap">for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>).</span></p>
+</div>
+</div>
+<p>A similar (weaker) statement holds if we replace <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal">u</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7651em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> with <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal">u</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>, since the time of the first honest vote for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span> necessarily precedes the time at which the signed <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord mathsf" style="margin-right:0.06944em;">proof</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2342em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">P</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> is submitted as a bft‑block, which necessarily <span style="white-space: nowrap">precedes <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>.</span> (Whether or not the notarization proof depends on the <em>first</em> honest vote for <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span>’s proposal <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>,</span> it must depend on some honest vote for that proposal that was not made earlier <span style="white-space: nowrap">than <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7651em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.)</span></p>
+<p>Furthermore, we have <span class="katex-display fleqn"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.461em;vertical-align:-0.9805em;"></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4805em;"><span style="top:-3.6405em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord textsf">ba-ctx</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">μ</span><span class="mclose">)</span></span></span><span style="top:-2.4026em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">ba</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">μ</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9805em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4805em;"><span style="top:-3.6405em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mspace" style="margin-right:-0.1667em;"></span><span class="mspace" style="margin-right:-0.1667em;"></span><span class="mspace" style="margin-right:-0.1667em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:-0.1667em;"></span><span class="mspace" style="margin-right:-0.1667em;"></span><span class="mspace" style="margin-right:-0.1667em;"></span></span></span><span style="top:-2.4026em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mspace" style="margin-right:-0.1667em;"></span><span class="mspace" style="margin-right:-0.1667em;"></span><span class="mspace" style="margin-right:-0.1667em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:-0.1667em;"></span><span class="mspace" style="margin-right:-0.1667em;"></span><span class="mspace" style="margin-right:-0.1667em;"></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9805em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4805em;"><span style="top:-3.6405em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord textsf">san-ctx</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathrm">fin</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">∣∣</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em;"><span style="top:-2.3987em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.1809em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3013em;"><span></span></span></span></span></span></span><span class="mclose">])</span></span></span><span style="top:-2.4026em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord textsf">context-txns</span></span><span class="mopen">(</span><span class="mord text"><span class="mord textsf">ba-ctx</span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">μ</span><span class="mclose">))</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9805em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span></span></span></span></span></p>
+<p>So in an execution of Crosslink 2 where <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has the <a href="./construction.html#prefix-consistency"><strong>Prefix Consistency</strong></a> property at confirmation <span style="white-space: nowrap">depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span></span></span></span>,</span> <span style="white-space: nowrap">if node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span></span> is honest <span style="white-space: nowrap">at time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span></span> then <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0836em;vertical-align:-0.3013em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em;"><span style="top:-2.3987em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.1809em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3013em;"><span></span></span></span></span></span></span></span></span></span> is also, <span style="white-space: nowrap">at any time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal">u</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>,</span> in the bc‑best‑chain of <span style="white-space: nowrap">every node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span></span> that is honest <span style="white-space: nowrap">at time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span>.</span></p>
+<p>If an execution of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has the <a href="./construction.html#prefix-consistency"><strong>Prefix Consistency</strong></a> property at confirmation <span style="white-space: nowrap">depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8304em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>,</span> then it necessarily also has it at confirmation <span style="white-space: nowrap">depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>.</span> Therefore we have:</p>
+<div id="admonition-theorem-on-snapshots-in-logba" class="admonition admonish-success">
+<div class="admonition-title">
+<p>Theorem: On snapshots in LOG<sub>ba</sub></p>
+<p><a class="admonition-anchor-link" href="#admonition-theorem-on-snapshots-in-logba"></a></p>
+</div>
+<div>
+<p>In an execution of Crosslink 2 where <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has the <strong>one‑third bound on non‑honest voting</strong> property and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has the <a href="./construction.html#prefix-consistency"><strong>Prefix Consistency</strong></a> property at confirmation <span style="white-space: nowrap">depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8304em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>,</span> every bc‑chain snapshot in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1792em;vertical-align:-0.3013em;"></span><span class="mord"><span class="mord mathrm">fin</span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">∣∣</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mopen">[</span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em;"><span style="top:-2.3987em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.1809em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3013em;"><span></span></span></span></span></span></span><span class="mclose">]</span></span></span></span> (and therefore every snapshot that contributes to <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">ba</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">μ</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span>)</span> is, <span style="white-space: nowrap">at any time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal">u</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>,</span> in the bc‑best‑chain of <span style="white-space: nowrap">every node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span></span> that is honest <span style="white-space: nowrap">at time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span>.</span></p>
+</div>
+</div>
+<p>Sketch: we also need the sequence of snapshots output from fin to only be extended in the view of any node. In that case we can infer that the node does not observe a rollback in LOG_ba.</p>
+<p>Recall that in the proof of safety for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>, we showed that in an execution of Crosslink 2 where <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> (or <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>) has <a href="./construction.html#final-agreement"><strong>Final Agreement</strong></a>, <span style="white-space: nowrap">for all times <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span></span> and <span style="white-space: nowrap">all nodes <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span></span> (potentially the same) such that <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span></span> and <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span>,</span> <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1279em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathrm">fin</span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:-0.5em;"></span></span><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">⪰</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1331em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord mathrm">fin</span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span></span></span></span>.</span></p>
+<p>What we want to show is that, under some conditions on executions, ...</p>
+<h2 id="disadvantages-of-crosslink"><a class="header" href="#disadvantages-of-crosslink">Disadvantages of Crosslink</a></h2>
+<h3 id="more-invasive-changes"><a class="header" href="#more-invasive-changes">More invasive changes</a></h3>
+<p>Unlike Snap‑and‑Chat, Crosslink 2 requires structural and consensus rule changes to both <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>. On the other hand, several of those changes are arguably necessary to fix a show‑stopping bug in Snap‑and‑Chat (not being able to spend some finalized outputs).</p>
+<h3 id="finalization-latency"><a class="header" href="#finalization-latency">Finalization latency</a></h3>
+<p>For a given choice of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>, the finalization latency is higher. The snapshot of the BFT chain used to obtain <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">μ</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span> is obtained from the block at depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span></span></span></span> on node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>’s best <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> chain, which will on average lead to a finalized view that is about <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> blocks back (in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>), rather than \sigma_{\sac}} blocks in Snap‑and‑Chat. This is essentially the cost of ensuring that safety is given by the stronger of the safety of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> (at <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span></span></span></span> confirmations) and the safety of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</p>
+<p>On the other hand, the relative increase in expected finalization latency is only \frac{\mu + 1 + \sigma}{\sigma_{\sac}}}, i.e. at most slightly more than a factor of 2 for the case \mu = \sigma = \sigma_{\sac}}.</p>
+<h3 id="more-involved-liveness-argument"><a class="header" href="#more-involved-liveness-argument">More involved liveness argument</a></h3>
+<p>See the Liveness section above.</p>
+<h2 id="every-rule-in-crosslink-2-is-individually-necessary"><a class="header" href="#every-rule-in-crosslink-2-is-individually-necessary">Every rule in Crosslink 2 is individually necessary</a></h2>
+<div id="admonition-warning" class="admonition admonish-warning">
+<div class="admonition-title">
+<p>Warning</p>
+<p><a class="admonition-anchor-link" href="#admonition-warning"></a></p>
+</div>
+<div>
+<p>In order to show that Crosslink 2 is at a <em>local</em> optimum in the security/complexity trade‑off space, for each rule we show attacks on safety and/or liveness that could be performed if that rule were omitted or simplified.</p>
+</div>
+</div>
+<p>Edit: some rules, e.g. the <a href="./construction.html#linearity-rule"><strong>Linearity rule</strong></a>, only contribute heuristically to security in the analysis so far.</p>
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+<h1 id="the-arguments-for-bounded-availability-and-finality-overrides"><a class="header" href="#the-arguments-for-bounded-availability-and-finality-overrides">The Arguments for Bounded Availability and Finality Overrides</a></h1>
+<p>This document considers disadvantages of allowing transactions to continue to be included at the chain tip while the gap from the last finalized block becomes unbounded, and our perspective on what should be done instead. This condition is allowed by Ebb‑and‑Flow protocols <a href="https://eprint.iacr.org/2020/1091.pdf">[NTT2020]</a>.</p>
+<p>We also argue that it is necessary to allow for the possibility of overriding finalization in order to respond to certain attacks, and that this should be explicitly modelled and subject to a well-defined governance process.</p>
+<p>This is a rewritten version of <a href="https://forum.zcashcommunity.com/t/the-trailing-finality-layer/45133/10">this forum post</a>, adapting the main argument to take into account the discussion of “tail‑thrashing attacks” and finalization availability from the Addendum. More details of how bounded availability could be implemented in the context of a Snap‑and‑Chat protocol are in <a href="./notes-on-snap-and-chat.html">Notes on Snap‑and‑Chat</a>.</p>
+<p>The proposed changes end up being significant enough to give our construction a new name: “<a href="./construction.html">Crosslink</a>”, referring to the cross-links between blocks of the BFT and best-chain protocols. Crosslink has evolved somewhat, and now includes other changes not covered in either this document or <a href="./notes-on-snap-and-chat.html">Notes on Snap‑and‑Chat</a>. The current version is called <a href="./construction.html">Crosslink 2</a>.</p>
+<h2 id="background"><a class="header" href="#background">Background</a></h2>
+<p>“Ebb‑and‑Flow”, as described in <a href="https://eprint.iacr.org/2020/1091.pdf">[NTT2020]</a> (<a href="https://arxiv.org/pdf/2009.04987.pdf">arXiv version</a>), is a security model for consensus protocols that provide two transaction logs, one with dynamic availability, and a prefix of it with finality.</p>
+<p>The paper proposes an instantiation of this security model called a “Snap‑and‑Chat” construction. It composes two consensus subprotocols, a BFT subprotocol and a best-chain subprotocol (it calls this the “longest chain protocol”). The above logs are obtained from the output of these subprotocols in a non-trivial way.</p>
+<p>This is claimed by the paper to “resolve” the tension between finality and dynamic availability. However, a necessary consequence is that in a situation where the “final” log stalls and the “available” log does not, the “finalization gap” between the finalization point and the chain tip can grow without bound. In particular, this means that transactions that spend funds can remain unfinalized for an arbitrary length of time.</p>
+<p>In this document, we argue that this is unacceptable, and that it is preferable to sacrifice strict dynamic availability. The main idea behind Ebb‑and‑Flow protocols is a good one, and allowing the chain tip to run ahead of the finalization point does make sense and has practical advantages. However, we also argue that it should not be possible to include transactions that spend funds in blocks that are too far ahead of the finalization point.</p>
+<div id="admonition-note" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Note</p>
+<p><a class="admonition-anchor-link" href="#admonition-note"></a></p>
+</div>
+<div>
+<p>Naive ways of preventing an unbounded finalization gap, such as stopping the chain completely in the case of a finalization stall, turn out to run into serious security problems — at least when the best-chain protocol uses Proof‑of‑Work. We’ll discuss those in detail in the section on <a href="#tail%E2%80%91thrashing-attacks">tail‑thrashing attacks</a>.</p>
+<p>Our proposed solution will be to require coinbase-only blocks during a long finalization stall. This solution has the advantage of not complicating the security analysis.</p>
+</div>
+</div>
+<p>We argue that losing strict dynamic availability in favour of “bounded availability” is preferable to the consequences of the unbounded finality gap, if/when a “long finalization stall” occurs.</p>
+<p>We also argue that it is beneficial to explicitly allow “finality overrides” under the control of a well-documented governance process. Such overrides allow long rollbacks that may be necessary in the case of an exploited security flaw. This is complementary to the argument for bounded availability, because the latter limits the period of user transactions that could be affected. The governance process can impose a limit on the length of this long rollback if desired.</p>
+<h2 id="finality--dynamic-availability--possibility-of-partition-implies-an-unbounded-finalization-gap"><a class="header" href="#finality--dynamic-availability--possibility-of-partition-implies-an-unbounded-finalization-gap">Finality + Dynamic availability + Possibility of Partition implies an unbounded finalization gap</a></h2>
+<p>Since partition between nodes sufficient for finalization cannot be prevented, loosely speaking the <a href="https://en.wikipedia.org/wiki/CAP_theorem">CAP theorem</a> implies that any consistent protocol (and therefore any protocol with finality) may stall for at least as long as the partition takes to heal.</p>
+<div id="admonition-background" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Background</p>
+<p><a class="admonition-anchor-link" href="#admonition-background"></a></p>
+</div>
+<div>
+<p>That “loosely speaking” is made precise by <a href="https://arxiv.org/pdf/2006.10698.pdf">[LR2020]</a>.</p>
+</div>
+</div>
+<p>Dynamic availability implies that the chain tip will continue to advance, and so the finalization gap increases without bound.</p>
+<p>Partition is not necessarily the only condition that could cause a finalization stall, it is just the one that most easily proves that this conclusion is impossible to avoid.</p>
+<h2 id="problems-with-allowing-spends-in-an-unbounded-finalization-gap"><a class="header" href="#problems-with-allowing-spends-in-an-unbounded-finalization-gap">Problems with allowing spends in an unbounded finalization gap</a></h2>
+<p>Both the available protocol, and the subprotocol that provides finality, will be used in practice — otherwise, one or both of them might as well not exist. There is always a risk that blocks may be rolled back to the finalization point, by definition.</p>
+<p>Suppose, then, that there is a long finalization stall. The final and available protocols are not separate: there is no duplication of tokens between protocols, but the rules about how to determine best-effort balance and guaranteed balance depend on both protocols, how they are composed, and how the history after the finalization point is interpreted.</p>
+<div id="admonition-discussion" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Discussion</p>
+<p><a class="admonition-anchor-link" href="#admonition-discussion"></a></p>
+</div>
+<div>
+<p>The guaranteed minimum balance of a given party is not just the minimum of their balance at the finalization point and their balance at the current tip. It is the minimum balance taken over all possible transaction histories that extend the finalized chain — taking into account that a party’s previously published transactions might be able to be reapplied in a different context without its explicit consent. The extent to which published transactions can be reapplied depends on technical choices that we must make, subject to some constraints (for example, we know that shielded transactions cannot be reapplied after their anchors have been invalidated). It may be desirable to further constrain re-use in order to make guaranteed minimum balances easier to compute.</p>
+</div>
+</div>
+<p>As the finalization gap increases, the negative consequences of rolling back user transactions that spend funds increase. (Coinbase transactions do not spend funds; they are a special case that we will discuss later.)</p>
+<p>There are several possible —not mutually exclusive— outcomes:</p>
+<ul>
+<li>Users of the currency start to consider the available protocol increasingly unreliable.</li>
+<li>Users start to consider a rollback to be untenable, and lobby to prevent it or cry foul if it occurs.</li>
+<li>Users start to consider finalization increasingly irrelevant. Services that depend on finality become unavailable.
+<ul>
+<li>There is no free lunch that would allow us to avoid availability problems for services that also depend on finality.</li>
+</ul>
+</li>
+<li>Service providers adopt temporary workarounds that may not have had adequate security analysis.</li>
+</ul>
+<p>Any of these might precipitate a crisis of confidence, and there are reasons to think this effect might be worse than if the chain had switched to a “Stalled Mode” designed to prevent loss of user funds. Any such crisis may have a negative effect on token prices and long-term adoption.</p>
+<p>Note that adding finalization using an Ebb‑and‑Flow protocol does not by itself increase the probability of a rollback in the available chain, provided the PoW remains as secure against rollbacks of a given length as before. But that is a big proviso. We have a design constraint (motivated by limiting token devaluation and by governance issues) to limit issuance to be no greater than that of the original Zcash protocol up to a given height. Since some of the issuance is likely needed to reward staking, the amount of money available for mining rewards is reduced, which may reduce overall hash rate and security of the PoW. Independently, there may be a temptation for design decisions to rely on finality in a way that reduces security of PoW (“risk compensation”). There is also pressure to reduce the energy usage of PoW, which necessarily reduces the global hash rate, and therefore the cost of performing an attack that depends on the adversary having any given proportion of global hash rate.</p>
+<p>It could be argued that the issue of availability of services that depend on finality is mainly one of avoiding over-claiming about what is possible. Nevertheless there are also real usability issues if balances as seen by those services can differ significantly and for long periods from balances at the chain tip.</p>
+<p>Regardless, incorrect assumptions about the extent to which the finalized and available states can differ are likely to be exposed if there is a finalization stall. And those who made the assumptions may (quite reasonably!) not accept “everything is fine, those assumptions were always wrong” as a satisfactory response.</p>
+<h2 id="what-is-bounded-availability"><a class="header" href="#what-is-bounded-availability">What is Bounded Availability?</a></h2>
+<p>An intuitive notion of “availability” for block‑chain protocols includes the ability to use the protocol as normal to spend funds. So, just to be clear, in a situation where that cannot happen we have lost availability, <em>even if</em> the block chain is advancing.</p>
+<p>Bounded availability is a weakening of dynamic availability <a href="https://arxiv.org/abs/2010.08154">[DKT2020]</a>. It means that we intentionally sacrifice availability when some potentially hazardous operation —a “hazard” for short— would occur too far after the current finalization point. For now, assume for simplicity that our only hazard is spending funds. More generally, the notion of bounded availability can be applied to a wider range of protocols by tailoring the definition of “hazard” to the protocol.</p>
+<div id="admonition-background-1" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Background</p>
+<p><a class="admonition-anchor-link" href="#admonition-background-1"></a></p>
+</div>
+<div>
+<p>This <a href="https://www.youtube.com/watch?v=SacUT_53Pg8">talk by Soubhik Deb</a> accompanying <a href="https://arxiv.org/abs/2010.08154">[DKT2020]</a> provides a good explanation of the advantages of dynamic availability. We do not define bounded availability formally in this document, but informally, we aim to preserve the ability to securely adapt to large changes in hash rate or total stake.</p>
+</div>
+</div>
+<h3 id="terminology-note"><a class="header" href="#terminology-note">Terminology note</a></h3>
+<p><a href="https://eprint.iacr.org/2020/1091.pdf">[NTT2020]</a> calls the dynamically available block‑chain protocol <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> that provides input to the rest of the contruction, the “longest chain” protocol. There are two reasons to avoid this terminology:</p>
+<ul>
+<li>In Bitcoin, Zcash, and most other PoW-based protocols, what is actually used by each node is not its longest observed chain, but its observed consensus-valid chain with most accumulated work. In Zcash this is called the node’s “<a href="https://zips.z.cash/protocol/protocol.pdf#blockchain">best valid block chain</a>”, which we shorten to “best chain”.</li>
+<li>As footnote 2 on page 3 of <a href="https://eprint.iacr.org/2020/1091.pdf">[NTT2020]</a> says, that paper does not require <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> to be a “longest chain” protocol anyway.</li>
+</ul>
+<div id="admonition-historical-note" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Historical note</p>
+<p><a class="admonition-anchor-link" href="#admonition-historical-note"></a></p>
+</div>
+<div>
+<p>The error in conflating the “longest chain” with the observed consensus-valid chain with most accumulated work, originates in the Bitcoin whitepaper. [<a href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.221.9986">Nakamoto2008</a>, page 3]</p>
+</div>
+</div>
+<p>We will use the term “best‑chain protocol” instead. Note that this corresponds roughly to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> in the Snap‑and‑Chat construction, although the Crosslink 2 protocol that we propose will end up having other significant differences from Snap-and-Chat.</p>
+<h3 id="how-to-block-hazards"><a class="header" href="#how-to-block-hazards">How to block hazards</a></h3>
+<p>We have not yet decided <em>how</em> to block hazards during a long finalization stall. We could do so directly, or by stopping block production in the more-available protocol. For reasons explained in the section on <a href="#tail%E2%80%91thrashing-attacks">tail‑thrashing attacks</a> below, it’s desirable <em>not</em> to stop block production. And so it’s consistent to have bounded availability together with another liveness property —which can be defined similarly to dynamic availability— that says the more-available protocol’s chain is still advancing. This is what we will aim for.</p>
+<p>We will call this method of blocking hazards, without stopping block production, “going into Stalled Mode”.</p>
+<div id="admonition-historical-note-1" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Historical note</p>
+<p><a class="admonition-anchor-link" href="#admonition-historical-note-1"></a></p>
+</div>
+<div>
+<p>This concept of Stalled Mode is very similar to a <a href="https://github.com/zcash/zcash/issues/3311">feature that was discussed early in the development of Zcash</a>, but never fully designed or implemented. (After originally being called “Stalled Mode”, it was at some point renamed to “Emergency Mode”, but then the latter term was used for <a href="https://electriccoin.co/blog/update-addressing-zcash-wallet-performance-issues/">something else</a>.)</p>
+</div>
+</div>
+<p>For Zcash, we propose that the main restriction of Stalled Mode should be to require coinbase-only blocks. This achieves a similar effect, for our purposes, as actually stalling the more-available protocol’s chain. Since funds cannot be spent in coinbase-only blocks, the vast majority of attacks that we are worried about would not be exploitable in this state.</p>
+<div id="admonition-info" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Info</p>
+<p><a class="admonition-anchor-link" href="#admonition-info"></a></p>
+</div>
+<div>
+<p>It is possible that a security flaw could affect coinbase transactions. We <em>might</em> want to turn off shielded coinbase for Stalled Mode blocks in order to reduce the chance of that.</p>
+<p>Also, mining rewards cannot be spent in a coinbase-only block; in particular, mining pools cannot distribute rewards. So there is a risk that an unscrupulous mining pool might try to do a rug-pull after mining of non-coinbase-only blocks resumes, if there were a very long finalization stall. But this approach works at least in the short term, and probably for long enough to allow manual intervention into the finalization protocol, or governance processes if needed.</p>
+</div>
+</div>
+<p>An analogy for the effect of this on availability that may be familiar to many people, is that it works like video streaming. All video streaming services use a buffer to paper over short-term interruptions or slow-downs of network access. In most cases, this buffer is bounded. This allows the video to be watched uninterrupted and at a constant rate in most circumstances. But if there is a longer-term network failure or insufficient sustained bandwidth, the playback will unavoidably stall. In our case, block production does not literally stall, but it’s the same as far as users’ ability to perform “hazardous” operations is concerned.</p>
+<h3 id="why-is-this-better"><a class="header" href="#why-is-this-better">Why is this better?</a></h3>
+<p>So, why do we advocate this over:</p>
+<ol>
+<li>A protocol that only provides dynamic availability;</li>
+<li>A protocol that only provides finality;</li>
+<li>An unmodified Ebb‑and‑Flow protocol?</li>
+</ol>
+<p>The reason to reject option 1 is straightforward: finality is a valuable security property that is necessary for some use cases.</p>
+<p>If a protocol only provides finality (option 2), then short-term availability is directly tied to finalization. It may be possible to make finalization stalls sufficiently rare or short-lived that this is tolerable. But that is more likely to be possible if and when there is a well-established staking ecosystem. Before that ecosystem is established, the protocol may be particularly vulnerable to stalls. Furthermore, it’s difficult to get to such a protocol from a pure PoW system like current Zcash.</p>
+<p>We argued in the previous section that allowing hazards in an unbounded finalization gap is bad. Option 3 entails an unbounded finalization gap that <em>will</em> allow hazards. However, that isn’t sufficient to argue that bounded availability is better. Perhaps there are no good solutions! What are we gaining from a bounded availability approach that would justify the complexity of a hybrid protocol without obtaining strict dynamic availability?</p>
+<p>The argument goes like this:</p>
+<ul>
+<li>It is likely that a high proportion of the situations in which a sustained finalization stall happens will require human intervention. If the finality protocol were going to recover without intervention, there is no reason to think that it wouldn’t do so in a relatively short time.</li>
+<li>When human intervention is required, the fact that the chain tip is still proceeding apace (in protocols with strict dynamic availability) makes restarting the finality protocol harder, for many potential causes of a finalization stall. This may be less difficult when only “non-hazardous” transactions are present— in particular, when only coinbase transactions (which are subject to fairly strict rules in Zcash and other Bitcoin-derived chains) are present. This argument carries even more force when the protocol also allows “finality overrides”, as discussed later in the <a href="#complementarity">Complementarity</a> section.</li>
+<li>Nothing about bounded availability prevents us from working hard to design a system that makes finalization stalls as infrequent and short-lived as possible, just as we would for any other option that provides finality.</li>
+<li>We want to optimistically minimize the finalization gap under good conditions, because this improves the usability of services that depend on finality. This argues against protocols that try to maintain a fixed gap, and motivates letting the gap vary.</li>
+<li>In practice, the likelihood of short finalization stalls is high enough that heuristically retaining availability in those situations is useful.</li>
+</ul>
+<p>The argument that it is difficult to completely prevent finalization stalls is supported by <a href="https://www.coindesk.com/tech/2023/05/17/ethereums-loss-of-finality-what-happened/">experience on Ethereum in May 2023</a>, when there were two stalls within 24 hours, <a href="https://offchain.medium.com/post-mortem-report-ethereum-mainnet-finality-05-11-2023-95e271dfd8b2">one for about 25 minutes and one for about 64 minutes</a>. This experience is consistent with our argument:</p>
+<ul>
+<li>Neither stall required short-term human intervention, and the network did in fact recover from them quickly.</li>
+<li>The stalls were caused by a resource exhaustion problem in the Prysm consensus client when handling attestations. It’s plausible to think that if this bug had been more serious, or possibly if Prysm clients had made up more of the network, then it would have required a hotfix release (and/or a significant proportion of nodes switching to another client) in order to resolve the stall. So this lines up with the hypothesis that longer stalls are likely to require manual intervention.</li>
+<li>A bounded availability protocol would very likely have resulted in either a shorter or no interruption in availability. If, say, the availability bound were set to be roughly an hour, then the first finalization stall would have been “papered over” and the second would have resulted in only a short loss of availability.</li>
+</ul>
+<p>Retaining short-term availability does not result in a risk compensation hazard:</p>
+<ul>
+<li>A finalization stall is still very visible, and directly affects applications relying on finality.</li>
+<li>Precisely because of the availability bound, it is obvious that it could affect all applications if it lasted long enough.</li>
+</ul>
+<p>A potential philosophical objection to lack of strict dynamic availability is that it creates a centralization risk to availability. That is, it becomes more likely that a coalition of validators can deliberately cause a denial of service. This objection may be more prevalent among people who would object to adding a finality layer or PoS at all.</p>
+<h2 id="finality-overrides"><a class="header" href="#finality-overrides">Finality Overrides</a></h2>
+<p>Consensus protocols sometimes fail. Potential causes of failure include:</p>
+<ul>
+<li>A design problem with the finality layer that causes a stall, or allows a stall to be provoked.</li>
+<li>A balance violation or spend authorization flaw that is being exploited or is sufficiently likely to be exploited.</li>
+<li>An implementation bug in a widely used node implementation that causes many nodes to diverge from consensus.</li>
+</ul>
+<p>In these situations, overriding finality may be better than any other alternative.</p>
+<p>An example is a <a href="https://bitcointalk.org/index.php?topic=822.0">balance violation flaw due to a 64-bit integer overflow</a> that was <a href="https://en.bitcoin.it/wiki/Value_overflow_incident">exploited on Bitcoin mainnet</a> on 15th August 2010. The response was to roll back the chain to before the exploitation, which is widely considered to have been the right decision. The time between the exploit (at block height 74638) and the forked chain overtaking the exploited chain (at block height 74691) was 53 blocks, or around 9 hours.</p>
+<p>Of course, Bitcoin used and still uses a pure‑PoW consensus. But the applicability of the example does not depend on that: the flaw was independent of the consensus mechanism.</p>
+<p>Another example of a situation that prompted this kind of override was the <a href="https://hackingdistributed.com/2016/06/18/analysis-of-the-dao-exploit/">DAO recursion exploit</a> on the Ethereum main chain <a href="https://www.coindesk.com/learn/understanding-the-dao-attack/">in June 2016</a>. The response to this was the forced balance adjustment hard fork on 20th July 2016 commonly known as the <a href="https://ethereum.org/en/history/#dao-fork">DAO fork</a>. Although this adjustment was not implemented as a rollback, and although Ethereum was using PoW at the time and did not make any formal finality guarantees, it did override transfers that would heuristically have been considered final at the fork height. Again, this flaw was independent of the consensus mechanism.</p>
+<p>The DAO fork was of course much more controversial than the Bitcoin fork, and a substantial minority of mining nodes split off to form Ethereum Classic. In any case, the point of this example is that it’s always possible to override finality in response to an exceptional situation, and that a chain’s community may decide to do so. The fact that Ethereum 2.0 now does claim a finality guarantee, would not in practice prevent a similar response in future that would override that guarantee.</p>
+<p>The question then is whether the procedure to override finality should be formalized or ad hoc. We argue that it should be formalized, including specifying the governance process to be used.</p>
+<p>This makes security analysis — of the consensus protocol per se, of the governance process, and of their interaction — much more feasible. Arguably a complete security analysis is not possible at all without it.</p>
+<p>It also front‑loads arguing about what procedure should be followed, and so it is more likely that stakeholders will agree to follow the process in any time‑critical incident.</p>
+<h3 id="a-way-of-modelling-overrides-that-is-insufficient"><a class="header" href="#a-way-of-modelling-overrides-that-is-insufficient">A way of modelling overrides that is insufficient</a></h3>
+<p>There is another possible way to model a protocol that claims finality but can be overridden in practice. We could say that the protocol after the override is a brand‑new protocol and chain (inheriting balances from the previous one, possibly modulo adjustments such as those that happened in the DAO fork).</p>
+<p>Although that would allow saying that the finality property has technically not been violated, it does not match how users think about an override situation. They are more likely to think of it as a protocol with finality that can be violated in exceptional cases — and they would reasonably want to know what those cases are and how they will be handled. It also does nothing to help with security analysis of such cases.</p>
+<h2 id="complementarity"><a class="header" href="#complementarity">Complementarity</a></h2>
+<p>Finality overrides and bounded availability are complementary in the following way: if a problem is urgent enough, then validators can be asked to stop validating. For genuinely harmful problems, it is likely to be in the interests of enough validators to stop that this causes a finalization stall. If this lasts longer than the availability bound then the protocol will go into Stalled Mode, giving time for the defined governance process to occur and decide what to do. And because the unfinalized consensus chain will contain only a limited period of user transactions that spend funds, the option of a long rollback remains realistically open.</p>
+<p>If, on the other hand, there is time pressure to make a governance decision about a rollback in order to reduce its length, that may result in a less well-considered decision.</p>
+<p>A possible objection is that there might be a coalition of validators who ignore the request to stop (possibly including the attacker or validators that an attacker can bribe), in which case the finalization stall would not happen. But that just means that we don’t gain the advantage of more time to make a governance decision; it isn’t actively a disadvantage relative to alternative designs. This outcome can also be thought of as a feature rather than a bug: going into Stalled Mode should be a last resort, and if the argument given for the request to stop failed to convince a sufficient number of validators that it was reason enough to do so, then perhaps it wasn’t a good enough reason.</p>
+<div id="admonition-rationale" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Rationale</p>
+<p><a class="admonition-anchor-link" href="#admonition-rationale"></a></p>
+</div>
+<div>
+<p>This resolves one of the main objections to the <a href="https://github.com/zcash/zcash/issues/3311">original Stalled Mode idea</a> that stopped us from implementing it in Zcash. The original proposal was to use a signature with a key held by ECC to trigger Stalled Mode, which would arguably have been too centralized. The Stalled Mode described in this document, on the other hand, can only be entered by consensus of a larger validator set, or if there is an availability failure of the finalization protocol.</p>
+</div>
+</div>
+<p>It is also possible to make the argument that the threshold of stake needed is imposed by technical properties of the finality protocol and by the resources of the attacker, which might not be ideal for the purpose described above. However, we would argue that it does not need to be ideal, and will be in the right ballpark in practice.</p>
+<p>There’s a caveat related to doing intentional rollbacks when using the Stalled Mode approach, where block production in the more-available protocol continues during a long finalization stall. What happens to incentives of block producers (miners in the case of Proof‑of‑Work), given that they know the consensus chain might be intentionally rolled back? They might reasonably conclude that it is less valuable to produce those blocks, leading to a reduction of hash rate or other violations of the security assumptions of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</p>
+<p>This is actually fairly easy to solve. We have the governance procedures say that if we do an intentional rollback, the coinbase-only mining rewards will be preserved. I.e. we produce a block or blocks that include those rewards paid to the same addresses (adjusting the consensus to allow them to be created from thin air if necessary), have everyone check it thoroughly, and require the chain to restart from that block. So as long as block producers believe that this governance procedure will be followed and that the chain will eventually recover at a reasonable coin price, they will still have incentive to produce on <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>, at least for a time.</p>
+<div id="admonition-rationale-1" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Rationale</p>
+<p><a class="admonition-anchor-link" href="#admonition-rationale-1"></a></p>
+</div>
+<div>
+<p>Although the community operating the governance procedures has already obtained the security benefit of mining done on the rolled-back chain by the time it creates the new chain, there is a strong incentive not to renege on the agreement with miners, because the same situation may happen again.</p>
+</div>
+</div>
+<h2 id="tail-thrashing-attacks"><a class="header" href="#tail-thrashing-attacks">Tail-thrashing attacks</a></h2>
+<p>Earlier we said that there were two possible approaches to preventing hazards during a long finalization stall:</p>
+<p>a) go into a Stalled Mode that directly disallows hazardous transactions (for example, by requiring <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> blocks to be coinbase-only in Zcash);</p>
+<p>b) temporarily cause the more-available chain to stall.</p>
+<p>This section describes an important class of potential attacks on approach b) that are difficult to resolve. They are based on the fact that when the unfinalized chain stalls, an adversary has more time to find blocks, and this might violate security assumptions of the more-available protocol. For instance, if the more-available protocol is PoW-based, then its security in the steady state is predicated on the fact that an adversary with a given proportion of hash power has only a limited time to use that power, before the rest of the network finds another block.</p>
+<div id="admonition-background-2" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Background</p>
+<p><a class="admonition-anchor-link" href="#admonition-background-2"></a></p>
+</div>
+<div>
+<p>For an analysis of the concrete security of Nakamoto-like protocols, see <a href="https://arxiv.org/abs/2005.10484">[DKT+2020]</a> and <a href="https://eprint.iacr.org/2020/661">[GKR2020]</a>. These papers confirm the intuition that the “private attack” —in which an adversary races privately against the rest of the network to construct a forking chain— is optimal, obtaining the same tight security bound independently using different techniques.</p>
+</div>
+</div>
+<p>During a chain stall, the adversary no longer has a limited time to construct a forking chain. If, say, the adversary has 10% hash power, then it can on average find a block <span style="white-space: nowrap">in 10 block times.</span> <span style="white-space: nowrap">And so in 100 block times</span> it can create <span style="white-space: nowrap">a 10-block fork.</span></p>
+<p>It may in fact be worse than this: once miners know that a finalization stall is happening, their incentive to continue mining is reduced, since they know that there is a greater chance that their blocks might be rolled back. So we would expect the global hash rate to fall —even before the finality gap bound is hit— and then the adversary would have a greater proportion of hash rate.</p>
+<div id="admonition-info-1" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Info</p>
+<p><a class="admonition-anchor-link" href="#admonition-info-1"></a></p>
+</div>
+<div>
+<p>Even in a pure Ebb‑and‑Flow protocol, a finalization stall could cause miners to infer that their blocks are more likely to be rolled back, but the fact that the chain is continuing would make that more difficult to exploit. This issue with the global hash rate is mostly specific to the more-available protocol being PoW: if it were PoS, then its validators might as well continue proposing blocks because it is cheap to do so. There might be other attacks when the more-available protocol is PoS; we haven’t spent much time analyzing that case.</p>
+</div>
+</div>
+<p>The problem is that the more-available chain does not necessarily <em>just halt</em> during a chain stall. In fact, for a finality gap bound of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> blocks, an adversary could cause <span style="white-space: nowrap">the <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span>-block “tail”</span> of the chain as seen by any given node to “thrash” between different chains. We will call this <span style="white-space: nowrap">a <strong>tail‑thrashing attack</strong>.</span></p>
+<p>If a protocol allowed such attacks then it would be a regression relative to the security we would normally expect from an otherwise similar PoW-based protocol. It only occurs during a finalization stall, but note that we cannot exclude the possibility of an adversary being able to provoke a finalization stall.</p>
+<p>Note that in the Snap‑and‑Chat construction, snapshots of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> are used as input to the BFT protocol. That implies that the tail‑thrashing problem could also affect the input to that protocol, which would be bad (not least for security analysis of availability, which seems somewhat intractable in that case).</p>
+<p>Also, when restarting <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>, we would need to take account of the fact that the adversary has had an arbitrary length of time to build long chains from every block that we could potentially restart from. It could be possible to invalidate those chains by requiring blocks after the restart to be dependent on fresh randomness, but that sounds quite tricky (especially given that we want to restart <em>without</em> manual intervention if possible), and there may be other attacks we haven’t thought of. This motivates using approach a) instead.</p>
+<p>Note that we have still glossed over precisely how consensus rules would change to enforce a). This will be covered later in <a href="./construction.html">The Crosslink 2 Construction</a>, but first we will discuss other issues with Snap-and-Chat.</p>
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+                <ol class="chapter"><li class="chapter-item expanded "><a href="../introduction.html"><strong aria-hidden="true">1.</strong> Introduction</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../introduction/a-path-to-pos-zcash.html"><strong aria-hidden="true">1.1.</strong> A Path to Proof-of-Stake Zcash</a></li><li class="chapter-item expanded "><a href="../introduction/trailing-finality-layer-in-a-nutshell.html"><strong aria-hidden="true">1.2.</strong> Trailing Finality Layer in a Nutshell</a></li><li class="chapter-item expanded "><a href="../introduction/status-and-next-steps.html"><strong aria-hidden="true">1.3.</strong> Status and Next Steps</a></li><li class="chapter-item expanded "><a href="../introduction/motivating-finality.html"><strong aria-hidden="true">1.4.</strong> Motivating Finality</a></li><li class="chapter-item expanded "><a href="../introduction/get-involved.html"><strong aria-hidden="true">1.5.</strong> Get Involved</a></li></ol></li><li class="chapter-item expanded "><a href="../terminology.html"><strong aria-hidden="true">2.</strong> Terminology</a></li><li class="chapter-item expanded "><a href="../design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/design-at-a-glance.html" class="active"><strong aria-hidden="true">3.1.</strong> Design at a Glance</a></li><li class="chapter-item expanded "><a href="../design/goals.html"><strong aria-hidden="true">3.2.</strong> Design Goals</a></li><li class="chapter-item expanded "><a href="../design/crosslink.html"><strong aria-hidden="true">3.3.</strong> Crosslink</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/crosslink/the-arguments-for-bounded-availability-and-finality-overrides.html"><strong aria-hidden="true">3.3.1.</strong> The Arguments for Bounded Availability and Finality Overrides</a></li><li class="chapter-item expanded "><a href="../design/crosslink/notes-on-snap-and-chat.html"><strong aria-hidden="true">3.3.2.</strong> Notes On Snap‑and‑Chat</a></li><li class="chapter-item expanded "><a href="../design/crosslink/construction.html"><strong aria-hidden="true">3.3.3.</strong> The Crosslink 2 Construction</a></li><li class="chapter-item expanded "><a href="../design/crosslink/security-analysis.html"><strong aria-hidden="true">3.3.4.</strong> Security Analysis of Crosslink 2</a></li><li class="chapter-item expanded "><a href="../design/crosslink/questions.html"><strong aria-hidden="true">3.3.5.</strong> Questions About Crosslink</a></li><li class="chapter-item expanded "><a href="../design/crosslink/potential-changes.html"><strong aria-hidden="true">3.3.6.</strong> Potential Changes to Crosslink</a></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">3.4.</strong> Model Code Architecture</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">3.5.</strong> Network Architecture</div></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">4.</strong> Protocol Specification</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">4.1.</strong> Abstract Protocol</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">4.1.1.</strong> Token Dynamics</div></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">4.2.</strong> Concrete Protocol</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">4.2.1.</strong> Token Dynamics</div></li></ol></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.</strong> Security</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.</strong> Abstract Analysis</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.1.</strong> Ebb-and-Flow analysis</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.2.</strong> Subprotocol Compromise Analysis</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.3.</strong> Model Code Architecture Analysis</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.4.</strong> Network Architecture Analysis</div></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.2.</strong> Concrete Analysis</div></li></ol></li><li class="chapter-item expanded "><a href="../references.html"><strong aria-hidden="true">6.</strong> References</a></li><li class="chapter-item expanded "><a href="../version-history.html"><strong aria-hidden="true">7.</strong> Version History</a></li></ol>
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+<h1 id="design-at-a-glance"><a class="header" href="#design-at-a-glance">Design at a Glance</a></h1>
+<p>The <a href="../terminology.html#definition-pow-tfl">PoW+TFL</a> consensus protocol is logically an extension of the Zcash consensus rules to introduce <a href="../terminology.html#definition-trailing-finality">trailing finality</a>. This is achieved by compartmentalizing the top-level PoW+TFL protocol into two <a href="../terminology.html#definition-consensus-subprotocols">consensus subprotocols</a>, one embodying most of the current consensus logic of Zcash and another the TFL. These protocols interact through a <a href="../terminology.html#definition-hybrid-construction">hybrid construction</a>, which specifies how the protocols interact, and what changes from &quot;off-the-shelf&quot; behavior, if any, need to be imposed on the subprotocols. Each of these components (the two subprotocols and the hybrid construction) are somewhat modular: different subprotocols or hybrid constructions may be combined (with some modification) to produce a candidate <a href="../terminology.html#definition-pow-tfl">PoW+TFL</a> protocol.</p>
+<p><strong>TODO:</strong> <a href="https://github.com/Electric-Coin-Company/tfl-book/issues/122">Add a protocol component diagram to &quot;Design at a Glance&quot; #122</a></p>
+<h2 id="hybrid-construction"><a class="header" href="#hybrid-construction">Hybrid Construction</a></h2>
+<p>The <a href="../terminology.html#definition-hybrid-construction">hybrid construction</a> is a major design component of the full consensus protocol which specifies how the subprotocols integrate. So far we have considered three candidates:</p>
+<ol>
+<li>The implied/loosely defined hybrid construction <a href="https://www.youtube.com/watch?v=qhMzMYeEPMM&amp;list=PL40dyJ0UYTLII7oQRQmNOFf0d2iKT35tL&amp;index=17">presented at Zcon4</a>.</li>
+<li>The <a href="../terminology.html#definition-snap-and-chat">Snap-and-Chat</a> from the <a href="https://eprint.iacr.org/2020/1091">Ebb-and-Flow paper</a>.</li>
+<li>The <a href="../terminology.html#definition-crosslink">Crosslink</a> construction.</li>
+</ol>
+<p>We believe <a href="../terminology.html#definition-crosslink">Crosslink</a> is the best candidate, due to its more rigorous <a href="./crosslink/construction.html">specification</a> and <a href="./crosslink/security-analysis.html">security analysis</a>, and due to the issues with Snap-and-Chat described in <a href="./crosslink/notes-on-snap-and-chat.html">Notes on Snap-and-Chat</a>.</p>
+<h2 id="subprotocols"><a class="header" href="#subprotocols">Subprotocols</a></h2>
+<p>The PoW+TFL hybrid consensus consists of two interacting subprotocols:</p>
+<ol>
+<li><a href="../terminology.html#definition-pow">PoW Subprotocol</a>: this subprotocol is very similar to NU5 or NU6 consensus. It is a design goal of the TFL design to minimize changes to this subprotocol. Note: the shorthand &quot;PoW&quot; is potentially misleading, because this subprotocol is also responsible for the bulk of all supply and transaction semantic consensus rules.</li>
+<li><a href="../terminology.html#definition-pos">PoS Subprotocol</a>: this is a new subprotocol which provides trailing finality via a finalizing PoS protocol.</li>
+</ol>
+<p><strong>TODO:</strong> <a href="https://github.com/Electric-Coin-Company/tfl-book/issues/119">Clarify the distinctions between pure PoW, the PoW subprotocol, NU6, and fork-choice vs all of transaction semantics. #119</a></p>
+<p>Note that the <a href="../terminology.html#definition-hybrid-construction">hybrid construction</a> may require modification to the &quot;off-the-shelf&quot; versions of these subprotocols. In particular <a href="../terminology.html#definition-crosslink">Crosslink</a> requires each protocol to refer to the state of the other to provide <a href="../terminology.html#definition-objective-validity">objective validity</a>.</p>
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+                <ol class="chapter"><li class="chapter-item expanded "><a href="../introduction.html"><strong aria-hidden="true">1.</strong> Introduction</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../introduction/a-path-to-pos-zcash.html"><strong aria-hidden="true">1.1.</strong> A Path to Proof-of-Stake Zcash</a></li><li class="chapter-item expanded "><a href="../introduction/trailing-finality-layer-in-a-nutshell.html"><strong aria-hidden="true">1.2.</strong> Trailing Finality Layer in a Nutshell</a></li><li class="chapter-item expanded "><a href="../introduction/status-and-next-steps.html"><strong aria-hidden="true">1.3.</strong> Status and Next Steps</a></li><li class="chapter-item expanded "><a href="../introduction/motivating-finality.html"><strong aria-hidden="true">1.4.</strong> Motivating Finality</a></li><li class="chapter-item expanded "><a href="../introduction/get-involved.html"><strong aria-hidden="true">1.5.</strong> Get Involved</a></li></ol></li><li class="chapter-item expanded "><a href="../terminology.html"><strong aria-hidden="true">2.</strong> Terminology</a></li><li class="chapter-item expanded "><a href="../design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/design-at-a-glance.html"><strong aria-hidden="true">3.1.</strong> Design at a Glance</a></li><li class="chapter-item expanded "><a href="../design/goals.html" class="active"><strong aria-hidden="true">3.2.</strong> Design Goals</a></li><li class="chapter-item expanded "><a href="../design/crosslink.html"><strong aria-hidden="true">3.3.</strong> Crosslink</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/crosslink/the-arguments-for-bounded-availability-and-finality-overrides.html"><strong aria-hidden="true">3.3.1.</strong> The Arguments for Bounded Availability and Finality Overrides</a></li><li class="chapter-item expanded "><a href="../design/crosslink/notes-on-snap-and-chat.html"><strong aria-hidden="true">3.3.2.</strong> Notes On Snap‑and‑Chat</a></li><li class="chapter-item expanded "><a href="../design/crosslink/construction.html"><strong aria-hidden="true">3.3.3.</strong> The Crosslink 2 Construction</a></li><li class="chapter-item expanded "><a href="../design/crosslink/security-analysis.html"><strong aria-hidden="true">3.3.4.</strong> Security Analysis of Crosslink 2</a></li><li class="chapter-item expanded "><a href="../design/crosslink/questions.html"><strong aria-hidden="true">3.3.5.</strong> Questions About Crosslink</a></li><li class="chapter-item expanded "><a href="../design/crosslink/potential-changes.html"><strong aria-hidden="true">3.3.6.</strong> Potential Changes to Crosslink</a></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">3.4.</strong> Model Code Architecture</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">3.5.</strong> Network Architecture</div></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">4.</strong> Protocol Specification</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">4.1.</strong> Abstract Protocol</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">4.1.1.</strong> Token Dynamics</div></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">4.2.</strong> Concrete Protocol</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">4.2.1.</strong> Token Dynamics</div></li></ol></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.</strong> Security</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.</strong> Abstract Analysis</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.1.</strong> Ebb-and-Flow analysis</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.2.</strong> Subprotocol Compromise Analysis</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.3.</strong> Model Code Architecture Analysis</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.4.</strong> Network Architecture Analysis</div></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.2.</strong> Concrete Analysis</div></li></ol></li><li class="chapter-item expanded "><a href="../references.html"><strong aria-hidden="true">6.</strong> References</a></li><li class="chapter-item expanded "><a href="../version-history.html"><strong aria-hidden="true">7.</strong> Version History</a></li></ol>
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+                    <h1 class="menu-title">Zcash Trailing Finality Layer - v0.1.0</h1>
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+                        <link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.12.0/dist/katex.min.css" integrity="sha384-AfEj0r4/OFrOo5t7NnNe46zW/tFgW6x/bCJG8FqQCEo3+Aro6EYUG4+cU+KJWu/X" crossorigin="anonymous">
+<h1 id="design-goals"><a class="header" href="#design-goals">Design Goals</a></h1>
+<p>Here we strive to lay out our high level TFL design goals.</p>
+<h2 id="goals-design-and-trade-offs"><a class="header" href="#goals-design-and-trade-offs">Goals, Design, and Trade-offs</a></h2>
+<p>Here we lay out <em>ideal</em> goals. As we develop a complete design, we are likely to inevitably encounter trade-offs some of which may preclude achieving the full idealized goals. Wherever possible, we motivate design decisions by these goals, and when goals are impacted by trade-offs we describe that impact and the rationale for the trade-off decision.</p>
+<p>For example, one ideal user experience goal below is to avoid disruption to existing wallets. However, the <a href="../terminology.html#definition-crosslink">Crosslink</a> construction may require wallets to alter their context of valid transactions differently from the current <a href="../terminology.html#definition-nu5nu6">NU5/NU6</a> consensus protocol.</p>
+<h2 id="user-experience-and-use-case-goals"><a class="header" href="#user-experience-and-use-case-goals">User Experience and Use Case Goals</a></h2>
+<p>We strive to start our protocol design process from user experience (UX) and use case considerations foremost, since at the end of the day all that matters in a protocol is what user needs it meets and how well.</p>
+<ul>
+<li>All currently supported wallet user experience should continue to operate seamlessly without change during or after protocol transitions. This covers the use of addresses, payment flow, transfers, ZEC supply cap and issuance rate, backup/restore, and other features users currently rely on.</li>
+<li>There must be no security or safety degradation due to wallet user behavior introduced by PoS transitions, assuming users follow their current behaviors unchanged and continue to use the same cognitive model of the impacts of their behaviors. This goal encompasses all of security and safety, including privacy and transparency or more explicit disclosures.</li>
+<li>The protocol should enable users of shielded mobile wallets to delegate ZEC to PoS consensus providers and earn a return on that ZEC coming via ZEC issuance or fees. Doing this may expose users to a risk of loss of delegated ZEC (such as through “slashing fees”). The protocol must guarantee that PoS consensus providers have no discretionary control over such delegated funds (including that they cannot steal those funds).</li>
+<li>For any hybrid PoW/PoS protocol (including the PoW+TFL protocol we’re proposing), the process and UX of mining remains unchanged except that the return on investment may be altered. This is true both of consensus-level block miners (i.e. mining pools and solo miners) and mining pool participants.</li>
+<li>Any hybrid PoW/PoS protocol (including PoW+TFL) block explorers will continue to function with the same UX through transitions as far as displaying information about transactions, the mempool, and blocks.</li>
+<li>Block explorers and other network metrics sites may require UX changes with respect to mining rewards and issuance calculations.</li>
+<li>Network metrics sites may require UX changes with respect to the p2p protocol or other network-specific information.</li>
+<li>Users can rely on <a href="../terminology.html#definition-assured-finality">assured finality</a> with an expected time‑to‑finality below 30 minutes.<sup class="footnote-reference"><a href="#req-ttf-30-min">1</a></sup></li>
+</ul>
+<h2 id="developer-experience-goals"><a class="header" href="#developer-experience-goals">Developer Experience Goals</a></h2>
+<p>For a full PoS transition, ecosystem developers for products such as consensus nodes, wallets, mining services, chain analytics, and more will certainly need to update their code to support transitions. However, we carve out a few goals as an exception to this for this category of users:</p>
+<ul>
+<li>Wallet developers should not be required to make any changes through protocol transitions as long as they rely solely on the lightwalletd protocol or a full node API (such as the zcashd RPC interface).</li>
+<li>For any hybrid PoW/PoS protocol (including PoW+TFL), mining pools and miners should not be required to make any software or protocol changes as long as they rely on zcashd-compatible GetBlockTemplate. One exception to this is software that bakes in assumptions about the block reward schedule, rather than relying on GetBlockTemplate solely.</li>
+</ul>
+<h2 id="safety-security-and-privacy-goals"><a class="header" href="#safety-security-and-privacy-goals">Safety, Security, and Privacy Goals</a></h2>
+<p>Zcash has always had exemplary safety, security, and privacy, and we aim to continue that tradition:</p>
+<ul>
+<li>For any hybrid PoW/PoS protocol (including PoW+TFL), the cost‑of‑attack for a 1-hour rollback should not be reduced, given a “reasonably rigorous” security argument.</li>
+<li>For any hybrid PoW/PoS protocol (including PoW+TFL), the cost‑of‑attack to halt the chain should be larger than the 24‑hour revenue of PoW mining rewards, given a “reasonably rigorous” security argument.</li>
+</ul>
+<p><strong>TODO:</strong> <a href="https://github.com/Electric-Coin-Company/tfl-book/issues/118">Define privacy goals of TFL #118</a></p>
+<p><strong>TODO:</strong> <a href="https://github.com/Electric-Coin-Company/tfl-book/issues/117">Define PoS Subprotocol desiderata which are distinct from Crosslink integration #117</a></p>
+<h2 id="design-conservatism-goals"><a class="header" href="#design-conservatism-goals">Design Conservatism Goals</a></h2>
+<p>We want to follow some conservative design heuristics to minimize risk and mistakes:</p>
+<ul>
+<li>Rely as much as possible on design components that are already proven in production environments.</li>
+<li>Rely as much as possible on design components with adequate theoretical underpinnings and security analyses.</li>
+<li>Minimize changes or variations on the above: strive to only alter existing work when necessary for overall design goals. For example, Zcash’s privacy or issuance constraints are likely less common among existing PoS designs.</li>
+</ul>
+<h1 id="non-goals"><a class="header" href="#non-goals">Non-goals</a></h1>
+<p>These are not goals of the TFL design, either to simplify the scope of the initial design (a.k.a. <a href="#out-of-scope-goals">Out-of-Scope Goals</a>), or because we believe some potential goal <em>should not</em> be supported (a.k.a. <a href="#anti-goals">Anti-goals</a>).</p>
+<h2 id="out-of-scope-goals"><a class="header" href="#out-of-scope-goals">Out-of-Scope Goals</a></h2>
+<p>While these desiderata may be common across the block‑chain consensus design space, they are not specific goals for the initial TFL design. Note that these may be goals for future protocol improvements.</p>
+<ul>
+<li>
+<p>Prioritizing minimal time-to-finality over other considerations (such as protocol simplicity, impact on existing use cases, or other goals above).</p>
+</li>
+<li>
+<p>In-protocol liquid staking derivatives.</p>
+</li>
+<li>
+<p>Maximizing the PoS staked-voter count ceiling. For example, <a href="https://github.com/tendermint/tendermint#research">Tendermint BFT</a> has a relatively low ceiling of ~hundreds of staked voters, whereas Ethereum’s <a href="https://ethereum.org/en/developers/docs/consensus-mechanisms/pos/gasper/">Gasper</a> supports hundreds of thousands of staked voters.</p>
+</li>
+<li>
+<p>Reducing energy usage. While this would presumably be a goal of a pure PoS transition, it likely cannot be achieved for hybrid PoW/PoS without loss of security.</p>
+</li>
+</ul>
+<h2 id="anti-goals"><a class="header" href="#anti-goals">Anti-Goals</a></h2>
+<p>Distinctly from <a href="#out-of-scope-goals">Out-of-Scope Goals</a> we track &quot;anti-goals&quot; which are potential goals that we <em>explicitly reject</em>, which are potential goals we aim to <em>not</em> support even in future protocol improvements.</p>
+<p>We currently have no defined anti-goals.</p>
+<h1 id="footnotes"><a class="header" href="#footnotes">Footnotes</a></h1>
+<div class="footnote-definition" id="req-ttf-30-min"><sup class="footnote-definition-label">1</sup>
+<p>This requirement comes from <a href="https://github.com/Electric-Coin-Company/tfl-book/issues/38">a request from a DEX developer</a>. While we have not yet surveyed DEX and Bridge designs, we're relying on this as a good starting point.</p>
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diff --git a/docs/diagrams/crosslink.png b/docs/diagrams/crosslink.png
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+       id="path1040-37-2-8-4-3"
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+    <path
+       style="fill:#77543b;stroke:#aa081d;stroke-width:0.265;stroke-dasharray:none;stroke-opacity:1;marker-start:url(#TriangleStart-6-4-1)"
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+       id="path1040-37-2-5"
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+    <text
+       xml:space="preserve"
+       style="font-size:3.175px;fill:#048108;fill-opacity:1;stroke:#048108;stroke-width:0.264999;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:1"
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+         x="49.426212"
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+    <text
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+         x="59.223118"
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diff --git a/docs/diagrams/pow-rollback.dot b/docs/diagrams/pow-rollback.dot
new file mode 100644
index 0000000..c1f75ac
--- /dev/null
+++ b/docs/diagrams/pow-rollback.dot
@@ -0,0 +1,38 @@
+digraph BlockHistory {
+    rankdir = RL
+    color = "lightgrey"
+
+    node [ shape=square ]
+
+    subgraph common_prefix {
+        e -> d
+        d -> c
+        c -> b
+        b -> a
+    }
+
+    subgraph cluster_rollback_branch {
+        label = "T=0"
+        node [ style = "dotted" ]
+
+        g0 [ label = "g" ]
+        f0 [ label = "f" ]
+
+        g0 -> f0
+    }
+
+    subgraph cluster_winning_branch {
+        label = "T=1"
+
+        h1 [ label = "h'" ]
+        g1 [ label = "g'" ]
+        f1 [ label = "f'" ]
+
+        h1 -> g1
+        g1 -> f1
+    }
+
+    f0 -> e
+    f1 -> e
+}
+
diff --git a/docs/diagrams/pow-single-branch.dot b/docs/diagrams/pow-single-branch.dot
new file mode 100644
index 0000000..0d2e614
--- /dev/null
+++ b/docs/diagrams/pow-single-branch.dot
@@ -0,0 +1,13 @@
+digraph BlockHistory {
+    rankdir = RL
+
+    node [ shape=square ]
+
+    g -> f
+    f -> e
+    e -> d
+    d -> c
+    c -> b
+    b -> a
+}
+
diff --git a/docs/diagrams/tf-starting-state.dot b/docs/diagrams/tf-starting-state.dot
new file mode 100644
index 0000000..205dcb4
--- /dev/null
+++ b/docs/diagrams/tf-starting-state.dot
@@ -0,0 +1,22 @@
+
+digraph BlockHistory {
+    rankdir = RL
+    color = "lightgrey"
+
+    subgraph cluster_pow {
+        label = "Unfinalized PoW Blocks"
+        node [ shape=square ]
+
+        f -> e
+        e -> d
+    }
+
+    subgraph cluster_final {
+        label = "Finalized Blocks"
+        node [ shape=Msquare ]
+
+        d -> c
+        c -> b
+        b -> a
+    }
+}
diff --git a/docs/diagrams/zcash-tech-tree-generic.dot b/docs/diagrams/zcash-tech-tree-generic.dot
new file mode 100644
index 0000000..08675d7
--- /dev/null
+++ b/docs/diagrams/zcash-tech-tree-generic.dot
@@ -0,0 +1,39 @@
+digraph ZcashTechTreeGeneric {
+    rankdir = RL
+    color = "lightgrey"
+
+    node [ shape=box ]
+
+    nu6 [ label = "NU6 Protocol" ]
+
+    a [ label = "Feature A"]
+    a -> nu6
+
+    b [ label = "Feature B"]
+    b -> a
+
+    c [ label = "Feature C"]
+    c -> nu6
+
+    d [ label = "Feature D"]
+    d -> c
+    d -> a
+
+    e [ label = "Feature E"]
+    e -> pos2
+    e -> d
+
+    pos1 [ label = "PoS Step 1"]
+    pos1 -> nu6
+
+    subgraph cluster_pos {
+        label = "PoS Transition"
+
+        pos2 [ label = "PoS Step 2"]
+        pos2 -> pos1
+
+        pos3 [ label = "PoS Step 3"]
+        pos3 -> pos2
+    }
+}
+
diff --git a/docs/diagrams/zcash-tech-tree-tfl-steps.dot b/docs/diagrams/zcash-tech-tree-tfl-steps.dot
new file mode 100644
index 0000000..a644d41
--- /dev/null
+++ b/docs/diagrams/zcash-tech-tree-tfl-steps.dot
@@ -0,0 +1,40 @@
+digraph ZcashTechTreeGeneric {
+    rankdir = RL
+    color = "lightgrey"
+
+    node [ shape=box ]
+
+    nu6 [ label = "NU6 Protocol" ]
+
+    a [ label = "Feature A"]
+    a -> nu6
+
+    b [ label = "Feature B"]
+    b -> a
+
+    c [ label = "Feature C"]
+    c -> nu6
+
+    d [ label = "Feature D"]
+    d -> c
+    d -> a
+
+    e [ label = "Feature E"]
+    e -> pos2
+    e -> d
+
+    pos1 [ label = "TFL v1"]
+    pos1 -> nu6
+
+    subgraph cluster_pos {
+        label = "PoS Transition"
+
+        pos2 [ label = "Pure PoS v1"]
+        pos2 -> pos1
+
+        pos3 [ label = "Pure PoS v2"]
+        pos3 -> pos2
+    }
+}
+
+
diff --git a/docs/elasticlunr.min.js b/docs/elasticlunr.min.js
new file mode 100644
index 0000000..94b20dd
--- /dev/null
+++ b/docs/elasticlunr.min.js
@@ -0,0 +1,10 @@
+/**
+ * elasticlunr - http://weixsong.github.io
+ * Lightweight full-text search engine in Javascript for browser search and offline search. - 0.9.5
+ *
+ * Copyright (C) 2017 Oliver Nightingale
+ * Copyright (C) 2017 Wei Song
+ * MIT Licensed
+ * @license
+ */
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+
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+      any Contribution intentionally submitted for inclusion in the Work
+      by You to the Licensor shall be under the terms and conditions of
+      this License, without any additional terms or conditions.
+      Notwithstanding the above, nothing herein shall supersede or modify
+      the terms of any separate license agreement you may have executed
+      with Licensor regarding such Contributions.
+
+   6. Trademarks. This License does not grant permission to use the trade
+      names, trademarks, service marks, or product names of the Licensor,
+      except as required for reasonable and customary use in describing the
+      origin of the Work and reproducing the content of the NOTICE file.
+
+   7. Disclaimer of Warranty. Unless required by applicable law or
+      agreed to in writing, Licensor provides the Work (and each
+      Contributor provides its Contributions) on an "AS IS" BASIS,
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+      appropriateness of using or redistributing the Work and assume any
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+
+   8. Limitation of Liability. In no event and under no legal theory,
+      whether in tort (including negligence), contract, or otherwise,
+      unless required by applicable law (such as deliberate and grossly
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+
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+
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+      To apply the Apache License to your work, attach the following
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+   Copyright [yyyy] [name of copyright owner]
+
+   Licensed under the Apache License, Version 2.0 (the "License");
+   you may not use this file except in compliance with the License.
+   You may obtain a copy of the License at
+
+       http://www.apache.org/licenses/LICENSE-2.0
+
+   Unless required by applicable law or agreed to in writing, software
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diff --git a/docs/fonts/SOURCE-CODE-PRO-LICENSE.txt b/docs/fonts/SOURCE-CODE-PRO-LICENSE.txt
new file mode 100644
index 0000000..366206f
--- /dev/null
+++ b/docs/fonts/SOURCE-CODE-PRO-LICENSE.txt
@@ -0,0 +1,93 @@
+Copyright 2010, 2012 Adobe Systems Incorporated (http://www.adobe.com/), with Reserved Font Name 'Source'. All Rights Reserved. Source is a trademark of Adobe Systems Incorporated in the United States and/or other countries.
+
+This Font Software is licensed under the SIL Open Font License, Version 1.1.
+This license is copied below, and is also available with a FAQ at:
+http://scripts.sil.org/OFL
+
+
+-----------------------------------------------------------
+SIL OPEN FONT LICENSE Version 1.1 - 26 February 2007
+-----------------------------------------------------------
+
+PREAMBLE
+The goals of the Open Font License (OFL) are to stimulate worldwide
+development of collaborative font projects, to support the font creation
+efforts of academic and linguistic communities, and to provide a free and
+open framework in which fonts may be shared and improved in partnership
+with others.
+
+The OFL allows the licensed fonts to be used, studied, modified and
+redistributed freely as long as they are not sold by themselves. The
+fonts, including any derivative works, can be bundled, embedded, 
+redistributed and/or sold with any software provided that any reserved
+names are not used by derivative works. The fonts and derivatives,
+however, cannot be released under any other type of license. The
+requirement for fonts to remain under this license does not apply
+to any document created using the fonts or their derivatives.
+
+DEFINITIONS
+"Font Software" refers to the set of files released by the Copyright
+Holder(s) under this license and clearly marked as such. This may
+include source files, build scripts and documentation.
+
+"Reserved Font Name" refers to any names specified as such after the
+copyright statement(s).
+
+"Original Version" refers to the collection of Font Software components as
+distributed by the Copyright Holder(s).
+
+"Modified Version" refers to any derivative made by adding to, deleting,
+or substituting -- in part or in whole -- any of the components of the
+Original Version, by changing formats or by porting the Font Software to a
+new environment.
+
+"Author" refers to any designer, engineer, programmer, technical
+writer or other person who contributed to the Font Software.
+
+PERMISSION & CONDITIONS
+Permission is hereby granted, free of charge, to any person obtaining
+a copy of the Font Software, to use, study, copy, merge, embed, modify,
+redistribute, and sell modified and unmodified copies of the Font
+Software, subject to the following conditions:
+
+1) Neither the Font Software nor any of its individual components,
+in Original or Modified Versions, may be sold by itself.
+
+2) Original or Modified Versions of the Font Software may be bundled,
+redistributed and/or sold with any software, provided that each copy
+contains the above copyright notice and this license. These can be
+included either as stand-alone text files, human-readable headers or
+in the appropriate machine-readable metadata fields within text or
+binary files as long as those fields can be easily viewed by the user.
+
+3) No Modified Version of the Font Software may use the Reserved Font
+Name(s) unless explicit written permission is granted by the corresponding
+Copyright Holder. This restriction only applies to the primary font name as
+presented to the users.
+
+4) The name(s) of the Copyright Holder(s) or the Author(s) of the Font
+Software shall not be used to promote, endorse or advertise any
+Modified Version, except to acknowledge the contribution(s) of the
+Copyright Holder(s) and the Author(s) or with their explicit written
+permission.
+
+5) The Font Software, modified or unmodified, in part or in whole,
+must be distributed entirely under this license, and must not be
+distributed under any other license. The requirement for fonts to
+remain under this license does not apply to any document created
+using the Font Software.
+
+TERMINATION
+This license becomes null and void if any of the above conditions are
+not met.
+
+DISCLAIMER
+THE FONT SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
+EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO ANY WARRANTIES OF
+MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT
+OF COPYRIGHT, PATENT, TRADEMARK, OR OTHER RIGHT. IN NO EVENT SHALL THE
+COPYRIGHT HOLDER BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY,
+INCLUDING ANY GENERAL, SPECIAL, INDIRECT, INCIDENTAL, OR CONSEQUENTIAL
+DAMAGES, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
+FROM, OUT OF THE USE OR INABILITY TO USE THE FONT SOFTWARE OR FROM
+OTHER DEALINGS IN THE FONT SOFTWARE.
diff --git a/docs/fonts/fonts.css b/docs/fonts/fonts.css
new file mode 100644
index 0000000..858efa5
--- /dev/null
+++ b/docs/fonts/fonts.css
@@ -0,0 +1,100 @@
+/* Open Sans is licensed under the Apache License, Version 2.0. See http://www.apache.org/licenses/LICENSE-2.0 */
+/* Source Code Pro is under the Open Font License. See https://scripts.sil.org/cms/scripts/page.php?site_id=nrsi&id=OFL */
+
+/* open-sans-300 - latin_vietnamese_latin-ext_greek-ext_greek_cyrillic-ext_cyrillic */
+@font-face {
+  font-family: 'Open Sans';
+  font-style: normal;
+  font-weight: 300;
+  src: local('Open Sans Light'), local('OpenSans-Light'),
+       url('open-sans-v17-all-charsets-300.woff2') format('woff2');
+}
+
+/* open-sans-300italic - latin_vietnamese_latin-ext_greek-ext_greek_cyrillic-ext_cyrillic */
+@font-face {
+  font-family: 'Open Sans';
+  font-style: italic;
+  font-weight: 300;
+  src: local('Open Sans Light Italic'), local('OpenSans-LightItalic'),
+       url('open-sans-v17-all-charsets-300italic.woff2') format('woff2');
+}
+
+/* open-sans-regular - latin_vietnamese_latin-ext_greek-ext_greek_cyrillic-ext_cyrillic */
+@font-face {
+  font-family: 'Open Sans';
+  font-style: normal;
+  font-weight: 400;
+  src: local('Open Sans Regular'), local('OpenSans-Regular'),
+       url('open-sans-v17-all-charsets-regular.woff2') format('woff2');
+}
+
+/* open-sans-italic - latin_vietnamese_latin-ext_greek-ext_greek_cyrillic-ext_cyrillic */
+@font-face {
+  font-family: 'Open Sans';
+  font-style: italic;
+  font-weight: 400;
+  src: local('Open Sans Italic'), local('OpenSans-Italic'),
+       url('open-sans-v17-all-charsets-italic.woff2') format('woff2');
+}
+
+/* open-sans-600 - latin_vietnamese_latin-ext_greek-ext_greek_cyrillic-ext_cyrillic */
+@font-face {
+  font-family: 'Open Sans';
+  font-style: normal;
+  font-weight: 600;
+  src: local('Open Sans SemiBold'), local('OpenSans-SemiBold'),
+       url('open-sans-v17-all-charsets-600.woff2') format('woff2');
+}
+
+/* open-sans-600italic - latin_vietnamese_latin-ext_greek-ext_greek_cyrillic-ext_cyrillic */
+@font-face {
+  font-family: 'Open Sans';
+  font-style: italic;
+  font-weight: 600;
+  src: local('Open Sans SemiBold Italic'), local('OpenSans-SemiBoldItalic'),
+       url('open-sans-v17-all-charsets-600italic.woff2') format('woff2');
+}
+
+/* open-sans-700 - latin_vietnamese_latin-ext_greek-ext_greek_cyrillic-ext_cyrillic */
+@font-face {
+  font-family: 'Open Sans';
+  font-style: normal;
+  font-weight: 700;
+  src: local('Open Sans Bold'), local('OpenSans-Bold'),
+       url('open-sans-v17-all-charsets-700.woff2') format('woff2');
+}
+
+/* open-sans-700italic - latin_vietnamese_latin-ext_greek-ext_greek_cyrillic-ext_cyrillic */
+@font-face {
+  font-family: 'Open Sans';
+  font-style: italic;
+  font-weight: 700;
+  src: local('Open Sans Bold Italic'), local('OpenSans-BoldItalic'),
+       url('open-sans-v17-all-charsets-700italic.woff2') format('woff2');
+}
+
+/* open-sans-800 - latin_vietnamese_latin-ext_greek-ext_greek_cyrillic-ext_cyrillic */
+@font-face {
+  font-family: 'Open Sans';
+  font-style: normal;
+  font-weight: 800;
+  src: local('Open Sans ExtraBold'), local('OpenSans-ExtraBold'),
+       url('open-sans-v17-all-charsets-800.woff2') format('woff2');
+}
+
+/* open-sans-800italic - latin_vietnamese_latin-ext_greek-ext_greek_cyrillic-ext_cyrillic */
+@font-face {
+  font-family: 'Open Sans';
+  font-style: italic;
+  font-weight: 800;
+  src: local('Open Sans ExtraBold Italic'), local('OpenSans-ExtraBoldItalic'),
+       url('open-sans-v17-all-charsets-800italic.woff2') format('woff2');
+}
+
+/* source-code-pro-500 - latin_vietnamese_latin-ext_greek_cyrillic-ext_cyrillic */
+@font-face {
+  font-family: 'Source Code Pro';
+  font-style: normal;
+  font-weight: 500;
+  src: url('source-code-pro-v11-all-charsets-500.woff2') format('woff2');
+}
diff --git a/docs/fonts/open-sans-v17-all-charsets-300.woff2 b/docs/fonts/open-sans-v17-all-charsets-300.woff2
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diff --git a/docs/highlight.css b/docs/highlight.css
new file mode 100644
index 0000000..ba57b82
--- /dev/null
+++ b/docs/highlight.css
@@ -0,0 +1,82 @@
+/*
+ * An increased contrast highlighting scheme loosely based on the
+ * "Base16 Atelier Dune Light" theme by Bram de Haan
+ * (http://atelierbram.github.io/syntax-highlighting/atelier-schemes/dune)
+ * Original Base16 color scheme by Chris Kempson
+ * (https://github.com/chriskempson/base16)
+ */
+
+/* Comment */
+.hljs-comment,
+.hljs-quote {
+  color: #575757;
+}
+
+/* Red */
+.hljs-variable,
+.hljs-template-variable,
+.hljs-attribute,
+.hljs-tag,
+.hljs-name,
+.hljs-regexp,
+.hljs-link,
+.hljs-name,
+.hljs-selector-id,
+.hljs-selector-class {
+  color: #d70025;
+}
+
+/* Orange */
+.hljs-number,
+.hljs-meta,
+.hljs-built_in,
+.hljs-builtin-name,
+.hljs-literal,
+.hljs-type,
+.hljs-params {
+  color: #b21e00;
+}
+
+/* Green */
+.hljs-string,
+.hljs-symbol,
+.hljs-bullet {
+  color: #008200;
+}
+
+/* Blue */
+.hljs-title,
+.hljs-section {
+  color: #0030f2;
+}
+
+/* Purple */
+.hljs-keyword,
+.hljs-selector-tag {
+  color: #9d00ec;
+}
+
+.hljs {
+  display: block;
+  overflow-x: auto;
+  background: #f6f7f6;
+  color: #000;
+}
+
+.hljs-emphasis {
+  font-style: italic;
+}
+
+.hljs-strong {
+  font-weight: bold;
+}
+
+.hljs-addition {
+  color: #22863a;
+  background-color: #f0fff4;
+}
+
+.hljs-deletion {
+  color: #b31d28;
+  background-color: #ffeef0;
+}
diff --git a/docs/highlight.js b/docs/highlight.js
new file mode 100644
index 0000000..180385b
--- /dev/null
+++ b/docs/highlight.js
@@ -0,0 +1,6 @@
+/*
+  Highlight.js 10.1.1 (93fd0d73)
+  License: BSD-3-Clause
+  Copyright (c) 2006-2020, Ivan Sagalaev
+*/
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index 0000000..fe755ce
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+<html lang="en" class="light" dir="ltr">
+    <head>
+        <!-- Book generated using mdBook -->
+        <meta charset="UTF-8">
+        <title>Introduction - Zcash Trailing Finality Layer - v0.1.0</title>
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+                <ol class="chapter"><li class="chapter-item expanded "><a href="introduction.html" class="active"><strong aria-hidden="true">1.</strong> Introduction</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="introduction/a-path-to-pos-zcash.html"><strong aria-hidden="true">1.1.</strong> A Path to Proof-of-Stake Zcash</a></li><li class="chapter-item expanded "><a href="introduction/trailing-finality-layer-in-a-nutshell.html"><strong aria-hidden="true">1.2.</strong> Trailing Finality Layer in a Nutshell</a></li><li class="chapter-item expanded "><a href="introduction/status-and-next-steps.html"><strong aria-hidden="true">1.3.</strong> Status and Next Steps</a></li><li class="chapter-item expanded "><a href="introduction/motivating-finality.html"><strong aria-hidden="true">1.4.</strong> Motivating Finality</a></li><li class="chapter-item expanded "><a href="introduction/get-involved.html"><strong aria-hidden="true">1.5.</strong> Get Involved</a></li></ol></li><li class="chapter-item expanded "><a href="terminology.html"><strong aria-hidden="true">2.</strong> Terminology</a></li><li class="chapter-item expanded "><a href="design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/design-at-a-glance.html"><strong aria-hidden="true">3.1.</strong> Design at a Glance</a></li><li class="chapter-item expanded "><a href="design/goals.html"><strong aria-hidden="true">3.2.</strong> Design Goals</a></li><li class="chapter-item expanded "><a href="design/crosslink.html"><strong aria-hidden="true">3.3.</strong> Crosslink</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/crosslink/the-arguments-for-bounded-availability-and-finality-overrides.html"><strong aria-hidden="true">3.3.1.</strong> The Arguments for Bounded Availability and Finality Overrides</a></li><li class="chapter-item expanded "><a href="design/crosslink/notes-on-snap-and-chat.html"><strong aria-hidden="true">3.3.2.</strong> Notes On Snap‑and‑Chat</a></li><li class="chapter-item expanded "><a href="design/crosslink/construction.html"><strong aria-hidden="true">3.3.3.</strong> The Crosslink 2 Construction</a></li><li class="chapter-item expanded "><a href="design/crosslink/security-analysis.html"><strong aria-hidden="true">3.3.4.</strong> Security Analysis of Crosslink 2</a></li><li class="chapter-item expanded "><a href="design/crosslink/questions.html"><strong aria-hidden="true">3.3.5.</strong> Questions About Crosslink</a></li><li class="chapter-item expanded "><a href="design/crosslink/potential-changes.html"><strong aria-hidden="true">3.3.6.</strong> Potential Changes to Crosslink</a></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">3.4.</strong> Model Code Architecture</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">3.5.</strong> Network Architecture</div></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">4.</strong> Protocol Specification</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">4.1.</strong> Abstract Protocol</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">4.1.1.</strong> Token Dynamics</div></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">4.2.</strong> Concrete Protocol</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">4.2.1.</strong> Token Dynamics</div></li></ol></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.</strong> Security</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.</strong> Abstract Analysis</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.1.</strong> Ebb-and-Flow analysis</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.2.</strong> Subprotocol Compromise Analysis</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.3.</strong> Model Code Architecture Analysis</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.4.</strong> Network Architecture Analysis</div></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.2.</strong> Concrete Analysis</div></li></ol></li><li class="chapter-item expanded "><a href="references.html"><strong aria-hidden="true">6.</strong> References</a></li><li class="chapter-item expanded "><a href="version-history.html"><strong aria-hidden="true">7.</strong> Version History</a></li></ol>
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+<h1 id="a-trailing-finality-layer-for-zcash"><a class="header" href="#a-trailing-finality-layer-for-zcash">A Trailing Finality Layer for Zcash</a></h1>
+<p>This book introduces and specifies a <a href="./terminology.html#definition-tfl"><em>Trailing Finality Layer</em></a> for the Zcash network. This is version <a href="./version-history.html#0_1_0-introducing-crosslink">0.1.0</a> of the book.</p>
+<p>This design augments the existing Zcash Proof‑of‑Work (<a href="./terminology.html#definition-pow">PoW</a>) network with a new consensus layer which provides <a href="./terminology.html#definition-trailing-finality"><em>trailing finality</em></a>. This layer enables transactions included via PoW to become <a href="./terminology.html#definition-final"><em>final</em></a> which assures that they cannot be reverted by the protocol. This enables safer and simpler wallets and other infrastructure, and aids trust-minimized cross-chain bridges. This consensus layer uses <a href="./terminology.html#definition-pos"><em>Proof-of-Stake</em></a> consensus, and enables ZEC holders to earn protocol rewards for contributing to the security of the Zcash network. By integrating a PoS layer with the current PoW Zcash protocol, this design specifies a <a href="./terminology.html#definition-hybrid-consensus"><em>hybrid consensus</em></a> protocol dubbed <a href="./terminology.html#definition-pow-tfl">PoW+TFL</a>.</p>
+<p>The rest of this introductory chapter is aimed at a general audience interested in the context of this proposal within Zcash development, status and next steps, motivations, a primer on finality, and tips to get involved.</p>
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+                <ol class="chapter"><li class="chapter-item expanded "><a href="introduction.html" class="active"><strong aria-hidden="true">1.</strong> Introduction</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="introduction/a-path-to-pos-zcash.html"><strong aria-hidden="true">1.1.</strong> A Path to Proof-of-Stake Zcash</a></li><li class="chapter-item expanded "><a href="introduction/trailing-finality-layer-in-a-nutshell.html"><strong aria-hidden="true">1.2.</strong> Trailing Finality Layer in a Nutshell</a></li><li class="chapter-item expanded "><a href="introduction/status-and-next-steps.html"><strong aria-hidden="true">1.3.</strong> Status and Next Steps</a></li><li class="chapter-item expanded "><a href="introduction/motivating-finality.html"><strong aria-hidden="true">1.4.</strong> Motivating Finality</a></li><li class="chapter-item expanded "><a href="introduction/get-involved.html"><strong aria-hidden="true">1.5.</strong> Get Involved</a></li></ol></li><li class="chapter-item expanded "><a href="terminology.html"><strong aria-hidden="true">2.</strong> Terminology</a></li><li class="chapter-item expanded "><a href="design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/design-at-a-glance.html"><strong aria-hidden="true">3.1.</strong> Design at a Glance</a></li><li class="chapter-item expanded "><a href="design/goals.html"><strong aria-hidden="true">3.2.</strong> Design Goals</a></li><li class="chapter-item expanded "><a href="design/crosslink.html"><strong aria-hidden="true">3.3.</strong> Crosslink</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/crosslink/the-arguments-for-bounded-availability-and-finality-overrides.html"><strong aria-hidden="true">3.3.1.</strong> The Arguments for Bounded Availability and Finality Overrides</a></li><li class="chapter-item expanded "><a href="design/crosslink/notes-on-snap-and-chat.html"><strong aria-hidden="true">3.3.2.</strong> Notes On Snap‑and‑Chat</a></li><li class="chapter-item expanded "><a href="design/crosslink/construction.html"><strong aria-hidden="true">3.3.3.</strong> The Crosslink 2 Construction</a></li><li class="chapter-item expanded "><a href="design/crosslink/security-analysis.html"><strong aria-hidden="true">3.3.4.</strong> Security Analysis of Crosslink 2</a></li><li class="chapter-item expanded "><a href="design/crosslink/questions.html"><strong aria-hidden="true">3.3.5.</strong> Questions About Crosslink</a></li><li class="chapter-item expanded "><a href="design/crosslink/potential-changes.html"><strong aria-hidden="true">3.3.6.</strong> Potential Changes to Crosslink</a></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">3.4.</strong> Model Code Architecture</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">3.5.</strong> Network Architecture</div></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">4.</strong> Protocol Specification</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">4.1.</strong> Abstract Protocol</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">4.1.1.</strong> Token Dynamics</div></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">4.2.</strong> Concrete Protocol</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">4.2.1.</strong> Token Dynamics</div></li></ol></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.</strong> Security</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.</strong> Abstract Analysis</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.1.</strong> Ebb-and-Flow analysis</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.2.</strong> Subprotocol Compromise Analysis</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.3.</strong> Model Code Architecture Analysis</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.4.</strong> Network Architecture Analysis</div></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.2.</strong> Concrete Analysis</div></li></ol></li><li class="chapter-item expanded "><a href="references.html"><strong aria-hidden="true">6.</strong> References</a></li><li class="chapter-item expanded "><a href="version-history.html"><strong aria-hidden="true">7.</strong> Version History</a></li></ol>
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+<h1 id="a-trailing-finality-layer-for-zcash"><a class="header" href="#a-trailing-finality-layer-for-zcash">A Trailing Finality Layer for Zcash</a></h1>
+<p>This book introduces and specifies a <a href="./terminology.html#definition-tfl"><em>Trailing Finality Layer</em></a> for the Zcash network. This is version <a href="./version-history.html#0_1_0-introducing-crosslink">0.1.0</a> of the book.</p>
+<p>This design augments the existing Zcash Proof‑of‑Work (<a href="./terminology.html#definition-pow">PoW</a>) network with a new consensus layer which provides <a href="./terminology.html#definition-trailing-finality"><em>trailing finality</em></a>. This layer enables transactions included via PoW to become <a href="./terminology.html#definition-final"><em>final</em></a> which assures that they cannot be reverted by the protocol. This enables safer and simpler wallets and other infrastructure, and aids trust-minimized cross-chain bridges. This consensus layer uses <a href="./terminology.html#definition-pos"><em>Proof-of-Stake</em></a> consensus, and enables ZEC holders to earn protocol rewards for contributing to the security of the Zcash network. By integrating a PoS layer with the current PoW Zcash protocol, this design specifies a <a href="./terminology.html#definition-hybrid-consensus"><em>hybrid consensus</em></a> protocol dubbed <a href="./terminology.html#definition-pow-tfl">PoW+TFL</a>.</p>
+<p>The rest of this introductory chapter is aimed at a general audience interested in the context of this proposal within Zcash development, status and next steps, motivations, a primer on finality, and tips to get involved.</p>
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+                <ol class="chapter"><li class="chapter-item expanded "><a href="../introduction.html"><strong aria-hidden="true">1.</strong> Introduction</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../introduction/a-path-to-pos-zcash.html" class="active"><strong aria-hidden="true">1.1.</strong> A Path to Proof-of-Stake Zcash</a></li><li class="chapter-item expanded "><a href="../introduction/trailing-finality-layer-in-a-nutshell.html"><strong aria-hidden="true">1.2.</strong> Trailing Finality Layer in a Nutshell</a></li><li class="chapter-item expanded "><a href="../introduction/status-and-next-steps.html"><strong aria-hidden="true">1.3.</strong> Status and Next Steps</a></li><li class="chapter-item expanded "><a href="../introduction/motivating-finality.html"><strong aria-hidden="true">1.4.</strong> Motivating Finality</a></li><li class="chapter-item expanded "><a href="../introduction/get-involved.html"><strong aria-hidden="true">1.5.</strong> Get Involved</a></li></ol></li><li class="chapter-item expanded "><a href="../terminology.html"><strong aria-hidden="true">2.</strong> Terminology</a></li><li class="chapter-item expanded "><a href="../design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/design-at-a-glance.html"><strong aria-hidden="true">3.1.</strong> Design at a Glance</a></li><li class="chapter-item expanded "><a href="../design/goals.html"><strong aria-hidden="true">3.2.</strong> Design Goals</a></li><li class="chapter-item expanded "><a href="../design/crosslink.html"><strong aria-hidden="true">3.3.</strong> Crosslink</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/crosslink/the-arguments-for-bounded-availability-and-finality-overrides.html"><strong aria-hidden="true">3.3.1.</strong> The Arguments for Bounded Availability and Finality Overrides</a></li><li class="chapter-item expanded "><a href="../design/crosslink/notes-on-snap-and-chat.html"><strong aria-hidden="true">3.3.2.</strong> Notes On Snap‑and‑Chat</a></li><li class="chapter-item expanded "><a href="../design/crosslink/construction.html"><strong aria-hidden="true">3.3.3.</strong> The Crosslink 2 Construction</a></li><li class="chapter-item expanded "><a href="../design/crosslink/security-analysis.html"><strong aria-hidden="true">3.3.4.</strong> Security Analysis of Crosslink 2</a></li><li class="chapter-item expanded "><a href="../design/crosslink/questions.html"><strong aria-hidden="true">3.3.5.</strong> Questions About Crosslink</a></li><li class="chapter-item expanded "><a href="../design/crosslink/potential-changes.html"><strong aria-hidden="true">3.3.6.</strong> Potential Changes to Crosslink</a></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">3.4.</strong> Model Code Architecture</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">3.5.</strong> Network Architecture</div></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">4.</strong> Protocol Specification</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">4.1.</strong> Abstract Protocol</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">4.1.1.</strong> Token Dynamics</div></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">4.2.</strong> Concrete Protocol</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">4.2.1.</strong> Token Dynamics</div></li></ol></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.</strong> Security</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.</strong> Abstract Analysis</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.1.</strong> Ebb-and-Flow analysis</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.2.</strong> Subprotocol Compromise Analysis</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.3.</strong> Model Code Architecture Analysis</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.4.</strong> Network Architecture Analysis</div></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.2.</strong> Concrete Analysis</div></li></ol></li><li class="chapter-item expanded "><a href="../references.html"><strong aria-hidden="true">6.</strong> References</a></li><li class="chapter-item expanded "><a href="../version-history.html"><strong aria-hidden="true">7.</strong> Version History</a></li></ol>
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+<h1 id="a-path-to-proof-of-stake-zcash"><a class="header" href="#a-path-to-proof-of-stake-zcash">A Path to Proof-of-Stake Zcash</a></h1>
+<p>The <a href="../terminology.html#definition-tfl">TFL</a> design provides a possible first step in transitioning <a href="https://z.cash">Zcash</a> to a <a href="../terminology.html#definition-pos">PoS</a> protocol. Here we describe how a transition to PoS relates to “the Zcash roadmap” and how TFL fits into one approach to a PoS transition.</p>
+<h2 id="the-zcash-tech-tree"><a class="header" href="#the-zcash-tech-tree">The Zcash Tech-Tree</a></h2>
+<p>There are multiple developer orgs working on different proposed features for Zcash. Some of these involve multiple large distinct upgrade steps, and several of these steps depend on other such steps. This could be represented as a directed acyclic graph. We have begun referring to this space of possible future improvements as the <em>Zcash Tech-Tree</em>, taking inspiration from an analogous concept in gaming.<sup class="footnote-reference"><a href="#tech-tree-history">1</a></sup></p>
+<p>We envision a <em>proof-of-stake transition path</em> as one of the potential paths within this tech-tree which is the primary protocol focus of this proposal. An example visualization of this Zcash Tech-Tree might look like this:</p>
+<p><img src="a_path_to_proof_of_stake_zcash_0.generated.svg" alt="" /></p>
+<h2 id="a-proof-of-stake-transition-path"><a class="header" href="#a-proof-of-stake-transition-path">A Proof-of-Stake Transition Path</a></h2>
+<p>Given that context, we envision a path within the Zcash Tech-Tree for transitioning Zcash to PoS. At the top level we propose that this path should include at least two major milestones:</p>
+<ol>
+<li>Transitioning from Zcash <a href="../terminology.html#definition-nu5nu6">NU5/NU6</a> <a href="../terminology.html#definition-pow">PoW</a> consensus to a PoW/PoS <a href="../terminology.html#definition-hybrid-consensus">hybrid consensus</a> protocol dubbed <a href="../terminology.html#definition-pow-tfl">PoW+TFL</a>.</li>
+<li>Transitioning from <a href="../terminology.html#definition-pow-tfl">PoW+TFL</a> to pure <a href="../terminology.html#definition-pow-tfl">PoS</a> protocol.</li>
+</ol>
+<p>After this transition to pure PoS, there are likely to be future improvements to the PoS protocol, or the consensus protocol more generally. This TFL book focuses almost exclusively on the first step in this sequence.</p>
+<p>Our primary motivation for proposing (at least) two steps is to minimize disruption to usability, safety, security, and the ecosystem during each step.</p>
+<p>This book primarily focuses this first step: the transition to <a href="../terminology.html#definition-pow-tfl">PoW+TFL</a>. To understand the specific goals for that, see <a href="../design/goals.html">Design Goals</a>.</p>
+<p>With this approach, the Zcash Tech Tree with the <a href="../terminology.html#definition-tfl">TFL</a> approach might look something like this:</p>
+<p><img src="a_path_to_proof_of_stake_zcash_1.generated.svg" alt="" /></p>
+<h2 id="why-two-steps"><a class="header" href="#why-two-steps">Why Two Steps?</a></h2>
+<p>One question we've gotten in proposing this approach is why take a two-step process with an intermediate <a href="../terminology.html#definition-hybrid-consensus">hybrid consensus</a> protocol, rather than a single transition directly to a <a href="../terminology.html#definition-pos">PoS</a> protocol?</p>
+<p>Here’s how we think about those trade-offs:</p>
+<h3 id="considering-single-transition-vs-hybrid-multi-step"><a class="header" href="#considering-single-transition-vs-hybrid-multi-step">Considering Single Transition (vs Hybrid Multi-Step)</a></h3>
+<h4 id="pros"><a class="header" href="#pros">Pros</a></h4>
+<ul>
+<li>We already understand the current PoW protocol well, and if we transition to an existing proven PoS protocol, then we could skip the complexity of an intermediate hybrid stage.</li>
+<li>The node implementation might be simpler.</li>
+<li>Explaining to people what is happening might be simpler. Something like “Zcash has been PoW since it launched, but on DATE (e.g. at block height <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span>) it will switch to PoS.”</li>
+<li>Given that the issuance in a given time period is bounded by the supply curve, the full amount that was previously allocated to mining rewards becomes immediately available for staking rewards at the switch-over, rather than having to share this amount between mining and staking during the hybrid stage.</li>
+</ul>
+<h4 id="cons"><a class="header" href="#cons">Cons</a></h4>
+<ul>
+<li>If there is any unforeseen show‑stopping problem in the new protocol or the transition process, we’d have to react to a network‑wide issue.</li>
+<li>It may be more likely to cause ecosystem disruption; unforeseen differences between PoW and PoS might cause various kinds of snags or papercuts throughout the ecosystem, and these would all pop up around the same time, which may lead to a loss of confidence/retention/adoption or at the very least inconvenience many users for some time.</li>
+<li>Losing miners: since the transition would be all at once, we may lose some number of miners, who are participants and users in the ecosystem. Miners may leave prior to the transition in order to take care of their own needs. If there is some showstopper in the transition, one possible short‑term mitigation would be to fall back on PoW which is well known, but if we’ve lost most miners, that may no longer be viable.</li>
+</ul>
+<h3 id="considering-hybrid-multi-step-approach-vs-single-step"><a class="header" href="#considering-hybrid-multi-step-approach-vs-single-step">Considering Hybrid Multi-Step approach (vs Single-Step):</a></h3>
+<p>Note: TFL is one instance of a multi‑step approach.</p>
+<h4 id="pros-1"><a class="header" href="#pros-1">Pros</a></h4>
+<ul>
+<li>We can hopefully be less disruptive across the ecosystem so that there are fewer snags and disruptions with each step.</li>
+<li>If there is a show-stopping flaw in any step, the fall‑back possibility seems more plausible. For example, if there is a show-stopper when transitioning from PoW to <a href="../terminology.html#definition-pow-tfl">PoW+TFL</a>, falling back to pure PoW seems more feasible, since both protocols rely on mainnet PoW infrastructure, so those participants will be present in either case.</li>
+<li>Retaining miners during a hybrid phase: while it is true that a hybrid protocol will lower miner revenue (since we aim to maintain the issuance schedule constraints), there is also more possibility and likelihood of keeping some of these users engaged. For example, they may begin participating in staking services (either as delegators or as infrastructure operators). If that is successful, then they’re also more likely to remain engaged in the subsequent transition to pure PoS.</li>
+<li>This general approach was demonstrated successfully by Ethereum, which is the largest or second largest cryptocurrency network for several important metrics (e.g. market cap, fees paid, user and developer activity, …). So we know this can be done well without disruptions.</li>
+</ul>
+<h4 id="cons-1"><a class="header" href="#cons-1">Cons</a></h4>
+<ul>
+<li>The intermediate hybrid step will be a more novel and less well understood protocol. (It will necessarily be fairly different from Ethereum’s Beacon chain era.)</li>
+<li>Consensus nodes will be more complex, involving logic for both sub‑protocols as well as their integration. (Ideally this complexity can be modularized so that the nodes are easier to maintain and improve.)</li>
+<li>This may be more complicated to explain to current and potential new users. Something like “Zcash launched as PoW, and on $DATE (block height X) it will transition to a hybrid system, then later to a pure PoS system.”</li>
+<li>The available issuance must be shared between mining and staking rewards during the hybrid stage. The security of the PoW layer and of the PoS layer during this stage is partially dependent on the funds allocated to issuance for each protocol, and it is not yet clear to what extent splitting rewards would affect overall security.</li>
+</ul>
+<h1 id="footnotes"><a class="header" href="#footnotes">Footnotes</a></h1>
+<div class="footnote-definition" id="tech-tree-history"><sup class="footnote-definition-label">1</sup>
+<p>See <a href="https://en.wikipedia.org/wiki/Technology_tree#History">Wikipedia’s Technology Tree - History</a> section for details.</p>
+</div>
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+        <nav id="sidebar" class="sidebar" aria-label="Table of contents">
+            <div class="sidebar-scrollbox">
+                <ol class="chapter"><li class="chapter-item expanded "><a href="../introduction.html"><strong aria-hidden="true">1.</strong> Introduction</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../introduction/a-path-to-pos-zcash.html"><strong aria-hidden="true">1.1.</strong> A Path to Proof-of-Stake Zcash</a></li><li class="chapter-item expanded "><a href="../introduction/trailing-finality-layer-in-a-nutshell.html"><strong aria-hidden="true">1.2.</strong> Trailing Finality Layer in a Nutshell</a></li><li class="chapter-item expanded "><a href="../introduction/status-and-next-steps.html"><strong aria-hidden="true">1.3.</strong> Status and Next Steps</a></li><li class="chapter-item expanded "><a href="../introduction/motivating-finality.html"><strong aria-hidden="true">1.4.</strong> Motivating Finality</a></li><li class="chapter-item expanded "><a href="../introduction/get-involved.html" class="active"><strong aria-hidden="true">1.5.</strong> Get Involved</a></li></ol></li><li class="chapter-item expanded "><a href="../terminology.html"><strong aria-hidden="true">2.</strong> Terminology</a></li><li class="chapter-item expanded "><a href="../design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/design-at-a-glance.html"><strong aria-hidden="true">3.1.</strong> Design at a Glance</a></li><li class="chapter-item expanded "><a href="../design/goals.html"><strong aria-hidden="true">3.2.</strong> Design Goals</a></li><li class="chapter-item expanded "><a href="../design/crosslink.html"><strong aria-hidden="true">3.3.</strong> Crosslink</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/crosslink/the-arguments-for-bounded-availability-and-finality-overrides.html"><strong aria-hidden="true">3.3.1.</strong> The Arguments for Bounded Availability and Finality Overrides</a></li><li class="chapter-item expanded "><a href="../design/crosslink/notes-on-snap-and-chat.html"><strong aria-hidden="true">3.3.2.</strong> Notes On Snap‑and‑Chat</a></li><li class="chapter-item expanded "><a href="../design/crosslink/construction.html"><strong aria-hidden="true">3.3.3.</strong> The Crosslink 2 Construction</a></li><li class="chapter-item expanded "><a href="../design/crosslink/security-analysis.html"><strong aria-hidden="true">3.3.4.</strong> Security Analysis of Crosslink 2</a></li><li class="chapter-item expanded "><a href="../design/crosslink/questions.html"><strong aria-hidden="true">3.3.5.</strong> Questions About Crosslink</a></li><li class="chapter-item expanded "><a href="../design/crosslink/potential-changes.html"><strong aria-hidden="true">3.3.6.</strong> Potential Changes to Crosslink</a></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">3.4.</strong> Model Code Architecture</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">3.5.</strong> Network Architecture</div></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">4.</strong> Protocol Specification</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">4.1.</strong> Abstract Protocol</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">4.1.1.</strong> Token Dynamics</div></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">4.2.</strong> Concrete Protocol</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">4.2.1.</strong> Token Dynamics</div></li></ol></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.</strong> Security</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.</strong> Abstract Analysis</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.1.</strong> Ebb-and-Flow analysis</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.2.</strong> Subprotocol Compromise Analysis</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.3.</strong> Model Code Architecture Analysis</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.4.</strong> Network Architecture Analysis</div></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.2.</strong> Concrete Analysis</div></li></ol></li><li class="chapter-item expanded "><a href="../references.html"><strong aria-hidden="true">6.</strong> References</a></li><li class="chapter-item expanded "><a href="../version-history.html"><strong aria-hidden="true">7.</strong> Version History</a></li></ol>
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+<h1 id="get-involved"><a class="header" href="#get-involved">Get Involved</a></h1>
+<p>We welcome contributions!</p>
+<p>There are a variety of ways to contribute to this project:</p>
+<h2 id="github"><a class="header" href="#github">GitHub</a></h2>
+<p>If you have a GitHub account, you can get hands-on via <a href="https://github.com/Electric-Coin-Company/tfl-book">the GitHub repository</a> for this book, including:</p>
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+<li><a href="https://github.com/Electric-Coin-Company/tfl-book/issues/new?labels=infrastructure&amp;projects=&amp;template=infrastructure.yml">Report Rendering / Infrastructure Issues</a> in case you're having trouble reading the content, viewing diagrams, rendering on your own computer, etc…</li>
+</ul>
+<h2 id="zcash-forum"><a class="header" href="#zcash-forum">Zcash Forum</a></h2>
+<p><a href="https://forum.zcashcommunity.com/">The Zcash Forum</a> is a hangout for many Zcash enthusiasts. This is a good spot for more open-ended discussion about this design proposal, alternatives, and other developments in Zcash.</p>
+<h2 id="zcash-rd-discord"><a class="header" href="#zcash-rd-discord">Zcash R&amp;D Discord</a></h2>
+<p>You can catch us on <a href="https://discord.gg/kCZf9NdneF">the Zcash R&amp;D Discord</a> in the <code>#proof-of-stake</code> channel.</p>
+<h2 id="zcash-arborist-calls"><a class="header" href="#zcash-arborist-calls">Zcash Arborist Calls</a></h2>
+<p>The <a href="https://zfnd.org/arborist-calls/">Zcash Arborist Calls</a> are bi-weekly Zcash protocol development calls, where proposals like this are discussed. Feel free to come lurk, ask questions, or provide feedback or suggestions.</p>
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+                <ol class="chapter"><li class="chapter-item expanded "><a href="../introduction.html"><strong aria-hidden="true">1.</strong> Introduction</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../introduction/a-path-to-pos-zcash.html"><strong aria-hidden="true">1.1.</strong> A Path to Proof-of-Stake Zcash</a></li><li class="chapter-item expanded "><a href="../introduction/trailing-finality-layer-in-a-nutshell.html"><strong aria-hidden="true">1.2.</strong> Trailing Finality Layer in a Nutshell</a></li><li class="chapter-item expanded "><a href="../introduction/status-and-next-steps.html"><strong aria-hidden="true">1.3.</strong> Status and Next Steps</a></li><li class="chapter-item expanded "><a href="../introduction/motivating-finality.html" class="active"><strong aria-hidden="true">1.4.</strong> Motivating Finality</a></li><li class="chapter-item expanded "><a href="../introduction/get-involved.html"><strong aria-hidden="true">1.5.</strong> Get Involved</a></li></ol></li><li class="chapter-item expanded "><a href="../terminology.html"><strong aria-hidden="true">2.</strong> Terminology</a></li><li class="chapter-item expanded "><a href="../design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/design-at-a-glance.html"><strong aria-hidden="true">3.1.</strong> Design at a Glance</a></li><li class="chapter-item expanded "><a href="../design/goals.html"><strong aria-hidden="true">3.2.</strong> Design Goals</a></li><li class="chapter-item expanded "><a href="../design/crosslink.html"><strong aria-hidden="true">3.3.</strong> Crosslink</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/crosslink/the-arguments-for-bounded-availability-and-finality-overrides.html"><strong aria-hidden="true">3.3.1.</strong> The Arguments for Bounded Availability and Finality Overrides</a></li><li class="chapter-item expanded "><a href="../design/crosslink/notes-on-snap-and-chat.html"><strong aria-hidden="true">3.3.2.</strong> Notes On Snap‑and‑Chat</a></li><li class="chapter-item expanded "><a href="../design/crosslink/construction.html"><strong aria-hidden="true">3.3.3.</strong> The Crosslink 2 Construction</a></li><li class="chapter-item expanded "><a href="../design/crosslink/security-analysis.html"><strong aria-hidden="true">3.3.4.</strong> Security Analysis of Crosslink 2</a></li><li class="chapter-item expanded "><a href="../design/crosslink/questions.html"><strong aria-hidden="true">3.3.5.</strong> Questions About Crosslink</a></li><li class="chapter-item expanded "><a href="../design/crosslink/potential-changes.html"><strong aria-hidden="true">3.3.6.</strong> Potential Changes to Crosslink</a></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">3.4.</strong> Model Code Architecture</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">3.5.</strong> Network Architecture</div></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">4.</strong> Protocol Specification</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">4.1.</strong> Abstract Protocol</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">4.1.1.</strong> Token Dynamics</div></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">4.2.</strong> Concrete Protocol</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">4.2.1.</strong> Token Dynamics</div></li></ol></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.</strong> Security</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.</strong> Abstract Analysis</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.1.</strong> Ebb-and-Flow analysis</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.2.</strong> Subprotocol Compromise Analysis</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.3.</strong> Model Code Architecture Analysis</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.4.</strong> Network Architecture Analysis</div></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.2.</strong> Concrete Analysis</div></li></ol></li><li class="chapter-item expanded "><a href="../references.html"><strong aria-hidden="true">6.</strong> References</a></li><li class="chapter-item expanded "><a href="../version-history.html"><strong aria-hidden="true">7.</strong> Version History</a></li></ol>
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+<h1 id="motivating-finality"><a class="header" href="#motivating-finality">Motivating Finality</a></h1>
+<p>In Zcash currently, consensus relies solely on PoW, which only provides <em>probabilistic finality</em>, rather than <a href="../terminology.html#definition-assured-finality"><em>assured finality</em></a>.<sup class="footnote-reference"><a href="#finality-qualifiers">1</a></sup> This style of consensus does not offer a guarantee that any given block may not be <em>rolled back</em> which may invalidate the transactions it contains. Instead, the probability that a block may be rolled back decreases as more blocks are mined subsquently on top of it.<sup class="footnote-reference"><a href="#pow-rollback-security-assumptions">2</a></sup></p>
+<p>Let’s walk through an example of how Zcash’s current PoW with probabilistic finality can impede important use cases. Consider a PoW node which sees this block sequence at time <code>T=0</code>:</p>
+<p><img src="motivating_finality_0.generated.svg" alt="" /></p>
+<p>When should a user, wallet, or other system choose to act based on a transaction in a block?</p>
+<p>For this example, let’s assume a bridging system may have received a deposit for <code>ZEC</code> in block <code>f</code> and issued a corresponding number of proxy tokens on a different network.</p>
+<p>At a later time, <code>T=1</code>, this same node may see a longer PoW chain which invalidates some previously seen blocks:</p>
+<p><img src="motivating_finality_1.generated.svg" alt="" /></p>
+<p>The node has observed a longer chain ending at block <code>h'</code>, so PoW consensus calls for that new sequence to be treated as consensus. The previously seen blocks <code>f</code> and <code>g</code> are no longer part of the consensus history, and have been rolled back.</p>
+<h2 id="impact-of-the-rollback"><a class="header" href="#impact-of-the-rollback">Impact of the Rollback</a></h2>
+<p>In our example, the bridging system acted in response to a transaction in the original block <code>f</code> at <code>T=0</code>. If the new sequence ending at <code>h'</code> no longer contains the deposit to the bridging system, the integrity of the bridge has been violated<sup class="footnote-reference"><a href="#txn-rollback">3</a></sup>; the associated proxy tokens may have already been used in a complex chain of Defi applications or deposited onto an exchange and sold, which would make any recovery impossible. The proxy tokens on the other network no longer correspond to the correct amount of <code>ZEC</code> on the Zcash network.</p>
+<h2 id="rollback-complications"><a class="header" href="#rollback-complications">Rollback Complications</a></h2>
+<p>This example demonstrates how a lack of assured finality can impede many useful real-world scenarios. In practice, systems and services which need greater assurances wait for more block confirmations.</p>
+<p>This has several drawbacks:</p>
+<ul>
+<li>It <em>doesn’t remove the vulnerability</em>, it only reduces the likelihood.</li>
+<li>Different applications/services may require different block depths, making it difficult to compose or chain together different applications/services.</li>
+<li>Different block depth policies potentially confuse users, i.e. “why do I have to wait one hour for my deposit in this exchange, but only 30 minutes on that exchange?”.</li>
+<li>It introduces a long delay which inhibits many useful applications.</li>
+</ul>
+<p>In addition to these user-facing and economic drawbacks, correctly handling rollbacks makes the code for nodes, wallets, and other infrastructure more complex. Worse still, many systems may not have correct behavior for rollbacks at different depths, and since large rollbacks are rarer, these implementation flaws may not surface until there is a large rollback. While a large rollback would be disruptive all by itself, it becomes even worse when previously undiscovered bugs exacerbate the situation.</p>
+<h2 id="trailing-finality-benefits"><a class="header" href="#trailing-finality-benefits">Trailing Finality Benefits</a></h2>
+<p>Trailing finality extends the existing PoW consensus so that older blocks become final, assuring they <em>cannot</em> be rolled back, and by extension neither can any of the transactions they contain.</p>
+<p>This directly addresses the first two flaws above: it completely removes the vulnerability, and it ensures all systems that need finality behave consistently with each other.</p>
+<p>As for delay, trailing finality also introduces delay since final blocks “trail behind” the most recent PoW blocks. This can be an improvement for some applications, depending on their latency requirements. For example, if the delay to finality averages around 10 minutes, then this would enable an improvement for an exchange that requires 60 minutes of PoW blocks for a deposit. On the other hand, it would not be an improvement for an application that needs finality faster than 10 minutes.</p>
+<p>Finally, implementations can be simplified by relying on the guarantee of finality. For example, a wallet can describe any transaction as pending or final, and does not need to provide difficult and potentially confusing UX (and the supporting database sophistication) for handling rollbacks.</p>
+<h1 id="footnotes"><a class="header" href="#footnotes">Footnotes</a></h1>
+<div class="footnote-definition" id="finality-qualifiers"><sup class="footnote-definition-label">1</sup>
+<p>Throughout this book, when we say <em>finality</em> or <em>final</em> without other qualifiers, we specifically are referring to <em>assured finality</em> or an <em>assured final</em> block. Where we call out <em>probabalistic finality</em> we always use that qualifier.</p>
+</div>
+<div class="footnote-definition" id="pow-rollback-security-assumptions"><sup class="footnote-definition-label">2</sup>
+<p>The estimated probability of a rollback relies on a variety of PoW security assumptions, and can be violated in various conditions, such as in mining efficiency breakthroughs, compromises of the PoW challenge algorithm (e.g. hash function collision resistance failure), difficulty-adjustment-algorithm failures, sudden/surprise mining capacity increases, and so on. So the estimated probability can be violated in potential “black swan” events.</p>
+</div>
+<div class="footnote-definition" id="txn-rollback"><sup class="footnote-definition-label">3</sup>
+<p>This discussion simplifies consideration of <em>transaction rollback</em> vs <em>block rollback</em>. When a block is rolled back, it is possible for some of the transactions contained in it to appear in new canonical blocks. The conditions when this can occur vs when it cannot are multifaceted and also subject to malicious influence, so for simplicity we assume all transactions within a rolled-back block are also rolled back.</p>
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+                <ol class="chapter"><li class="chapter-item expanded "><a href="../introduction.html"><strong aria-hidden="true">1.</strong> Introduction</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../introduction/a-path-to-pos-zcash.html"><strong aria-hidden="true">1.1.</strong> A Path to Proof-of-Stake Zcash</a></li><li class="chapter-item expanded "><a href="../introduction/trailing-finality-layer-in-a-nutshell.html"><strong aria-hidden="true">1.2.</strong> Trailing Finality Layer in a Nutshell</a></li><li class="chapter-item expanded "><a href="../introduction/status-and-next-steps.html" class="active"><strong aria-hidden="true">1.3.</strong> Status and Next Steps</a></li><li class="chapter-item expanded "><a href="../introduction/motivating-finality.html"><strong aria-hidden="true">1.4.</strong> Motivating Finality</a></li><li class="chapter-item expanded "><a href="../introduction/get-involved.html"><strong aria-hidden="true">1.5.</strong> Get Involved</a></li></ol></li><li class="chapter-item expanded "><a href="../terminology.html"><strong aria-hidden="true">2.</strong> Terminology</a></li><li class="chapter-item expanded "><a href="../design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/design-at-a-glance.html"><strong aria-hidden="true">3.1.</strong> Design at a Glance</a></li><li class="chapter-item expanded "><a href="../design/goals.html"><strong aria-hidden="true">3.2.</strong> Design Goals</a></li><li class="chapter-item expanded "><a href="../design/crosslink.html"><strong aria-hidden="true">3.3.</strong> Crosslink</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/crosslink/the-arguments-for-bounded-availability-and-finality-overrides.html"><strong aria-hidden="true">3.3.1.</strong> The Arguments for Bounded Availability and Finality Overrides</a></li><li class="chapter-item expanded "><a href="../design/crosslink/notes-on-snap-and-chat.html"><strong aria-hidden="true">3.3.2.</strong> Notes On Snap‑and‑Chat</a></li><li class="chapter-item expanded "><a href="../design/crosslink/construction.html"><strong aria-hidden="true">3.3.3.</strong> The Crosslink 2 Construction</a></li><li class="chapter-item expanded "><a href="../design/crosslink/security-analysis.html"><strong aria-hidden="true">3.3.4.</strong> Security Analysis of Crosslink 2</a></li><li class="chapter-item expanded "><a href="../design/crosslink/questions.html"><strong aria-hidden="true">3.3.5.</strong> Questions About Crosslink</a></li><li class="chapter-item expanded "><a href="../design/crosslink/potential-changes.html"><strong aria-hidden="true">3.3.6.</strong> Potential Changes to Crosslink</a></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">3.4.</strong> Model Code Architecture</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">3.5.</strong> Network Architecture</div></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">4.</strong> Protocol Specification</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">4.1.</strong> Abstract Protocol</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">4.1.1.</strong> Token Dynamics</div></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">4.2.</strong> Concrete Protocol</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">4.2.1.</strong> Token Dynamics</div></li></ol></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.</strong> Security</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.</strong> Abstract Analysis</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.1.</strong> Ebb-and-Flow analysis</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.2.</strong> Subprotocol Compromise Analysis</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.3.</strong> Model Code Architecture Analysis</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.4.</strong> Network Architecture Analysis</div></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.2.</strong> Concrete Analysis</div></li></ol></li><li class="chapter-item expanded "><a href="../references.html"><strong aria-hidden="true">6.</strong> References</a></li><li class="chapter-item expanded "><a href="../version-history.html"><strong aria-hidden="true">7.</strong> Version History</a></li></ol>
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+                    <h1 class="menu-title">Zcash Trailing Finality Layer - v0.1.0</h1>
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+                        <link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.12.0/dist/katex.min.css" integrity="sha384-AfEj0r4/OFrOo5t7NnNe46zW/tFgW6x/bCJG8FqQCEo3+Aro6EYUG4+cU+KJWu/X" crossorigin="anonymous">
+<h1 id="status-and-next-steps"><a class="header" href="#status-and-next-steps">Status and Next Steps</a></h1>
+<p>This is an early and incomplete protocol design proposal. It has not been well vetted for feasibility and safety. It has not had broad review from the Zcash community, so its status on any Zcash roadmap is undetermined.</p>
+<h2 id="current-components"><a class="header" href="#current-components">Current Components</a></h2>
+<h3 id="this-book"><a class="header" href="#this-book">This Book</a></h3>
+<p>This book is intended to become both a high-level overview and introduction to <a href="../terminology.html#definition-tfl">TFL</a>, and a full specification.</p>
+<h3 id="crosslink"><a class="header" href="#crosslink">Crosslink</a></h3>
+<p>The current heart of the design work is an in-progress hybrid consensus protocol construction called <a href="../terminology.html#definition-crosslink">Crosslink (definition)</a>. This is defined in the <a href="../design/crosslink.html">Crosslink chapter</a>.</p>
+<h3 id="simtfl"><a class="header" href="#simtfl"><code>simtfl</code></a></h3>
+<p>We've begun creating a simulator called <a href="../terminology.html#definition-simtfl"><code>simtfl</code></a> which we will use to model security and abstract performance concerns. Its development is tracked at <a href="https://github.com/zcash/simtfl">https://github.com/zcash/simtfl</a>.</p>
+<h2 id="major-missing-components"><a class="header" href="#major-missing-components">Major Missing Components</a></h2>
+<ul>
+<li>PoS subprotocol selection,</li>
+<li>Issuance and supply mechanics, such as how much ZEC stakers may earn,</li>
+<li>Integrated Zcash transaction semantics,</li>
+<li>A transition plan from current Zcash mainnet to this protocol design,</li>
+<li><a href="../terminology.html#definition-zip">ZIP</a>s specifying the above to the level of specificity required by ZIPs,</li>
+<li>Security and safety analyses,</li>
+<li>Economic analyses.</li>
+</ul>
+<p>This list may be incomplete, and as the design matures the need for major new components may be revealed.</p>
+<h2 id="next-steps"><a class="header" href="#next-steps">Next Steps</a></h2>
+<p>This design proposal is being developed by <a href="https://electriccoin.co/">Electric Coin Company</a> and <a href="https://x.com/shieldedlabs">Shielded Labs</a> as the first major milestone in our focus of deploying Proof-of-Stake to the Zcash protocol. Our rough near-term plan for this proposal is as follows:</p>
+<ol>
+<li>Complete the <a href="../terminology.html#definition-crosslink">Crosslink</a> description.</li>
+<li>Complete core security arguments for <a href="../terminology.html#definition-crosslink">Crosslink</a>.</li>
+<li>Define the <a href="#major-missing-components">Major Missing Components</a> above, including considerations such as issuance mechanics and Proof-of-Stake mechanisms.</li>
+<li>Complete auxillary security arguments and analyses, such as specific attack scenarios, game-theoretic security, and so forth.</li>
+<li>Mature <a href="../terminology.html#definition-simtfl"><code>simtfl</code></a> to analyze all cases of interest.</li>
+<li>Follow the general Zcash process for proposal/review/refinement, including proposing one or more <a href="../terminology.html#definition-zip">ZIP</a>s.</li>
+<li>Follow the general Zcash governance process for proposal review and refinement.</li>
+<li>If accepted, productionize the proposal in ECC products and collaborate with other implementors who implement the proposal.</li>
+<li>Celebrate when and if the proposal is activated on Mainnet. 🎉</li>
+</ol>
+<p>The fine-grained day-to-day goals and tasks for this project are present in <a href="https://zcash.github.io/developers">the Zcash Developers Hub</a> in the <a href="https://zcash.github.io/developers/zcash-tfl-dag">TFL-focused DAG</a>.</p>
+<p>Please also see <a href="./get-involved.html">Get Involved</a> if you are interested in tracking this progress more closely, or in contributing.</p>
+
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+                <ol class="chapter"><li class="chapter-item expanded "><a href="../introduction.html"><strong aria-hidden="true">1.</strong> Introduction</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../introduction/a-path-to-pos-zcash.html"><strong aria-hidden="true">1.1.</strong> A Path to Proof-of-Stake Zcash</a></li><li class="chapter-item expanded "><a href="../introduction/trailing-finality-layer-in-a-nutshell.html" class="active"><strong aria-hidden="true">1.2.</strong> Trailing Finality Layer in a Nutshell</a></li><li class="chapter-item expanded "><a href="../introduction/status-and-next-steps.html"><strong aria-hidden="true">1.3.</strong> Status and Next Steps</a></li><li class="chapter-item expanded "><a href="../introduction/motivating-finality.html"><strong aria-hidden="true">1.4.</strong> Motivating Finality</a></li><li class="chapter-item expanded "><a href="../introduction/get-involved.html"><strong aria-hidden="true">1.5.</strong> Get Involved</a></li></ol></li><li class="chapter-item expanded "><a href="../terminology.html"><strong aria-hidden="true">2.</strong> Terminology</a></li><li class="chapter-item expanded "><a href="../design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/design-at-a-glance.html"><strong aria-hidden="true">3.1.</strong> Design at a Glance</a></li><li class="chapter-item expanded "><a href="../design/goals.html"><strong aria-hidden="true">3.2.</strong> Design Goals</a></li><li class="chapter-item expanded "><a href="../design/crosslink.html"><strong aria-hidden="true">3.3.</strong> Crosslink</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/crosslink/the-arguments-for-bounded-availability-and-finality-overrides.html"><strong aria-hidden="true">3.3.1.</strong> The Arguments for Bounded Availability and Finality Overrides</a></li><li class="chapter-item expanded "><a href="../design/crosslink/notes-on-snap-and-chat.html"><strong aria-hidden="true">3.3.2.</strong> Notes On Snap‑and‑Chat</a></li><li class="chapter-item expanded "><a href="../design/crosslink/construction.html"><strong aria-hidden="true">3.3.3.</strong> The Crosslink 2 Construction</a></li><li class="chapter-item expanded "><a href="../design/crosslink/security-analysis.html"><strong aria-hidden="true">3.3.4.</strong> Security Analysis of Crosslink 2</a></li><li class="chapter-item expanded "><a href="../design/crosslink/questions.html"><strong aria-hidden="true">3.3.5.</strong> Questions About Crosslink</a></li><li class="chapter-item expanded "><a href="../design/crosslink/potential-changes.html"><strong aria-hidden="true">3.3.6.</strong> Potential Changes to Crosslink</a></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">3.4.</strong> Model Code Architecture</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">3.5.</strong> Network Architecture</div></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">4.</strong> Protocol Specification</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">4.1.</strong> Abstract Protocol</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">4.1.1.</strong> Token Dynamics</div></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">4.2.</strong> Concrete Protocol</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">4.2.1.</strong> Token Dynamics</div></li></ol></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.</strong> Security</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.</strong> Abstract Analysis</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.1.</strong> Ebb-and-Flow analysis</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.2.</strong> Subprotocol Compromise Analysis</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.3.</strong> Model Code Architecture Analysis</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.4.</strong> Network Architecture Analysis</div></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.2.</strong> Concrete Analysis</div></li></ol></li><li class="chapter-item expanded "><a href="../references.html"><strong aria-hidden="true">6.</strong> References</a></li><li class="chapter-item expanded "><a href="../version-history.html"><strong aria-hidden="true">7.</strong> Version History</a></li></ol>
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+                    <h1 class="menu-title">Zcash Trailing Finality Layer - v0.1.0</h1>
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+<h1 id="trailing-finality-layer-in-a-nutshell"><a class="header" href="#trailing-finality-layer-in-a-nutshell">Trailing Finality Layer in a Nutshell</a></h1>
+<p>The hybrid PoW/PoS consensus protocol proposed in this book is similar to today’s Zcash NU5/NU6 consensus protocol with the addition of a <em>Trailing Finality Layer</em>:</p>
+<p><strong>TODO:</strong> <a href="https://github.com/Electric-Coin-Company/tfl-book/issues/124">Add network topology / software subcomponent diagram #124</a></p>
+<p>The Zcash Trailing Finality Layer refers to a new subprotocol of a new hybrid PoW/PoS protocol, which we refer to as <em>PoW+TFL</em>. This subprotocol introduces <a href="../terminology.html#definition-assured-finality">assured finality</a> for the Zcash block chain, ensuring that <em>final</em> blocks (and the transactions within them) may never be rolled back.</p>
+<p>We use the term “layer” because we can understand this design as introducing a new layer to the Zcash network, making only minimal changes to the existing network and consensus protocol. This modular separation is present in the consensus rules, the network protocol, and the code architecture.</p>
+<h2 id="why-should-users-care"><a class="header" href="#why-should-users-care">Why Should Users Care?</a></h2>
+<p>There are three categories of users this proposed TFL protocol would impact:</p>
+<h3 id="current-zec-users"><a class="header" href="#current-zec-users">Current ZEC Users</a></h3>
+<p>Existing ZEC users who are primarily concerned with storing or receiving ZEC, whether private or transparent may benefit from this change in the short or medium term, because it may help lower delay times for some services, such as exchange deposits. As exchanges come to rely on the new finality guarantee, they can often reduce their deposit wait times. Other services with similar confirmation-depth-based wait times can be improved in a similar way to lower these wait times. Other than this improvement, these users should notice no other changes.</p>
+<p>In the longer term, providing finality will be useful in establishing trust‑minimized bridges to other block‑chain protocols. We anticipate this can enable better connecting ZEC to the Defi ecosystem, and with the introduction of Zcash Shielded Assets, this can enable other assets to connect to the Zcash shielded pool.</p>
+<h3 id="proof-of-stake-users"><a class="header" href="#proof-of-stake-users">Proof-of-Stake Users</a></h3>
+<p>Users who are interested in providing finality infrastructure, or users who want to delegate ZEC towards finality, will be able to earn rewards from the protocol for doing so, while also taking on some risk to their funds (to prevent malicious abuse of the protocol). This may be an important new category of ZEC users and use cases.</p>
+<h3 id="miners"><a class="header" href="#miners">Miners</a></h3>
+<p>Miners who provide Proof‑of‑Work security will necessarily see some reduction in their block rewards, since this proposal maintains the same issuance schedule and supply cap of ZEC while also spending some rewards on finality.</p>
+<p><strong>Important Note:</strong> The proportion and details of how much mining rewards will be impacted, and conversely how much finality/PoS providers will earn, are not yet specified in this proposal.</p>
+<h2 id="why-is-this-a-good-approach-to-a-pos-transition"><a class="header" href="#why-is-this-a-good-approach-to-a-pos-transition">Why is this a Good Approach to a PoS Transition?</a></h2>
+<p>This design is appealing as a safer first step in transitioning the Zcash protocol for multiple reasons:</p>
+<h3 id="it-enables-proof-of-stake-mechanisms-conservatively"><a class="header" href="#it-enables-proof-of-stake-mechanisms-conservatively">It Enables Proof-of-Stake Mechanisms Conservatively</a></h3>
+<p>This transition would enable PoS mechanisms, including the ability to operate PoS infratructure and delegate ZEC towards those providers to earn protocol rewards. While all PoS transitions would accomplish this, this approach does so in a conservative manner: it introduces these mechanics while striving to minimize the impact on existing use cases and protocol security.</p>
+<p>In a sense, we can think of this approach as enabling the Zcash community to “dip our toes in the PoS waters” rather than diving in. If the results pan out well, it gives us confidence for further transitions. If we discover challenges, flaws, or risks, we anticipate their impact will be more limited since this is a more cautious transition step.</p>
+<h3 id="minimal-use-case-disruption"><a class="header" href="#minimal-use-case-disruption">Minimal Use-Case Disruption</a></h3>
+<p>In many cases, existing products, services, and tools can continue using the mainnet chain with no changes to code assuming they rely on existing consensus nodes. We view this as a major benefit which allows Zcash’s existing user network effect to continue safely unperturbed.</p>
+<p>There will be certain narrow exceptional areas if those products, services, or tools need to be precise in areas where the protocol has changed, such as mining/staking reward calculations, transaction formats (in particular any new PoS-related fields or logic), or chain rollback logic.</p>
+<h3 id="modular-design"><a class="header" href="#modular-design">Modular Design</a></h3>
+<p>By conceptualizing the TFL as a distinct “layer” or subprotocol, the consensus rules can be described in terms of two <a href="../terminology.html#definition-consensus-subprotocols">consensus subprotocols</a>, one embodying most of the current consensus logic of Zcash and another the TFL. These protocols interact through a <a href="../terminology.html#definition-hybrid-construction">hybrid construction</a>. See <a href="../design/design-at-a-glance.html">Design at a Glance</a> to learn more about these distinct subprotocols.</p>
+<p>Reasoning about the whole protocol can leverage analysis and understanding of each subprotocol and the hybrid construction somewhat independently due to this modular design. Note that although this design is modular, the hybrid construction may require modifications to the [PoW] and/or [PoS] subprotocols to protect safety and liveness properties. Nevertheless, the modularity still improves analysis and reasoning compared to a monolithic design.</p>
+<p>Finally, since one subprotocol is very similar to the existing Zcash <a href="../terminology.html#definition-nu5nu6">NU5/NU6</a> consensus protocol, this lessens risk that the consensus properties within that subprotocol compromise current NU5/NU6 properties.</p>
+<h3 id="modular-implementation"><a class="header" href="#modular-implementation">Modular Implementation</a></h3>
+<p>In addition to the other benefits of protocol design modularity, we anticipate actual implementations can realize this modularity in code. This can help makes implementations more robust, easier to maintain, and more interoperable.</p>
+<p>For example, we can envision a standardized interface between PoW &amp; TFL consensus components, enabling different development teams to provide these different components and for “full node” packagers to mix and match them. This is somewhat reminiscent of Ethereum’s execution/consensus layer separation which we believe has shown great success in implementation team and product diversity.</p>
+<h2 id="cracking-the-nutshell"><a class="header" href="#cracking-the-nutshell">Cracking the Nutshell</a></h2>
+<p>In the rest of the introductory section of this book, we describe the status and next steps for the TFL proposal, provide a motivation for finality, and suggestions for getting involved.</p>
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diff --git a/docs/print.html b/docs/print.html
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+                <ol class="chapter"><li class="chapter-item expanded "><a href="introduction.html"><strong aria-hidden="true">1.</strong> Introduction</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="introduction/a-path-to-pos-zcash.html"><strong aria-hidden="true">1.1.</strong> A Path to Proof-of-Stake Zcash</a></li><li class="chapter-item expanded "><a href="introduction/trailing-finality-layer-in-a-nutshell.html"><strong aria-hidden="true">1.2.</strong> Trailing Finality Layer in a Nutshell</a></li><li class="chapter-item expanded "><a href="introduction/status-and-next-steps.html"><strong aria-hidden="true">1.3.</strong> Status and Next Steps</a></li><li class="chapter-item expanded "><a href="introduction/motivating-finality.html"><strong aria-hidden="true">1.4.</strong> Motivating Finality</a></li><li class="chapter-item expanded "><a href="introduction/get-involved.html"><strong aria-hidden="true">1.5.</strong> Get Involved</a></li></ol></li><li class="chapter-item expanded "><a href="terminology.html"><strong aria-hidden="true">2.</strong> Terminology</a></li><li class="chapter-item expanded "><a href="design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/design-at-a-glance.html"><strong aria-hidden="true">3.1.</strong> Design at a Glance</a></li><li class="chapter-item expanded "><a href="design/goals.html"><strong aria-hidden="true">3.2.</strong> Design Goals</a></li><li class="chapter-item expanded "><a href="design/crosslink.html"><strong aria-hidden="true">3.3.</strong> Crosslink</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/crosslink/the-arguments-for-bounded-availability-and-finality-overrides.html"><strong aria-hidden="true">3.3.1.</strong> The Arguments for Bounded Availability and Finality Overrides</a></li><li class="chapter-item expanded "><a href="design/crosslink/notes-on-snap-and-chat.html"><strong aria-hidden="true">3.3.2.</strong> Notes On Snap‑and‑Chat</a></li><li class="chapter-item expanded "><a href="design/crosslink/construction.html"><strong aria-hidden="true">3.3.3.</strong> The Crosslink 2 Construction</a></li><li class="chapter-item expanded "><a href="design/crosslink/security-analysis.html"><strong aria-hidden="true">3.3.4.</strong> Security Analysis of Crosslink 2</a></li><li class="chapter-item expanded "><a href="design/crosslink/questions.html"><strong aria-hidden="true">3.3.5.</strong> Questions About Crosslink</a></li><li class="chapter-item expanded "><a href="design/crosslink/potential-changes.html"><strong aria-hidden="true">3.3.6.</strong> Potential Changes to Crosslink</a></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">3.4.</strong> Model Code Architecture</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">3.5.</strong> Network Architecture</div></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">4.</strong> Protocol Specification</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">4.1.</strong> Abstract Protocol</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">4.1.1.</strong> Token Dynamics</div></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">4.2.</strong> Concrete Protocol</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">4.2.1.</strong> Token Dynamics</div></li></ol></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.</strong> Security</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.</strong> Abstract Analysis</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.1.</strong> Ebb-and-Flow analysis</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.2.</strong> Subprotocol Compromise Analysis</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.3.</strong> Model Code Architecture Analysis</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.4.</strong> Network Architecture Analysis</div></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.2.</strong> Concrete Analysis</div></li></ol></li><li class="chapter-item expanded "><a href="references.html"><strong aria-hidden="true">6.</strong> References</a></li><li class="chapter-item expanded "><a href="version-history.html"><strong aria-hidden="true">7.</strong> Version History</a></li></ol>
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+<h1 id="a-trailing-finality-layer-for-zcash"><a class="header" href="#a-trailing-finality-layer-for-zcash">A Trailing Finality Layer for Zcash</a></h1>
+<p>This book introduces and specifies a <a href="./terminology.html#definition-tfl"><em>Trailing Finality Layer</em></a> for the Zcash network. This is version <a href="./version-history.html#0_1_0-introducing-crosslink">0.1.0</a> of the book.</p>
+<p>This design augments the existing Zcash Proof‑of‑Work (<a href="./terminology.html#definition-pow">PoW</a>) network with a new consensus layer which provides <a href="./terminology.html#definition-trailing-finality"><em>trailing finality</em></a>. This layer enables transactions included via PoW to become <a href="./terminology.html#definition-final"><em>final</em></a> which assures that they cannot be reverted by the protocol. This enables safer and simpler wallets and other infrastructure, and aids trust-minimized cross-chain bridges. This consensus layer uses <a href="./terminology.html#definition-pos"><em>Proof-of-Stake</em></a> consensus, and enables ZEC holders to earn protocol rewards for contributing to the security of the Zcash network. By integrating a PoS layer with the current PoW Zcash protocol, this design specifies a <a href="./terminology.html#definition-hybrid-consensus"><em>hybrid consensus</em></a> protocol dubbed <a href="./terminology.html#definition-pow-tfl">PoW+TFL</a>.</p>
+<p>The rest of this introductory chapter is aimed at a general audience interested in the context of this proposal within Zcash development, status and next steps, motivations, a primer on finality, and tips to get involved.</p>
+<div style="break-before: page; page-break-before: always;"></div><link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.12.0/dist/katex.min.css" integrity="sha384-AfEj0r4/OFrOo5t7NnNe46zW/tFgW6x/bCJG8FqQCEo3+Aro6EYUG4+cU+KJWu/X" crossorigin="anonymous">
+<h1 id="a-path-to-proof-of-stake-zcash"><a class="header" href="#a-path-to-proof-of-stake-zcash">A Path to Proof-of-Stake Zcash</a></h1>
+<p>The <a href="introduction/../terminology.html#definition-tfl">TFL</a> design provides a possible first step in transitioning <a href="https://z.cash">Zcash</a> to a <a href="introduction/../terminology.html#definition-pos">PoS</a> protocol. Here we describe how a transition to PoS relates to “the Zcash roadmap” and how TFL fits into one approach to a PoS transition.</p>
+<h2 id="the-zcash-tech-tree"><a class="header" href="#the-zcash-tech-tree">The Zcash Tech-Tree</a></h2>
+<p>There are multiple developer orgs working on different proposed features for Zcash. Some of these involve multiple large distinct upgrade steps, and several of these steps depend on other such steps. This could be represented as a directed acyclic graph. We have begun referring to this space of possible future improvements as the <em>Zcash Tech-Tree</em>, taking inspiration from an analogous concept in gaming.<sup class="footnote-reference"><a href="#tech-tree-history">1</a></sup></p>
+<p>We envision a <em>proof-of-stake transition path</em> as one of the potential paths within this tech-tree which is the primary protocol focus of this proposal. An example visualization of this Zcash Tech-Tree might look like this:</p>
+<p><img src="introduction/a_path_to_proof_of_stake_zcash_0.generated.svg" alt="" /></p>
+<h2 id="a-proof-of-stake-transition-path"><a class="header" href="#a-proof-of-stake-transition-path">A Proof-of-Stake Transition Path</a></h2>
+<p>Given that context, we envision a path within the Zcash Tech-Tree for transitioning Zcash to PoS. At the top level we propose that this path should include at least two major milestones:</p>
+<ol>
+<li>Transitioning from Zcash <a href="introduction/../terminology.html#definition-nu5nu6">NU5/NU6</a> <a href="introduction/../terminology.html#definition-pow">PoW</a> consensus to a PoW/PoS <a href="introduction/../terminology.html#definition-hybrid-consensus">hybrid consensus</a> protocol dubbed <a href="introduction/../terminology.html#definition-pow-tfl">PoW+TFL</a>.</li>
+<li>Transitioning from <a href="introduction/../terminology.html#definition-pow-tfl">PoW+TFL</a> to pure <a href="introduction/../terminology.html#definition-pow-tfl">PoS</a> protocol.</li>
+</ol>
+<p>After this transition to pure PoS, there are likely to be future improvements to the PoS protocol, or the consensus protocol more generally. This TFL book focuses almost exclusively on the first step in this sequence.</p>
+<p>Our primary motivation for proposing (at least) two steps is to minimize disruption to usability, safety, security, and the ecosystem during each step.</p>
+<p>This book primarily focuses this first step: the transition to <a href="introduction/../terminology.html#definition-pow-tfl">PoW+TFL</a>. To understand the specific goals for that, see <a href="introduction/../design/goals.html">Design Goals</a>.</p>
+<p>With this approach, the Zcash Tech Tree with the <a href="introduction/../terminology.html#definition-tfl">TFL</a> approach might look something like this:</p>
+<p><img src="introduction/a_path_to_proof_of_stake_zcash_1.generated.svg" alt="" /></p>
+<h2 id="why-two-steps"><a class="header" href="#why-two-steps">Why Two Steps?</a></h2>
+<p>One question we've gotten in proposing this approach is why take a two-step process with an intermediate <a href="introduction/../terminology.html#definition-hybrid-consensus">hybrid consensus</a> protocol, rather than a single transition directly to a <a href="introduction/../terminology.html#definition-pos">PoS</a> protocol?</p>
+<p>Here’s how we think about those trade-offs:</p>
+<h3 id="considering-single-transition-vs-hybrid-multi-step"><a class="header" href="#considering-single-transition-vs-hybrid-multi-step">Considering Single Transition (vs Hybrid Multi-Step)</a></h3>
+<h4 id="pros"><a class="header" href="#pros">Pros</a></h4>
+<ul>
+<li>We already understand the current PoW protocol well, and if we transition to an existing proven PoS protocol, then we could skip the complexity of an intermediate hybrid stage.</li>
+<li>The node implementation might be simpler.</li>
+<li>Explaining to people what is happening might be simpler. Something like “Zcash has been PoW since it launched, but on DATE (e.g. at block height <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span>) it will switch to PoS.”</li>
+<li>Given that the issuance in a given time period is bounded by the supply curve, the full amount that was previously allocated to mining rewards becomes immediately available for staking rewards at the switch-over, rather than having to share this amount between mining and staking during the hybrid stage.</li>
+</ul>
+<h4 id="cons"><a class="header" href="#cons">Cons</a></h4>
+<ul>
+<li>If there is any unforeseen show‑stopping problem in the new protocol or the transition process, we’d have to react to a network‑wide issue.</li>
+<li>It may be more likely to cause ecosystem disruption; unforeseen differences between PoW and PoS might cause various kinds of snags or papercuts throughout the ecosystem, and these would all pop up around the same time, which may lead to a loss of confidence/retention/adoption or at the very least inconvenience many users for some time.</li>
+<li>Losing miners: since the transition would be all at once, we may lose some number of miners, who are participants and users in the ecosystem. Miners may leave prior to the transition in order to take care of their own needs. If there is some showstopper in the transition, one possible short‑term mitigation would be to fall back on PoW which is well known, but if we’ve lost most miners, that may no longer be viable.</li>
+</ul>
+<h3 id="considering-hybrid-multi-step-approach-vs-single-step"><a class="header" href="#considering-hybrid-multi-step-approach-vs-single-step">Considering Hybrid Multi-Step approach (vs Single-Step):</a></h3>
+<p>Note: TFL is one instance of a multi‑step approach.</p>
+<h4 id="pros-1"><a class="header" href="#pros-1">Pros</a></h4>
+<ul>
+<li>We can hopefully be less disruptive across the ecosystem so that there are fewer snags and disruptions with each step.</li>
+<li>If there is a show-stopping flaw in any step, the fall‑back possibility seems more plausible. For example, if there is a show-stopper when transitioning from PoW to <a href="introduction/../terminology.html#definition-pow-tfl">PoW+TFL</a>, falling back to pure PoW seems more feasible, since both protocols rely on mainnet PoW infrastructure, so those participants will be present in either case.</li>
+<li>Retaining miners during a hybrid phase: while it is true that a hybrid protocol will lower miner revenue (since we aim to maintain the issuance schedule constraints), there is also more possibility and likelihood of keeping some of these users engaged. For example, they may begin participating in staking services (either as delegators or as infrastructure operators). If that is successful, then they’re also more likely to remain engaged in the subsequent transition to pure PoS.</li>
+<li>This general approach was demonstrated successfully by Ethereum, which is the largest or second largest cryptocurrency network for several important metrics (e.g. market cap, fees paid, user and developer activity, …). So we know this can be done well without disruptions.</li>
+</ul>
+<h4 id="cons-1"><a class="header" href="#cons-1">Cons</a></h4>
+<ul>
+<li>The intermediate hybrid step will be a more novel and less well understood protocol. (It will necessarily be fairly different from Ethereum’s Beacon chain era.)</li>
+<li>Consensus nodes will be more complex, involving logic for both sub‑protocols as well as their integration. (Ideally this complexity can be modularized so that the nodes are easier to maintain and improve.)</li>
+<li>This may be more complicated to explain to current and potential new users. Something like “Zcash launched as PoW, and on $DATE (block height X) it will transition to a hybrid system, then later to a pure PoS system.”</li>
+<li>The available issuance must be shared between mining and staking rewards during the hybrid stage. The security of the PoW layer and of the PoS layer during this stage is partially dependent on the funds allocated to issuance for each protocol, and it is not yet clear to what extent splitting rewards would affect overall security.</li>
+</ul>
+<h1 id="footnotes"><a class="header" href="#footnotes">Footnotes</a></h1>
+<div class="footnote-definition" id="tech-tree-history"><sup class="footnote-definition-label">1</sup>
+<p>See <a href="https://en.wikipedia.org/wiki/Technology_tree#History">Wikipedia’s Technology Tree - History</a> section for details.</p>
+</div>
+<div style="break-before: page; page-break-before: always;"></div><link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.12.0/dist/katex.min.css" integrity="sha384-AfEj0r4/OFrOo5t7NnNe46zW/tFgW6x/bCJG8FqQCEo3+Aro6EYUG4+cU+KJWu/X" crossorigin="anonymous">
+<h1 id="trailing-finality-layer-in-a-nutshell"><a class="header" href="#trailing-finality-layer-in-a-nutshell">Trailing Finality Layer in a Nutshell</a></h1>
+<p>The hybrid PoW/PoS consensus protocol proposed in this book is similar to today’s Zcash NU5/NU6 consensus protocol with the addition of a <em>Trailing Finality Layer</em>:</p>
+<p><strong>TODO:</strong> <a href="https://github.com/Electric-Coin-Company/tfl-book/issues/124">Add network topology / software subcomponent diagram #124</a></p>
+<p>The Zcash Trailing Finality Layer refers to a new subprotocol of a new hybrid PoW/PoS protocol, which we refer to as <em>PoW+TFL</em>. This subprotocol introduces <a href="introduction/../terminology.html#definition-assured-finality">assured finality</a> for the Zcash block chain, ensuring that <em>final</em> blocks (and the transactions within them) may never be rolled back.</p>
+<p>We use the term “layer” because we can understand this design as introducing a new layer to the Zcash network, making only minimal changes to the existing network and consensus protocol. This modular separation is present in the consensus rules, the network protocol, and the code architecture.</p>
+<h2 id="why-should-users-care"><a class="header" href="#why-should-users-care">Why Should Users Care?</a></h2>
+<p>There are three categories of users this proposed TFL protocol would impact:</p>
+<h3 id="current-zec-users"><a class="header" href="#current-zec-users">Current ZEC Users</a></h3>
+<p>Existing ZEC users who are primarily concerned with storing or receiving ZEC, whether private or transparent may benefit from this change in the short or medium term, because it may help lower delay times for some services, such as exchange deposits. As exchanges come to rely on the new finality guarantee, they can often reduce their deposit wait times. Other services with similar confirmation-depth-based wait times can be improved in a similar way to lower these wait times. Other than this improvement, these users should notice no other changes.</p>
+<p>In the longer term, providing finality will be useful in establishing trust‑minimized bridges to other block‑chain protocols. We anticipate this can enable better connecting ZEC to the Defi ecosystem, and with the introduction of Zcash Shielded Assets, this can enable other assets to connect to the Zcash shielded pool.</p>
+<h3 id="proof-of-stake-users"><a class="header" href="#proof-of-stake-users">Proof-of-Stake Users</a></h3>
+<p>Users who are interested in providing finality infrastructure, or users who want to delegate ZEC towards finality, will be able to earn rewards from the protocol for doing so, while also taking on some risk to their funds (to prevent malicious abuse of the protocol). This may be an important new category of ZEC users and use cases.</p>
+<h3 id="miners"><a class="header" href="#miners">Miners</a></h3>
+<p>Miners who provide Proof‑of‑Work security will necessarily see some reduction in their block rewards, since this proposal maintains the same issuance schedule and supply cap of ZEC while also spending some rewards on finality.</p>
+<p><strong>Important Note:</strong> The proportion and details of how much mining rewards will be impacted, and conversely how much finality/PoS providers will earn, are not yet specified in this proposal.</p>
+<h2 id="why-is-this-a-good-approach-to-a-pos-transition"><a class="header" href="#why-is-this-a-good-approach-to-a-pos-transition">Why is this a Good Approach to a PoS Transition?</a></h2>
+<p>This design is appealing as a safer first step in transitioning the Zcash protocol for multiple reasons:</p>
+<h3 id="it-enables-proof-of-stake-mechanisms-conservatively"><a class="header" href="#it-enables-proof-of-stake-mechanisms-conservatively">It Enables Proof-of-Stake Mechanisms Conservatively</a></h3>
+<p>This transition would enable PoS mechanisms, including the ability to operate PoS infratructure and delegate ZEC towards those providers to earn protocol rewards. While all PoS transitions would accomplish this, this approach does so in a conservative manner: it introduces these mechanics while striving to minimize the impact on existing use cases and protocol security.</p>
+<p>In a sense, we can think of this approach as enabling the Zcash community to “dip our toes in the PoS waters” rather than diving in. If the results pan out well, it gives us confidence for further transitions. If we discover challenges, flaws, or risks, we anticipate their impact will be more limited since this is a more cautious transition step.</p>
+<h3 id="minimal-use-case-disruption"><a class="header" href="#minimal-use-case-disruption">Minimal Use-Case Disruption</a></h3>
+<p>In many cases, existing products, services, and tools can continue using the mainnet chain with no changes to code assuming they rely on existing consensus nodes. We view this as a major benefit which allows Zcash’s existing user network effect to continue safely unperturbed.</p>
+<p>There will be certain narrow exceptional areas if those products, services, or tools need to be precise in areas where the protocol has changed, such as mining/staking reward calculations, transaction formats (in particular any new PoS-related fields or logic), or chain rollback logic.</p>
+<h3 id="modular-design"><a class="header" href="#modular-design">Modular Design</a></h3>
+<p>By conceptualizing the TFL as a distinct “layer” or subprotocol, the consensus rules can be described in terms of two <a href="introduction/../terminology.html#definition-consensus-subprotocols">consensus subprotocols</a>, one embodying most of the current consensus logic of Zcash and another the TFL. These protocols interact through a <a href="introduction/../terminology.html#definition-hybrid-construction">hybrid construction</a>. See <a href="introduction/../design/design-at-a-glance.html">Design at a Glance</a> to learn more about these distinct subprotocols.</p>
+<p>Reasoning about the whole protocol can leverage analysis and understanding of each subprotocol and the hybrid construction somewhat independently due to this modular design. Note that although this design is modular, the hybrid construction may require modifications to the [PoW] and/or [PoS] subprotocols to protect safety and liveness properties. Nevertheless, the modularity still improves analysis and reasoning compared to a monolithic design.</p>
+<p>Finally, since one subprotocol is very similar to the existing Zcash <a href="introduction/../terminology.html#definition-nu5nu6">NU5/NU6</a> consensus protocol, this lessens risk that the consensus properties within that subprotocol compromise current NU5/NU6 properties.</p>
+<h3 id="modular-implementation"><a class="header" href="#modular-implementation">Modular Implementation</a></h3>
+<p>In addition to the other benefits of protocol design modularity, we anticipate actual implementations can realize this modularity in code. This can help makes implementations more robust, easier to maintain, and more interoperable.</p>
+<p>For example, we can envision a standardized interface between PoW &amp; TFL consensus components, enabling different development teams to provide these different components and for “full node” packagers to mix and match them. This is somewhat reminiscent of Ethereum’s execution/consensus layer separation which we believe has shown great success in implementation team and product diversity.</p>
+<h2 id="cracking-the-nutshell"><a class="header" href="#cracking-the-nutshell">Cracking the Nutshell</a></h2>
+<p>In the rest of the introductory section of this book, we describe the status and next steps for the TFL proposal, provide a motivation for finality, and suggestions for getting involved.</p>
+<div style="break-before: page; page-break-before: always;"></div><link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.12.0/dist/katex.min.css" integrity="sha384-AfEj0r4/OFrOo5t7NnNe46zW/tFgW6x/bCJG8FqQCEo3+Aro6EYUG4+cU+KJWu/X" crossorigin="anonymous">
+<h1 id="status-and-next-steps"><a class="header" href="#status-and-next-steps">Status and Next Steps</a></h1>
+<p>This is an early and incomplete protocol design proposal. It has not been well vetted for feasibility and safety. It has not had broad review from the Zcash community, so its status on any Zcash roadmap is undetermined.</p>
+<h2 id="current-components"><a class="header" href="#current-components">Current Components</a></h2>
+<h3 id="this-book"><a class="header" href="#this-book">This Book</a></h3>
+<p>This book is intended to become both a high-level overview and introduction to <a href="introduction/../terminology.html#definition-tfl">TFL</a>, and a full specification.</p>
+<h3 id="crosslink"><a class="header" href="#crosslink">Crosslink</a></h3>
+<p>The current heart of the design work is an in-progress hybrid consensus protocol construction called <a href="introduction/../terminology.html#definition-crosslink">Crosslink (definition)</a>. This is defined in the <a href="introduction/../design/crosslink.html">Crosslink chapter</a>.</p>
+<h3 id="simtfl"><a class="header" href="#simtfl"><code>simtfl</code></a></h3>
+<p>We've begun creating a simulator called <a href="introduction/../terminology.html#definition-simtfl"><code>simtfl</code></a> which we will use to model security and abstract performance concerns. Its development is tracked at <a href="https://github.com/zcash/simtfl">https://github.com/zcash/simtfl</a>.</p>
+<h2 id="major-missing-components"><a class="header" href="#major-missing-components">Major Missing Components</a></h2>
+<ul>
+<li>PoS subprotocol selection,</li>
+<li>Issuance and supply mechanics, such as how much ZEC stakers may earn,</li>
+<li>Integrated Zcash transaction semantics,</li>
+<li>A transition plan from current Zcash mainnet to this protocol design,</li>
+<li><a href="introduction/../terminology.html#definition-zip">ZIP</a>s specifying the above to the level of specificity required by ZIPs,</li>
+<li>Security and safety analyses,</li>
+<li>Economic analyses.</li>
+</ul>
+<p>This list may be incomplete, and as the design matures the need for major new components may be revealed.</p>
+<h2 id="next-steps"><a class="header" href="#next-steps">Next Steps</a></h2>
+<p>This design proposal is being developed by <a href="https://electriccoin.co/">Electric Coin Company</a> and <a href="https://x.com/shieldedlabs">Shielded Labs</a> as the first major milestone in our focus of deploying Proof-of-Stake to the Zcash protocol. Our rough near-term plan for this proposal is as follows:</p>
+<ol>
+<li>Complete the <a href="introduction/../terminology.html#definition-crosslink">Crosslink</a> description.</li>
+<li>Complete core security arguments for <a href="introduction/../terminology.html#definition-crosslink">Crosslink</a>.</li>
+<li>Define the <a href="introduction/status-and-next-steps.html#major-missing-components">Major Missing Components</a> above, including considerations such as issuance mechanics and Proof-of-Stake mechanisms.</li>
+<li>Complete auxillary security arguments and analyses, such as specific attack scenarios, game-theoretic security, and so forth.</li>
+<li>Mature <a href="introduction/../terminology.html#definition-simtfl"><code>simtfl</code></a> to analyze all cases of interest.</li>
+<li>Follow the general Zcash process for proposal/review/refinement, including proposing one or more <a href="introduction/../terminology.html#definition-zip">ZIP</a>s.</li>
+<li>Follow the general Zcash governance process for proposal review and refinement.</li>
+<li>If accepted, productionize the proposal in ECC products and collaborate with other implementors who implement the proposal.</li>
+<li>Celebrate when and if the proposal is activated on Mainnet. 🎉</li>
+</ol>
+<p>The fine-grained day-to-day goals and tasks for this project are present in <a href="https://zcash.github.io/developers">the Zcash Developers Hub</a> in the <a href="https://zcash.github.io/developers/zcash-tfl-dag">TFL-focused DAG</a>.</p>
+<p>Please also see <a href="introduction/./get-involved.html">Get Involved</a> if you are interested in tracking this progress more closely, or in contributing.</p>
+<div style="break-before: page; page-break-before: always;"></div><link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.12.0/dist/katex.min.css" integrity="sha384-AfEj0r4/OFrOo5t7NnNe46zW/tFgW6x/bCJG8FqQCEo3+Aro6EYUG4+cU+KJWu/X" crossorigin="anonymous">
+<h1 id="motivating-finality"><a class="header" href="#motivating-finality">Motivating Finality</a></h1>
+<p>In Zcash currently, consensus relies solely on PoW, which only provides <em>probabilistic finality</em>, rather than <a href="introduction/../terminology.html#definition-assured-finality"><em>assured finality</em></a>.<sup class="footnote-reference"><a href="#finality-qualifiers">1</a></sup> This style of consensus does not offer a guarantee that any given block may not be <em>rolled back</em> which may invalidate the transactions it contains. Instead, the probability that a block may be rolled back decreases as more blocks are mined subsquently on top of it.<sup class="footnote-reference"><a href="#pow-rollback-security-assumptions">2</a></sup></p>
+<p>Let’s walk through an example of how Zcash’s current PoW with probabilistic finality can impede important use cases. Consider a PoW node which sees this block sequence at time <code>T=0</code>:</p>
+<p><img src="introduction/motivating_finality_0.generated.svg" alt="" /></p>
+<p>When should a user, wallet, or other system choose to act based on a transaction in a block?</p>
+<p>For this example, let’s assume a bridging system may have received a deposit for <code>ZEC</code> in block <code>f</code> and issued a corresponding number of proxy tokens on a different network.</p>
+<p>At a later time, <code>T=1</code>, this same node may see a longer PoW chain which invalidates some previously seen blocks:</p>
+<p><img src="introduction/motivating_finality_1.generated.svg" alt="" /></p>
+<p>The node has observed a longer chain ending at block <code>h'</code>, so PoW consensus calls for that new sequence to be treated as consensus. The previously seen blocks <code>f</code> and <code>g</code> are no longer part of the consensus history, and have been rolled back.</p>
+<h2 id="impact-of-the-rollback"><a class="header" href="#impact-of-the-rollback">Impact of the Rollback</a></h2>
+<p>In our example, the bridging system acted in response to a transaction in the original block <code>f</code> at <code>T=0</code>. If the new sequence ending at <code>h'</code> no longer contains the deposit to the bridging system, the integrity of the bridge has been violated<sup class="footnote-reference"><a href="#txn-rollback">3</a></sup>; the associated proxy tokens may have already been used in a complex chain of Defi applications or deposited onto an exchange and sold, which would make any recovery impossible. The proxy tokens on the other network no longer correspond to the correct amount of <code>ZEC</code> on the Zcash network.</p>
+<h2 id="rollback-complications"><a class="header" href="#rollback-complications">Rollback Complications</a></h2>
+<p>This example demonstrates how a lack of assured finality can impede many useful real-world scenarios. In practice, systems and services which need greater assurances wait for more block confirmations.</p>
+<p>This has several drawbacks:</p>
+<ul>
+<li>It <em>doesn’t remove the vulnerability</em>, it only reduces the likelihood.</li>
+<li>Different applications/services may require different block depths, making it difficult to compose or chain together different applications/services.</li>
+<li>Different block depth policies potentially confuse users, i.e. “why do I have to wait one hour for my deposit in this exchange, but only 30 minutes on that exchange?”.</li>
+<li>It introduces a long delay which inhibits many useful applications.</li>
+</ul>
+<p>In addition to these user-facing and economic drawbacks, correctly handling rollbacks makes the code for nodes, wallets, and other infrastructure more complex. Worse still, many systems may not have correct behavior for rollbacks at different depths, and since large rollbacks are rarer, these implementation flaws may not surface until there is a large rollback. While a large rollback would be disruptive all by itself, it becomes even worse when previously undiscovered bugs exacerbate the situation.</p>
+<h2 id="trailing-finality-benefits"><a class="header" href="#trailing-finality-benefits">Trailing Finality Benefits</a></h2>
+<p>Trailing finality extends the existing PoW consensus so that older blocks become final, assuring they <em>cannot</em> be rolled back, and by extension neither can any of the transactions they contain.</p>
+<p>This directly addresses the first two flaws above: it completely removes the vulnerability, and it ensures all systems that need finality behave consistently with each other.</p>
+<p>As for delay, trailing finality also introduces delay since final blocks “trail behind” the most recent PoW blocks. This can be an improvement for some applications, depending on their latency requirements. For example, if the delay to finality averages around 10 minutes, then this would enable an improvement for an exchange that requires 60 minutes of PoW blocks for a deposit. On the other hand, it would not be an improvement for an application that needs finality faster than 10 minutes.</p>
+<p>Finally, implementations can be simplified by relying on the guarantee of finality. For example, a wallet can describe any transaction as pending or final, and does not need to provide difficult and potentially confusing UX (and the supporting database sophistication) for handling rollbacks.</p>
+<h1 id="footnotes-1"><a class="header" href="#footnotes-1">Footnotes</a></h1>
+<div class="footnote-definition" id="finality-qualifiers"><sup class="footnote-definition-label">1</sup>
+<p>Throughout this book, when we say <em>finality</em> or <em>final</em> without other qualifiers, we specifically are referring to <em>assured finality</em> or an <em>assured final</em> block. Where we call out <em>probabalistic finality</em> we always use that qualifier.</p>
+</div>
+<div class="footnote-definition" id="pow-rollback-security-assumptions"><sup class="footnote-definition-label">2</sup>
+<p>The estimated probability of a rollback relies on a variety of PoW security assumptions, and can be violated in various conditions, such as in mining efficiency breakthroughs, compromises of the PoW challenge algorithm (e.g. hash function collision resistance failure), difficulty-adjustment-algorithm failures, sudden/surprise mining capacity increases, and so on. So the estimated probability can be violated in potential “black swan” events.</p>
+</div>
+<div class="footnote-definition" id="txn-rollback"><sup class="footnote-definition-label">3</sup>
+<p>This discussion simplifies consideration of <em>transaction rollback</em> vs <em>block rollback</em>. When a block is rolled back, it is possible for some of the transactions contained in it to appear in new canonical blocks. The conditions when this can occur vs when it cannot are multifaceted and also subject to malicious influence, so for simplicity we assume all transactions within a rolled-back block are also rolled back.</p>
+</div>
+<div style="break-before: page; page-break-before: always;"></div><link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.12.0/dist/katex.min.css" integrity="sha384-AfEj0r4/OFrOo5t7NnNe46zW/tFgW6x/bCJG8FqQCEo3+Aro6EYUG4+cU+KJWu/X" crossorigin="anonymous">
+<h1 id="get-involved"><a class="header" href="#get-involved">Get Involved</a></h1>
+<p>We welcome contributions!</p>
+<p>There are a variety of ways to contribute to this project:</p>
+<h2 id="github"><a class="header" href="#github">GitHub</a></h2>
+<p>If you have a GitHub account, you can get hands-on via <a href="https://github.com/Electric-Coin-Company/tfl-book">the GitHub repository</a> for this book, including:</p>
+<ul>
+<li><a href="https://github.com/Electric-Coin-Company/tfl-book/issues/new?labels=question&amp;projects=&amp;template=question.yml">Ask a Question</a> - all questions welcome from basics to in-depth.</li>
+<li><a href="https://github.com/Electric-Coin-Company/tfl-book/issues/new?labels=improvement&amp;projects=&amp;template=improvement.yml">Suggest an Improvement</a> to the content: anything from typo fixes to major design change proposals.</li>
+<li><a href="https://github.com/Electric-Coin-Company/tfl-book/issues/new?labels=infrastructure&amp;projects=&amp;template=infrastructure.yml">Report Rendering / Infrastructure Issues</a> in case you're having trouble reading the content, viewing diagrams, rendering on your own computer, etc…</li>
+</ul>
+<h2 id="zcash-forum"><a class="header" href="#zcash-forum">Zcash Forum</a></h2>
+<p><a href="https://forum.zcashcommunity.com/">The Zcash Forum</a> is a hangout for many Zcash enthusiasts. This is a good spot for more open-ended discussion about this design proposal, alternatives, and other developments in Zcash.</p>
+<h2 id="zcash-rd-discord"><a class="header" href="#zcash-rd-discord">Zcash R&amp;D Discord</a></h2>
+<p>You can catch us on <a href="https://discord.gg/kCZf9NdneF">the Zcash R&amp;D Discord</a> in the <code>#proof-of-stake</code> channel.</p>
+<h2 id="zcash-arborist-calls"><a class="header" href="#zcash-arborist-calls">Zcash Arborist Calls</a></h2>
+<p>The <a href="https://zfnd.org/arborist-calls/">Zcash Arborist Calls</a> are bi-weekly Zcash protocol development calls, where proposals like this are discussed. Feel free to come lurk, ask questions, or provide feedback or suggestions.</p>
+<div style="break-before: page; page-break-before: always;"></div><link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.12.0/dist/katex.min.css" integrity="sha384-AfEj0r4/OFrOo5t7NnNe46zW/tFgW6x/bCJG8FqQCEo3+Aro6EYUG4+cU+KJWu/X" crossorigin="anonymous">
+<h1 id="terminology"><a class="header" href="#terminology">Terminology</a></h1>
+<p>This book relies on the following terminology. We strive to keep these definitions as precise as possible and their definitions may deviate from uses elsewhere.</p>
+<p>Definitions are sorted alphabetically.</p>
+<h2 id="terms"><a class="header" href="#terms">Terms</a></h2>
+<!-- Unlike true html, it seems `mdbook` requires the span tags to be empty and immediately pre-fix the intended anchor target. -->
+<p><span id="definition-assured-finality"></span>
+<strong>Assured Finality</strong>: A protocol property that assures that transactions cannot be reverted by that protocol. As with all protocol guarantees, a protocol assumes certain conditions must be met. A transaction may either be final or not: transactions which are not final may not become final, whereas once transactions do achieve finality they retain that property indefinitely (so long as protocol requirements are met).</p>
+<p>Importantly, it is not feasible for any protocol to prevent reversing final transactions &quot;out of band&quot; from the protocol, such as if a sufficiently large and motivated group of users forks the network to include a specific new validity rule reverting transactions. In some cases this might be desirable, for example to mitigate exploitation of a security flaw. We are investigating the implications for governance and how to incorporate such situations into our security model. In any case, for this reason we eschew the term &quot;absolute finality&quot; sometimes used in technical discussions about consensus protocols.</p>
+<p><span id="definition-consensus-subprotocols"></span>
+<strong>Consensus Subprotocols</strong>: The <a href="terminology.html#definition-pow">PoW</a> and <a href="terminology.html#defintion-pos">PoS</a> subprotocols in <a href="terminology.html#definition-pow-tfl">PoW+TFL</a> or other <a href="terminology.html#definition-hybrid-consensus">hybrid protocols</a>.</p>
+<p><span id="definition-crosslink"></span>
+<strong>Crosslink</strong>: A <a href="terminology.html#definition-hybrid-construction">hybrid construction</a> consensus protocol striving to implement the <a href="terminology.html#definition-tfl">TFL</a> design goals. See <a href="./introduction/status-and-next-steps.html#current-components">Status and Next Steps: Current Components</a> for current status.</p>
+<p><span id="definition-final"></span>
+<strong>Final</strong>: A protocol property of transactions. In this book, this always implies <a href="terminology.html#definition-assured-finality">assured finality</a>, in contrast to concepts like &quot;probabilistic finality&quot; provided by <a href="terminology.html#definition-pow">PoW</a>.</p>
+<p><span id="definition-hybrid-consensus"></span>
+<strong>Hybrid Consensus</strong>: A consensus protocol that integrates more than one consensus subprotocol. <a href="terminology.html#definition-pow-tfl">PoW+TFL</a> is an instance of a hybrid protocol integrating <a href="terminology.html#definition-pow">PoW</a> and <a href="terminology.html#definition-pos">PoS</a> protocols.</p>
+<p><span id="definition-hybrid-construction"></span>
+<strong>Hybrid Construction</strong>: The design component of a <a href="terminology.html#defintion-hybrid-consensus">hybrid consensus</a> which specifies how to integrate <a href="terminology.html#definition-consensus-subprotocols">subprotocols</a> and what modifications, if any, those subprotocols need to be safely integrated. Examples include <a href="terminology.html#definition-crosslink">Crosslink</a> and <a href="terminology.html#definition-snap-and-chat">Snap-and-Chat</a>.</p>
+<p><span id="definition-liveness"></span>
+<strong>Liveness</strong>: The property of a distributed protocol which ensures that the protocol may progress provided liveness requirements are met. <strong>TODO:</strong> <a href="https://github.com/Electric-Coin-Company/tfl-book/issues/120">Fix the definition of Liveness #120</a></p>
+<p><span id="definition-nu5nu6"></span>
+<strong>NU5/NU6</strong>: The Zcash consensus protocol as of NU5 or NU6 (which do not differ significantly in terms of the base block chain layer).<sup class="footnote-reference"><a href="#new-mainnet-precursors">1</a></sup></p>
+<p><span id="definition-objective-validity"></span>
+<strong>Objective Validity</strong>: A validity property of a protocol history (such as a ledger) which can be computed purely from that history with no other context. Objective validity is needed to define consensus rules that will lead to the same protocol state being eventually agreed on by all nodes.</p>
+<p><span id="definition-pos"></span>
+<strong>Proof-of-Stake</strong>: A PoS protocol achieves consensus on transaction status by taking into account the weighting of staking tokens. PoS protocols exist under a large umbrella and may or may not provide <a href="terminology.html#definition-assured-finality">assured finality</a> or other properties this design requires of <a href="terminology.html#definition-tfl">TFL</a>.</p>
+<p><span id="definition-pow"></span>
+<strong>Proof‑of‑Work</strong>: A PoW protocol uses Nakamoto consensus pioneered by Bitcoin. The PoW subprotocol within PoW+TFL is a different consensus protocol from <a href="terminology.html#definition-nu5nu6">NU5/NU6</a> and encompasses more than narrow Nakamoto PoW consensus, including transaction semantics such as for shielded transfers.</p>
+<p><span id="definition-pow-tfl"></span>
+<strong>PoW+TFL</strong>: the overall complete, integrated consensus protocol specified in this book.</p>
+<p><span id="definition-safety"></span>
+<strong>Safety</strong>: The property of a distributed protocol that guarantees a participant may safely rely on a consistent local state, provided safety requirements are met. <strong>TODO:</strong> <a href="https://github.com/Electric-Coin-Company/tfl-book/issues/121">Provide a rigorous definition of Safety #121</a></p>
+<p><span id="definition-simtfl"></span>
+<strong><code>simtfl</code></strong>: a protocol simulator for analyzing <a href="terminology.html#definition-tfl">TFL</a> security and abstract performance. Development lives at <a href="https://github.com/zcash/simtfl">https://github.com/zcash/simtfl</a></span>. See <a href="./introduction/status-and-next-steps.html#current-components">Status and Next Steps: Current Components</a> for current status.</p>
+<p><span id="definition-snap-and-chat"></span>
+<strong>Snap-and-Chat</strong>: A <a href="terminology.html#definition-hybrid-construction">hybrid construction</a> consensus protocol introduced in <a href="./references.html#ebb-and-flow-protocols">Ebb-and-Flow Protocols</a>.</p>
+<p><span id="definition-tfl"></span>
+<strong>TFL</strong>: The <em>Trailing Finality Layer</em> subprotocol within PoW+TFL. This is a new <a href="terminology.html#definition-pos">PoS</a> subprotocol which provides <a href="terminology.html#definition-assured-finality">assured finality</a> for Zcash.</p>
+<p><span id="definition-trailing-finality"></span>
+<strong>Trailing Finality</strong>: A protocol property wherein transactions become final some time after first appearing in <a href="terminology.html#definition-pow">PoW</a> blocks.</p>
+<p><span id="definition-zip"></span>
+<strong>ZIP</strong>: a Zcash Improvement Proposal is the protocol development process the Zcash community uses to safely define potential protocol improvements. See <a href="https://zips.z.cash">https://zips.z.cash</a></span>.</p>
+<p><strong>TODO:</strong> <a href="https://github.com/Electric-Coin-Company/tfl-book/issues/119">Clarify the distinctions between pure PoW, the PoW subprotocol, NU6, and fork-choice vs all of transaction semantics. #119</a></p>
+<h1 id="footnotes-2"><a class="header" href="#footnotes-2">Footnotes</a></h1>
+<div class="footnote-definition" id="new-mainnet-precursors"><sup class="footnote-definition-label">1</sup>
+<p>If new consensus changes are deployed to Zcash mainnet prior to PoW+TFL design finalization, this design must be updated to refer to the new delta (e.g. by reanalyzing all changes against NU6 or NU7, etc…)</p>
+</div>
+<!-- This trailing whitespace ensures that readers who follow a link to a definition will always see that term at the top of their view. -->
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+<div style="break-before: page; page-break-before: always;"></div><link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.12.0/dist/katex.min.css" integrity="sha384-AfEj0r4/OFrOo5t7NnNe46zW/tFgW6x/bCJG8FqQCEo3+Aro6EYUG4+cU+KJWu/X" crossorigin="anonymous">
+<h1 id="design-overview"><a class="header" href="#design-overview">Design Overview</a></h1>
+<p>This design augments the existing Zcash Proof‑of‑Work (PoW) network with a new consensus layer which provides <em>trailing finality</em>, called the <em>Trailing Finality Layer (TFL)</em>.</p>
+<p>This layer enables blocks produced via PoW to become <em>final</em> which ensures they may never be rolled back. This enables safer and simpler wallets and other infrastructure, and aids trust-minimized cross-chain bridges.</p>
+<p>This consensus layer uses a finalizing <em>Proof-of-Stake (PoS)</em> consensus protocol, and enables ZEC holders to earn protocol rewards for contributing to the security of the Zcash network. By integrating a PoS layer with the current PoW Zcash protocol, this design specifies a <em>hybrid consensus protocol</em>.</p>
+<p>The integration of the current PoW consensus with the TFL produces a new top-level consensus protocol referred to as <em>PoW+TFL</em>.</p>
+<p>In the following subchapters we introduce the <a href="./design/design-at-a-glance.html">Design at a Glance</a>, then provide an overview of the major components of the design.</p>
+<p>Following this overview chapter, we proceed into a detailed <a href="">Protocol Specification (TODO)</a>.</p>
+<div style="break-before: page; page-break-before: always;"></div><link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.12.0/dist/katex.min.css" integrity="sha384-AfEj0r4/OFrOo5t7NnNe46zW/tFgW6x/bCJG8FqQCEo3+Aro6EYUG4+cU+KJWu/X" crossorigin="anonymous">
+<h1 id="design-at-a-glance"><a class="header" href="#design-at-a-glance">Design at a Glance</a></h1>
+<p>The <a href="design/../terminology.html#definition-pow-tfl">PoW+TFL</a> consensus protocol is logically an extension of the Zcash consensus rules to introduce <a href="design/../terminology.html#definition-trailing-finality">trailing finality</a>. This is achieved by compartmentalizing the top-level PoW+TFL protocol into two <a href="design/../terminology.html#definition-consensus-subprotocols">consensus subprotocols</a>, one embodying most of the current consensus logic of Zcash and another the TFL. These protocols interact through a <a href="design/../terminology.html#definition-hybrid-construction">hybrid construction</a>, which specifies how the protocols interact, and what changes from &quot;off-the-shelf&quot; behavior, if any, need to be imposed on the subprotocols. Each of these components (the two subprotocols and the hybrid construction) are somewhat modular: different subprotocols or hybrid constructions may be combined (with some modification) to produce a candidate <a href="design/../terminology.html#definition-pow-tfl">PoW+TFL</a> protocol.</p>
+<p><strong>TODO:</strong> <a href="https://github.com/Electric-Coin-Company/tfl-book/issues/122">Add a protocol component diagram to &quot;Design at a Glance&quot; #122</a></p>
+<h2 id="hybrid-construction"><a class="header" href="#hybrid-construction">Hybrid Construction</a></h2>
+<p>The <a href="design/../terminology.html#definition-hybrid-construction">hybrid construction</a> is a major design component of the full consensus protocol which specifies how the subprotocols integrate. So far we have considered three candidates:</p>
+<ol>
+<li>The implied/loosely defined hybrid construction <a href="https://www.youtube.com/watch?v=qhMzMYeEPMM&amp;list=PL40dyJ0UYTLII7oQRQmNOFf0d2iKT35tL&amp;index=17">presented at Zcon4</a>.</li>
+<li>The <a href="design/../terminology.html#definition-snap-and-chat">Snap-and-Chat</a> from the <a href="https://eprint.iacr.org/2020/1091">Ebb-and-Flow paper</a>.</li>
+<li>The <a href="design/../terminology.html#definition-crosslink">Crosslink</a> construction.</li>
+</ol>
+<p>We believe <a href="design/../terminology.html#definition-crosslink">Crosslink</a> is the best candidate, due to its more rigorous <a href="design/./crosslink/construction.html">specification</a> and <a href="design/./crosslink/security-analysis.html">security analysis</a>, and due to the issues with Snap-and-Chat described in <a href="design/./crosslink/notes-on-snap-and-chat.html">Notes on Snap-and-Chat</a>.</p>
+<h2 id="subprotocols"><a class="header" href="#subprotocols">Subprotocols</a></h2>
+<p>The PoW+TFL hybrid consensus consists of two interacting subprotocols:</p>
+<ol>
+<li><a href="design/../terminology.html#definition-pow">PoW Subprotocol</a>: this subprotocol is very similar to NU5 or NU6 consensus. It is a design goal of the TFL design to minimize changes to this subprotocol. Note: the shorthand &quot;PoW&quot; is potentially misleading, because this subprotocol is also responsible for the bulk of all supply and transaction semantic consensus rules.</li>
+<li><a href="design/../terminology.html#definition-pos">PoS Subprotocol</a>: this is a new subprotocol which provides trailing finality via a finalizing PoS protocol.</li>
+</ol>
+<p><strong>TODO:</strong> <a href="https://github.com/Electric-Coin-Company/tfl-book/issues/119">Clarify the distinctions between pure PoW, the PoW subprotocol, NU6, and fork-choice vs all of transaction semantics. #119</a></p>
+<p>Note that the <a href="design/../terminology.html#definition-hybrid-construction">hybrid construction</a> may require modification to the &quot;off-the-shelf&quot; versions of these subprotocols. In particular <a href="design/../terminology.html#definition-crosslink">Crosslink</a> requires each protocol to refer to the state of the other to provide <a href="design/../terminology.html#definition-objective-validity">objective validity</a>.</p>
+<div style="break-before: page; page-break-before: always;"></div><link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.12.0/dist/katex.min.css" integrity="sha384-AfEj0r4/OFrOo5t7NnNe46zW/tFgW6x/bCJG8FqQCEo3+Aro6EYUG4+cU+KJWu/X" crossorigin="anonymous">
+<h1 id="design-goals"><a class="header" href="#design-goals">Design Goals</a></h1>
+<p>Here we strive to lay out our high level TFL design goals.</p>
+<h2 id="goals-design-and-trade-offs"><a class="header" href="#goals-design-and-trade-offs">Goals, Design, and Trade-offs</a></h2>
+<p>Here we lay out <em>ideal</em> goals. As we develop a complete design, we are likely to inevitably encounter trade-offs some of which may preclude achieving the full idealized goals. Wherever possible, we motivate design decisions by these goals, and when goals are impacted by trade-offs we describe that impact and the rationale for the trade-off decision.</p>
+<p>For example, one ideal user experience goal below is to avoid disruption to existing wallets. However, the <a href="design/../terminology.html#definition-crosslink">Crosslink</a> construction may require wallets to alter their context of valid transactions differently from the current <a href="design/../terminology.html#definition-nu5nu6">NU5/NU6</a> consensus protocol.</p>
+<h2 id="user-experience-and-use-case-goals"><a class="header" href="#user-experience-and-use-case-goals">User Experience and Use Case Goals</a></h2>
+<p>We strive to start our protocol design process from user experience (UX) and use case considerations foremost, since at the end of the day all that matters in a protocol is what user needs it meets and how well.</p>
+<ul>
+<li>All currently supported wallet user experience should continue to operate seamlessly without change during or after protocol transitions. This covers the use of addresses, payment flow, transfers, ZEC supply cap and issuance rate, backup/restore, and other features users currently rely on.</li>
+<li>There must be no security or safety degradation due to wallet user behavior introduced by PoS transitions, assuming users follow their current behaviors unchanged and continue to use the same cognitive model of the impacts of their behaviors. This goal encompasses all of security and safety, including privacy and transparency or more explicit disclosures.</li>
+<li>The protocol should enable users of shielded mobile wallets to delegate ZEC to PoS consensus providers and earn a return on that ZEC coming via ZEC issuance or fees. Doing this may expose users to a risk of loss of delegated ZEC (such as through “slashing fees”). The protocol must guarantee that PoS consensus providers have no discretionary control over such delegated funds (including that they cannot steal those funds).</li>
+<li>For any hybrid PoW/PoS protocol (including the PoW+TFL protocol we’re proposing), the process and UX of mining remains unchanged except that the return on investment may be altered. This is true both of consensus-level block miners (i.e. mining pools and solo miners) and mining pool participants.</li>
+<li>Any hybrid PoW/PoS protocol (including PoW+TFL) block explorers will continue to function with the same UX through transitions as far as displaying information about transactions, the mempool, and blocks.</li>
+<li>Block explorers and other network metrics sites may require UX changes with respect to mining rewards and issuance calculations.</li>
+<li>Network metrics sites may require UX changes with respect to the p2p protocol or other network-specific information.</li>
+<li>Users can rely on <a href="design/../terminology.html#definition-assured-finality">assured finality</a> with an expected time‑to‑finality below 30 minutes.<sup class="footnote-reference"><a href="#req-ttf-30-min">1</a></sup></li>
+</ul>
+<h2 id="developer-experience-goals"><a class="header" href="#developer-experience-goals">Developer Experience Goals</a></h2>
+<p>For a full PoS transition, ecosystem developers for products such as consensus nodes, wallets, mining services, chain analytics, and more will certainly need to update their code to support transitions. However, we carve out a few goals as an exception to this for this category of users:</p>
+<ul>
+<li>Wallet developers should not be required to make any changes through protocol transitions as long as they rely solely on the lightwalletd protocol or a full node API (such as the zcashd RPC interface).</li>
+<li>For any hybrid PoW/PoS protocol (including PoW+TFL), mining pools and miners should not be required to make any software or protocol changes as long as they rely on zcashd-compatible GetBlockTemplate. One exception to this is software that bakes in assumptions about the block reward schedule, rather than relying on GetBlockTemplate solely.</li>
+</ul>
+<h2 id="safety-security-and-privacy-goals"><a class="header" href="#safety-security-and-privacy-goals">Safety, Security, and Privacy Goals</a></h2>
+<p>Zcash has always had exemplary safety, security, and privacy, and we aim to continue that tradition:</p>
+<ul>
+<li>For any hybrid PoW/PoS protocol (including PoW+TFL), the cost‑of‑attack for a 1-hour rollback should not be reduced, given a “reasonably rigorous” security argument.</li>
+<li>For any hybrid PoW/PoS protocol (including PoW+TFL), the cost‑of‑attack to halt the chain should be larger than the 24‑hour revenue of PoW mining rewards, given a “reasonably rigorous” security argument.</li>
+</ul>
+<p><strong>TODO:</strong> <a href="https://github.com/Electric-Coin-Company/tfl-book/issues/118">Define privacy goals of TFL #118</a></p>
+<p><strong>TODO:</strong> <a href="https://github.com/Electric-Coin-Company/tfl-book/issues/117">Define PoS Subprotocol desiderata which are distinct from Crosslink integration #117</a></p>
+<h2 id="design-conservatism-goals"><a class="header" href="#design-conservatism-goals">Design Conservatism Goals</a></h2>
+<p>We want to follow some conservative design heuristics to minimize risk and mistakes:</p>
+<ul>
+<li>Rely as much as possible on design components that are already proven in production environments.</li>
+<li>Rely as much as possible on design components with adequate theoretical underpinnings and security analyses.</li>
+<li>Minimize changes or variations on the above: strive to only alter existing work when necessary for overall design goals. For example, Zcash’s privacy or issuance constraints are likely less common among existing PoS designs.</li>
+</ul>
+<h1 id="non-goals"><a class="header" href="#non-goals">Non-goals</a></h1>
+<p>These are not goals of the TFL design, either to simplify the scope of the initial design (a.k.a. <a href="design/goals.html#out-of-scope-goals">Out-of-Scope Goals</a>), or because we believe some potential goal <em>should not</em> be supported (a.k.a. <a href="design/goals.html#anti-goals">Anti-goals</a>).</p>
+<h2 id="out-of-scope-goals"><a class="header" href="#out-of-scope-goals">Out-of-Scope Goals</a></h2>
+<p>While these desiderata may be common across the block‑chain consensus design space, they are not specific goals for the initial TFL design. Note that these may be goals for future protocol improvements.</p>
+<ul>
+<li>
+<p>Prioritizing minimal time-to-finality over other considerations (such as protocol simplicity, impact on existing use cases, or other goals above).</p>
+</li>
+<li>
+<p>In-protocol liquid staking derivatives.</p>
+</li>
+<li>
+<p>Maximizing the PoS staked-voter count ceiling. For example, <a href="https://github.com/tendermint/tendermint#research">Tendermint BFT</a> has a relatively low ceiling of ~hundreds of staked voters, whereas Ethereum’s <a href="https://ethereum.org/en/developers/docs/consensus-mechanisms/pos/gasper/">Gasper</a> supports hundreds of thousands of staked voters.</p>
+</li>
+<li>
+<p>Reducing energy usage. While this would presumably be a goal of a pure PoS transition, it likely cannot be achieved for hybrid PoW/PoS without loss of security.</p>
+</li>
+</ul>
+<h2 id="anti-goals"><a class="header" href="#anti-goals">Anti-Goals</a></h2>
+<p>Distinctly from <a href="design/goals.html#out-of-scope-goals">Out-of-Scope Goals</a> we track &quot;anti-goals&quot; which are potential goals that we <em>explicitly reject</em>, which are potential goals we aim to <em>not</em> support even in future protocol improvements.</p>
+<p>We currently have no defined anti-goals.</p>
+<h1 id="footnotes-3"><a class="header" href="#footnotes-3">Footnotes</a></h1>
+<div class="footnote-definition" id="req-ttf-30-min"><sup class="footnote-definition-label">1</sup>
+<p>This requirement comes from <a href="https://github.com/Electric-Coin-Company/tfl-book/issues/38">a request from a DEX developer</a>. While we have not yet surveyed DEX and Bridge designs, we're relying on this as a good starting point.</p>
+</div>
+<div style="break-before: page; page-break-before: always;"></div><link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.12.0/dist/katex.min.css" integrity="sha384-AfEj0r4/OFrOo5t7NnNe46zW/tFgW6x/bCJG8FqQCEo3+Aro6EYUG4+cU+KJWu/X" crossorigin="anonymous">
+<h1 id="crosslink-1"><a class="header" href="#crosslink-1">Crosslink</a></h1>
+<p>Crosslink is the proposed <a href="design/../terminology.html#definition-hybrid-construction">hybrid construction</a> for the Trailing Finality Layer. The current version is Crosslink 2.</p>
+<h2 id="contents"><a class="header" href="#contents">Contents</a></h2>
+<ul>
+<li><a href="design/./crosslink/the-arguments-for-bounded-availability-and-finality-overrides.html">The Arguments for Bounded Availability and Finality Overrides</a></li>
+<li><a href="design/./crosslink/notes-on-snap-and-chat.html">Notes On Snap‑and‑Chat</a></li>
+<li><a href="design/./crosslink/construction.html">The Crosslink 2 Construction</a></li>
+<li><a href="design/./crosslink/security-analysis.html">Security Analysis of Crosslink 2</a></li>
+<li><a href="design/./crosslink/questions.html">Questions About Crosslink</a></li>
+<li><a href="design/./crosslink/potential-changes.html">Potential Changes to Crosslink</a></li>
+</ul>
+<div style="break-before: page; page-break-before: always;"></div><link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.12.0/dist/katex.min.css" integrity="sha384-AfEj0r4/OFrOo5t7NnNe46zW/tFgW6x/bCJG8FqQCEo3+Aro6EYUG4+cU+KJWu/X" crossorigin="anonymous">
+<h1 id="the-arguments-for-bounded-availability-and-finality-overrides"><a class="header" href="#the-arguments-for-bounded-availability-and-finality-overrides">The Arguments for Bounded Availability and Finality Overrides</a></h1>
+<p>This document considers disadvantages of allowing transactions to continue to be included at the chain tip while the gap from the last finalized block becomes unbounded, and our perspective on what should be done instead. This condition is allowed by Ebb‑and‑Flow protocols <a href="https://eprint.iacr.org/2020/1091.pdf">[NTT2020]</a>.</p>
+<p>We also argue that it is necessary to allow for the possibility of overriding finalization in order to respond to certain attacks, and that this should be explicitly modelled and subject to a well-defined governance process.</p>
+<p>This is a rewritten version of <a href="https://forum.zcashcommunity.com/t/the-trailing-finality-layer/45133/10">this forum post</a>, adapting the main argument to take into account the discussion of “tail‑thrashing attacks” and finalization availability from the Addendum. More details of how bounded availability could be implemented in the context of a Snap‑and‑Chat protocol are in <a href="design/crosslink/./notes-on-snap-and-chat.html">Notes on Snap‑and‑Chat</a>.</p>
+<p>The proposed changes end up being significant enough to give our construction a new name: “<a href="design/crosslink/./construction.html">Crosslink</a>”, referring to the cross-links between blocks of the BFT and best-chain protocols. Crosslink has evolved somewhat, and now includes other changes not covered in either this document or <a href="design/crosslink/./notes-on-snap-and-chat.html">Notes on Snap‑and‑Chat</a>. The current version is called <a href="design/crosslink/./construction.html">Crosslink 2</a>.</p>
+<h2 id="background"><a class="header" href="#background">Background</a></h2>
+<p>“Ebb‑and‑Flow”, as described in <a href="https://eprint.iacr.org/2020/1091.pdf">[NTT2020]</a> (<a href="https://arxiv.org/pdf/2009.04987.pdf">arXiv version</a>), is a security model for consensus protocols that provide two transaction logs, one with dynamic availability, and a prefix of it with finality.</p>
+<p>The paper proposes an instantiation of this security model called a “Snap‑and‑Chat” construction. It composes two consensus subprotocols, a BFT subprotocol and a best-chain subprotocol (it calls this the “longest chain protocol”). The above logs are obtained from the output of these subprotocols in a non-trivial way.</p>
+<p>This is claimed by the paper to “resolve” the tension between finality and dynamic availability. However, a necessary consequence is that in a situation where the “final” log stalls and the “available” log does not, the “finalization gap” between the finalization point and the chain tip can grow without bound. In particular, this means that transactions that spend funds can remain unfinalized for an arbitrary length of time.</p>
+<p>In this document, we argue that this is unacceptable, and that it is preferable to sacrifice strict dynamic availability. The main idea behind Ebb‑and‑Flow protocols is a good one, and allowing the chain tip to run ahead of the finalization point does make sense and has practical advantages. However, we also argue that it should not be possible to include transactions that spend funds in blocks that are too far ahead of the finalization point.</p>
+<div id="admonition-note" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Note</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/the-arguments-for-bounded-availability-and-finality-overrides.html#admonition-note"></a></p>
+</div>
+<div>
+<p>Naive ways of preventing an unbounded finalization gap, such as stopping the chain completely in the case of a finalization stall, turn out to run into serious security problems — at least when the best-chain protocol uses Proof‑of‑Work. We’ll discuss those in detail in the section on <a href="design/crosslink/the-arguments-for-bounded-availability-and-finality-overrides.html#tail%E2%80%91thrashing-attacks">tail‑thrashing attacks</a>.</p>
+<p>Our proposed solution will be to require coinbase-only blocks during a long finalization stall. This solution has the advantage of not complicating the security analysis.</p>
+</div>
+</div>
+<p>We argue that losing strict dynamic availability in favour of “bounded availability” is preferable to the consequences of the unbounded finality gap, if/when a “long finalization stall” occurs.</p>
+<p>We also argue that it is beneficial to explicitly allow “finality overrides” under the control of a well-documented governance process. Such overrides allow long rollbacks that may be necessary in the case of an exploited security flaw. This is complementary to the argument for bounded availability, because the latter limits the period of user transactions that could be affected. The governance process can impose a limit on the length of this long rollback if desired.</p>
+<h2 id="finality--dynamic-availability--possibility-of-partition-implies-an-unbounded-finalization-gap"><a class="header" href="#finality--dynamic-availability--possibility-of-partition-implies-an-unbounded-finalization-gap">Finality + Dynamic availability + Possibility of Partition implies an unbounded finalization gap</a></h2>
+<p>Since partition between nodes sufficient for finalization cannot be prevented, loosely speaking the <a href="https://en.wikipedia.org/wiki/CAP_theorem">CAP theorem</a> implies that any consistent protocol (and therefore any protocol with finality) may stall for at least as long as the partition takes to heal.</p>
+<div id="admonition-background" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Background</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/the-arguments-for-bounded-availability-and-finality-overrides.html#admonition-background"></a></p>
+</div>
+<div>
+<p>That “loosely speaking” is made precise by <a href="https://arxiv.org/pdf/2006.10698.pdf">[LR2020]</a>.</p>
+</div>
+</div>
+<p>Dynamic availability implies that the chain tip will continue to advance, and so the finalization gap increases without bound.</p>
+<p>Partition is not necessarily the only condition that could cause a finalization stall, it is just the one that most easily proves that this conclusion is impossible to avoid.</p>
+<h2 id="problems-with-allowing-spends-in-an-unbounded-finalization-gap"><a class="header" href="#problems-with-allowing-spends-in-an-unbounded-finalization-gap">Problems with allowing spends in an unbounded finalization gap</a></h2>
+<p>Both the available protocol, and the subprotocol that provides finality, will be used in practice — otherwise, one or both of them might as well not exist. There is always a risk that blocks may be rolled back to the finalization point, by definition.</p>
+<p>Suppose, then, that there is a long finalization stall. The final and available protocols are not separate: there is no duplication of tokens between protocols, but the rules about how to determine best-effort balance and guaranteed balance depend on both protocols, how they are composed, and how the history after the finalization point is interpreted.</p>
+<div id="admonition-discussion" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Discussion</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/the-arguments-for-bounded-availability-and-finality-overrides.html#admonition-discussion"></a></p>
+</div>
+<div>
+<p>The guaranteed minimum balance of a given party is not just the minimum of their balance at the finalization point and their balance at the current tip. It is the minimum balance taken over all possible transaction histories that extend the finalized chain — taking into account that a party’s previously published transactions might be able to be reapplied in a different context without its explicit consent. The extent to which published transactions can be reapplied depends on technical choices that we must make, subject to some constraints (for example, we know that shielded transactions cannot be reapplied after their anchors have been invalidated). It may be desirable to further constrain re-use in order to make guaranteed minimum balances easier to compute.</p>
+</div>
+</div>
+<p>As the finalization gap increases, the negative consequences of rolling back user transactions that spend funds increase. (Coinbase transactions do not spend funds; they are a special case that we will discuss later.)</p>
+<p>There are several possible —not mutually exclusive— outcomes:</p>
+<ul>
+<li>Users of the currency start to consider the available protocol increasingly unreliable.</li>
+<li>Users start to consider a rollback to be untenable, and lobby to prevent it or cry foul if it occurs.</li>
+<li>Users start to consider finalization increasingly irrelevant. Services that depend on finality become unavailable.
+<ul>
+<li>There is no free lunch that would allow us to avoid availability problems for services that also depend on finality.</li>
+</ul>
+</li>
+<li>Service providers adopt temporary workarounds that may not have had adequate security analysis.</li>
+</ul>
+<p>Any of these might precipitate a crisis of confidence, and there are reasons to think this effect might be worse than if the chain had switched to a “Stalled Mode” designed to prevent loss of user funds. Any such crisis may have a negative effect on token prices and long-term adoption.</p>
+<p>Note that adding finalization using an Ebb‑and‑Flow protocol does not by itself increase the probability of a rollback in the available chain, provided the PoW remains as secure against rollbacks of a given length as before. But that is a big proviso. We have a design constraint (motivated by limiting token devaluation and by governance issues) to limit issuance to be no greater than that of the original Zcash protocol up to a given height. Since some of the issuance is likely needed to reward staking, the amount of money available for mining rewards is reduced, which may reduce overall hash rate and security of the PoW. Independently, there may be a temptation for design decisions to rely on finality in a way that reduces security of PoW (“risk compensation”). There is also pressure to reduce the energy usage of PoW, which necessarily reduces the global hash rate, and therefore the cost of performing an attack that depends on the adversary having any given proportion of global hash rate.</p>
+<p>It could be argued that the issue of availability of services that depend on finality is mainly one of avoiding over-claiming about what is possible. Nevertheless there are also real usability issues if balances as seen by those services can differ significantly and for long periods from balances at the chain tip.</p>
+<p>Regardless, incorrect assumptions about the extent to which the finalized and available states can differ are likely to be exposed if there is a finalization stall. And those who made the assumptions may (quite reasonably!) not accept “everything is fine, those assumptions were always wrong” as a satisfactory response.</p>
+<h2 id="what-is-bounded-availability"><a class="header" href="#what-is-bounded-availability">What is Bounded Availability?</a></h2>
+<p>An intuitive notion of “availability” for block‑chain protocols includes the ability to use the protocol as normal to spend funds. So, just to be clear, in a situation where that cannot happen we have lost availability, <em>even if</em> the block chain is advancing.</p>
+<p>Bounded availability is a weakening of dynamic availability <a href="https://arxiv.org/abs/2010.08154">[DKT2020]</a>. It means that we intentionally sacrifice availability when some potentially hazardous operation —a “hazard” for short— would occur too far after the current finalization point. For now, assume for simplicity that our only hazard is spending funds. More generally, the notion of bounded availability can be applied to a wider range of protocols by tailoring the definition of “hazard” to the protocol.</p>
+<div id="admonition-background-1" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Background</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/the-arguments-for-bounded-availability-and-finality-overrides.html#admonition-background-1"></a></p>
+</div>
+<div>
+<p>This <a href="https://www.youtube.com/watch?v=SacUT_53Pg8">talk by Soubhik Deb</a> accompanying <a href="https://arxiv.org/abs/2010.08154">[DKT2020]</a> provides a good explanation of the advantages of dynamic availability. We do not define bounded availability formally in this document, but informally, we aim to preserve the ability to securely adapt to large changes in hash rate or total stake.</p>
+</div>
+</div>
+<h3 id="terminology-note"><a class="header" href="#terminology-note">Terminology note</a></h3>
+<p><a href="https://eprint.iacr.org/2020/1091.pdf">[NTT2020]</a> calls the dynamically available block‑chain protocol <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> that provides input to the rest of the contruction, the “longest chain” protocol. There are two reasons to avoid this terminology:</p>
+<ul>
+<li>In Bitcoin, Zcash, and most other PoW-based protocols, what is actually used by each node is not its longest observed chain, but its observed consensus-valid chain with most accumulated work. In Zcash this is called the node’s “<a href="https://zips.z.cash/protocol/protocol.pdf#blockchain">best valid block chain</a>”, which we shorten to “best chain”.</li>
+<li>As footnote 2 on page 3 of <a href="https://eprint.iacr.org/2020/1091.pdf">[NTT2020]</a> says, that paper does not require <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> to be a “longest chain” protocol anyway.</li>
+</ul>
+<div id="admonition-historical-note" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Historical note</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/the-arguments-for-bounded-availability-and-finality-overrides.html#admonition-historical-note"></a></p>
+</div>
+<div>
+<p>The error in conflating the “longest chain” with the observed consensus-valid chain with most accumulated work, originates in the Bitcoin whitepaper. [<a href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.221.9986">Nakamoto2008</a>, page 3]</p>
+</div>
+</div>
+<p>We will use the term “best‑chain protocol” instead. Note that this corresponds roughly to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> in the Snap‑and‑Chat construction, although the Crosslink 2 protocol that we propose will end up having other significant differences from Snap-and-Chat.</p>
+<h3 id="how-to-block-hazards"><a class="header" href="#how-to-block-hazards">How to block hazards</a></h3>
+<p>We have not yet decided <em>how</em> to block hazards during a long finalization stall. We could do so directly, or by stopping block production in the more-available protocol. For reasons explained in the section on <a href="design/crosslink/the-arguments-for-bounded-availability-and-finality-overrides.html#tail%E2%80%91thrashing-attacks">tail‑thrashing attacks</a> below, it’s desirable <em>not</em> to stop block production. And so it’s consistent to have bounded availability together with another liveness property —which can be defined similarly to dynamic availability— that says the more-available protocol’s chain is still advancing. This is what we will aim for.</p>
+<p>We will call this method of blocking hazards, without stopping block production, “going into Stalled Mode”.</p>
+<div id="admonition-historical-note-1" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Historical note</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/the-arguments-for-bounded-availability-and-finality-overrides.html#admonition-historical-note-1"></a></p>
+</div>
+<div>
+<p>This concept of Stalled Mode is very similar to a <a href="https://github.com/zcash/zcash/issues/3311">feature that was discussed early in the development of Zcash</a>, but never fully designed or implemented. (After originally being called “Stalled Mode”, it was at some point renamed to “Emergency Mode”, but then the latter term was used for <a href="https://electriccoin.co/blog/update-addressing-zcash-wallet-performance-issues/">something else</a>.)</p>
+</div>
+</div>
+<p>For Zcash, we propose that the main restriction of Stalled Mode should be to require coinbase-only blocks. This achieves a similar effect, for our purposes, as actually stalling the more-available protocol’s chain. Since funds cannot be spent in coinbase-only blocks, the vast majority of attacks that we are worried about would not be exploitable in this state.</p>
+<div id="admonition-info" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Info</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/the-arguments-for-bounded-availability-and-finality-overrides.html#admonition-info"></a></p>
+</div>
+<div>
+<p>It is possible that a security flaw could affect coinbase transactions. We <em>might</em> want to turn off shielded coinbase for Stalled Mode blocks in order to reduce the chance of that.</p>
+<p>Also, mining rewards cannot be spent in a coinbase-only block; in particular, mining pools cannot distribute rewards. So there is a risk that an unscrupulous mining pool might try to do a rug-pull after mining of non-coinbase-only blocks resumes, if there were a very long finalization stall. But this approach works at least in the short term, and probably for long enough to allow manual intervention into the finalization protocol, or governance processes if needed.</p>
+</div>
+</div>
+<p>An analogy for the effect of this on availability that may be familiar to many people, is that it works like video streaming. All video streaming services use a buffer to paper over short-term interruptions or slow-downs of network access. In most cases, this buffer is bounded. This allows the video to be watched uninterrupted and at a constant rate in most circumstances. But if there is a longer-term network failure or insufficient sustained bandwidth, the playback will unavoidably stall. In our case, block production does not literally stall, but it’s the same as far as users’ ability to perform “hazardous” operations is concerned.</p>
+<h3 id="why-is-this-better"><a class="header" href="#why-is-this-better">Why is this better?</a></h3>
+<p>So, why do we advocate this over:</p>
+<ol>
+<li>A protocol that only provides dynamic availability;</li>
+<li>A protocol that only provides finality;</li>
+<li>An unmodified Ebb‑and‑Flow protocol?</li>
+</ol>
+<p>The reason to reject option 1 is straightforward: finality is a valuable security property that is necessary for some use cases.</p>
+<p>If a protocol only provides finality (option 2), then short-term availability is directly tied to finalization. It may be possible to make finalization stalls sufficiently rare or short-lived that this is tolerable. But that is more likely to be possible if and when there is a well-established staking ecosystem. Before that ecosystem is established, the protocol may be particularly vulnerable to stalls. Furthermore, it’s difficult to get to such a protocol from a pure PoW system like current Zcash.</p>
+<p>We argued in the previous section that allowing hazards in an unbounded finalization gap is bad. Option 3 entails an unbounded finalization gap that <em>will</em> allow hazards. However, that isn’t sufficient to argue that bounded availability is better. Perhaps there are no good solutions! What are we gaining from a bounded availability approach that would justify the complexity of a hybrid protocol without obtaining strict dynamic availability?</p>
+<p>The argument goes like this:</p>
+<ul>
+<li>It is likely that a high proportion of the situations in which a sustained finalization stall happens will require human intervention. If the finality protocol were going to recover without intervention, there is no reason to think that it wouldn’t do so in a relatively short time.</li>
+<li>When human intervention is required, the fact that the chain tip is still proceeding apace (in protocols with strict dynamic availability) makes restarting the finality protocol harder, for many potential causes of a finalization stall. This may be less difficult when only “non-hazardous” transactions are present— in particular, when only coinbase transactions (which are subject to fairly strict rules in Zcash and other Bitcoin-derived chains) are present. This argument carries even more force when the protocol also allows “finality overrides”, as discussed later in the <a href="design/crosslink/the-arguments-for-bounded-availability-and-finality-overrides.html#complementarity">Complementarity</a> section.</li>
+<li>Nothing about bounded availability prevents us from working hard to design a system that makes finalization stalls as infrequent and short-lived as possible, just as we would for any other option that provides finality.</li>
+<li>We want to optimistically minimize the finalization gap under good conditions, because this improves the usability of services that depend on finality. This argues against protocols that try to maintain a fixed gap, and motivates letting the gap vary.</li>
+<li>In practice, the likelihood of short finalization stalls is high enough that heuristically retaining availability in those situations is useful.</li>
+</ul>
+<p>The argument that it is difficult to completely prevent finalization stalls is supported by <a href="https://www.coindesk.com/tech/2023/05/17/ethereums-loss-of-finality-what-happened/">experience on Ethereum in May 2023</a>, when there were two stalls within 24 hours, <a href="https://offchain.medium.com/post-mortem-report-ethereum-mainnet-finality-05-11-2023-95e271dfd8b2">one for about 25 minutes and one for about 64 minutes</a>. This experience is consistent with our argument:</p>
+<ul>
+<li>Neither stall required short-term human intervention, and the network did in fact recover from them quickly.</li>
+<li>The stalls were caused by a resource exhaustion problem in the Prysm consensus client when handling attestations. It’s plausible to think that if this bug had been more serious, or possibly if Prysm clients had made up more of the network, then it would have required a hotfix release (and/or a significant proportion of nodes switching to another client) in order to resolve the stall. So this lines up with the hypothesis that longer stalls are likely to require manual intervention.</li>
+<li>A bounded availability protocol would very likely have resulted in either a shorter or no interruption in availability. If, say, the availability bound were set to be roughly an hour, then the first finalization stall would have been “papered over” and the second would have resulted in only a short loss of availability.</li>
+</ul>
+<p>Retaining short-term availability does not result in a risk compensation hazard:</p>
+<ul>
+<li>A finalization stall is still very visible, and directly affects applications relying on finality.</li>
+<li>Precisely because of the availability bound, it is obvious that it could affect all applications if it lasted long enough.</li>
+</ul>
+<p>A potential philosophical objection to lack of strict dynamic availability is that it creates a centralization risk to availability. That is, it becomes more likely that a coalition of validators can deliberately cause a denial of service. This objection may be more prevalent among people who would object to adding a finality layer or PoS at all.</p>
+<h2 id="finality-overrides"><a class="header" href="#finality-overrides">Finality Overrides</a></h2>
+<p>Consensus protocols sometimes fail. Potential causes of failure include:</p>
+<ul>
+<li>A design problem with the finality layer that causes a stall, or allows a stall to be provoked.</li>
+<li>A balance violation or spend authorization flaw that is being exploited or is sufficiently likely to be exploited.</li>
+<li>An implementation bug in a widely used node implementation that causes many nodes to diverge from consensus.</li>
+</ul>
+<p>In these situations, overriding finality may be better than any other alternative.</p>
+<p>An example is a <a href="https://bitcointalk.org/index.php?topic=822.0">balance violation flaw due to a 64-bit integer overflow</a> that was <a href="https://en.bitcoin.it/wiki/Value_overflow_incident">exploited on Bitcoin mainnet</a> on 15th August 2010. The response was to roll back the chain to before the exploitation, which is widely considered to have been the right decision. The time between the exploit (at block height 74638) and the forked chain overtaking the exploited chain (at block height 74691) was 53 blocks, or around 9 hours.</p>
+<p>Of course, Bitcoin used and still uses a pure‑PoW consensus. But the applicability of the example does not depend on that: the flaw was independent of the consensus mechanism.</p>
+<p>Another example of a situation that prompted this kind of override was the <a href="https://hackingdistributed.com/2016/06/18/analysis-of-the-dao-exploit/">DAO recursion exploit</a> on the Ethereum main chain <a href="https://www.coindesk.com/learn/understanding-the-dao-attack/">in June 2016</a>. The response to this was the forced balance adjustment hard fork on 20th July 2016 commonly known as the <a href="https://ethereum.org/en/history/#dao-fork">DAO fork</a>. Although this adjustment was not implemented as a rollback, and although Ethereum was using PoW at the time and did not make any formal finality guarantees, it did override transfers that would heuristically have been considered final at the fork height. Again, this flaw was independent of the consensus mechanism.</p>
+<p>The DAO fork was of course much more controversial than the Bitcoin fork, and a substantial minority of mining nodes split off to form Ethereum Classic. In any case, the point of this example is that it’s always possible to override finality in response to an exceptional situation, and that a chain’s community may decide to do so. The fact that Ethereum 2.0 now does claim a finality guarantee, would not in practice prevent a similar response in future that would override that guarantee.</p>
+<p>The question then is whether the procedure to override finality should be formalized or ad hoc. We argue that it should be formalized, including specifying the governance process to be used.</p>
+<p>This makes security analysis — of the consensus protocol per se, of the governance process, and of their interaction — much more feasible. Arguably a complete security analysis is not possible at all without it.</p>
+<p>It also front‑loads arguing about what procedure should be followed, and so it is more likely that stakeholders will agree to follow the process in any time‑critical incident.</p>
+<h3 id="a-way-of-modelling-overrides-that-is-insufficient"><a class="header" href="#a-way-of-modelling-overrides-that-is-insufficient">A way of modelling overrides that is insufficient</a></h3>
+<p>There is another possible way to model a protocol that claims finality but can be overridden in practice. We could say that the protocol after the override is a brand‑new protocol and chain (inheriting balances from the previous one, possibly modulo adjustments such as those that happened in the DAO fork).</p>
+<p>Although that would allow saying that the finality property has technically not been violated, it does not match how users think about an override situation. They are more likely to think of it as a protocol with finality that can be violated in exceptional cases — and they would reasonably want to know what those cases are and how they will be handled. It also does nothing to help with security analysis of such cases.</p>
+<h2 id="complementarity"><a class="header" href="#complementarity">Complementarity</a></h2>
+<p>Finality overrides and bounded availability are complementary in the following way: if a problem is urgent enough, then validators can be asked to stop validating. For genuinely harmful problems, it is likely to be in the interests of enough validators to stop that this causes a finalization stall. If this lasts longer than the availability bound then the protocol will go into Stalled Mode, giving time for the defined governance process to occur and decide what to do. And because the unfinalized consensus chain will contain only a limited period of user transactions that spend funds, the option of a long rollback remains realistically open.</p>
+<p>If, on the other hand, there is time pressure to make a governance decision about a rollback in order to reduce its length, that may result in a less well-considered decision.</p>
+<p>A possible objection is that there might be a coalition of validators who ignore the request to stop (possibly including the attacker or validators that an attacker can bribe), in which case the finalization stall would not happen. But that just means that we don’t gain the advantage of more time to make a governance decision; it isn’t actively a disadvantage relative to alternative designs. This outcome can also be thought of as a feature rather than a bug: going into Stalled Mode should be a last resort, and if the argument given for the request to stop failed to convince a sufficient number of validators that it was reason enough to do so, then perhaps it wasn’t a good enough reason.</p>
+<div id="admonition-rationale" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Rationale</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/the-arguments-for-bounded-availability-and-finality-overrides.html#admonition-rationale"></a></p>
+</div>
+<div>
+<p>This resolves one of the main objections to the <a href="https://github.com/zcash/zcash/issues/3311">original Stalled Mode idea</a> that stopped us from implementing it in Zcash. The original proposal was to use a signature with a key held by ECC to trigger Stalled Mode, which would arguably have been too centralized. The Stalled Mode described in this document, on the other hand, can only be entered by consensus of a larger validator set, or if there is an availability failure of the finalization protocol.</p>
+</div>
+</div>
+<p>It is also possible to make the argument that the threshold of stake needed is imposed by technical properties of the finality protocol and by the resources of the attacker, which might not be ideal for the purpose described above. However, we would argue that it does not need to be ideal, and will be in the right ballpark in practice.</p>
+<p>There’s a caveat related to doing intentional rollbacks when using the Stalled Mode approach, where block production in the more-available protocol continues during a long finalization stall. What happens to incentives of block producers (miners in the case of Proof‑of‑Work), given that they know the consensus chain might be intentionally rolled back? They might reasonably conclude that it is less valuable to produce those blocks, leading to a reduction of hash rate or other violations of the security assumptions of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</p>
+<p>This is actually fairly easy to solve. We have the governance procedures say that if we do an intentional rollback, the coinbase-only mining rewards will be preserved. I.e. we produce a block or blocks that include those rewards paid to the same addresses (adjusting the consensus to allow them to be created from thin air if necessary), have everyone check it thoroughly, and require the chain to restart from that block. So as long as block producers believe that this governance procedure will be followed and that the chain will eventually recover at a reasonable coin price, they will still have incentive to produce on <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>, at least for a time.</p>
+<div id="admonition-rationale-1" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Rationale</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/the-arguments-for-bounded-availability-and-finality-overrides.html#admonition-rationale-1"></a></p>
+</div>
+<div>
+<p>Although the community operating the governance procedures has already obtained the security benefit of mining done on the rolled-back chain by the time it creates the new chain, there is a strong incentive not to renege on the agreement with miners, because the same situation may happen again.</p>
+</div>
+</div>
+<h2 id="tail-thrashing-attacks"><a class="header" href="#tail-thrashing-attacks">Tail-thrashing attacks</a></h2>
+<p>Earlier we said that there were two possible approaches to preventing hazards during a long finalization stall:</p>
+<p>a) go into a Stalled Mode that directly disallows hazardous transactions (for example, by requiring <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> blocks to be coinbase-only in Zcash);</p>
+<p>b) temporarily cause the more-available chain to stall.</p>
+<p>This section describes an important class of potential attacks on approach b) that are difficult to resolve. They are based on the fact that when the unfinalized chain stalls, an adversary has more time to find blocks, and this might violate security assumptions of the more-available protocol. For instance, if the more-available protocol is PoW-based, then its security in the steady state is predicated on the fact that an adversary with a given proportion of hash power has only a limited time to use that power, before the rest of the network finds another block.</p>
+<div id="admonition-background-2" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Background</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/the-arguments-for-bounded-availability-and-finality-overrides.html#admonition-background-2"></a></p>
+</div>
+<div>
+<p>For an analysis of the concrete security of Nakamoto-like protocols, see <a href="https://arxiv.org/abs/2005.10484">[DKT+2020]</a> and <a href="https://eprint.iacr.org/2020/661">[GKR2020]</a>. These papers confirm the intuition that the “private attack” —in which an adversary races privately against the rest of the network to construct a forking chain— is optimal, obtaining the same tight security bound independently using different techniques.</p>
+</div>
+</div>
+<p>During a chain stall, the adversary no longer has a limited time to construct a forking chain. If, say, the adversary has 10% hash power, then it can on average find a block <span style="white-space: nowrap">in 10 block times.</span> <span style="white-space: nowrap">And so in 100 block times</span> it can create <span style="white-space: nowrap">a 10-block fork.</span></p>
+<p>It may in fact be worse than this: once miners know that a finalization stall is happening, their incentive to continue mining is reduced, since they know that there is a greater chance that their blocks might be rolled back. So we would expect the global hash rate to fall —even before the finality gap bound is hit— and then the adversary would have a greater proportion of hash rate.</p>
+<div id="admonition-info-1" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Info</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/the-arguments-for-bounded-availability-and-finality-overrides.html#admonition-info-1"></a></p>
+</div>
+<div>
+<p>Even in a pure Ebb‑and‑Flow protocol, a finalization stall could cause miners to infer that their blocks are more likely to be rolled back, but the fact that the chain is continuing would make that more difficult to exploit. This issue with the global hash rate is mostly specific to the more-available protocol being PoW: if it were PoS, then its validators might as well continue proposing blocks because it is cheap to do so. There might be other attacks when the more-available protocol is PoS; we haven’t spent much time analyzing that case.</p>
+</div>
+</div>
+<p>The problem is that the more-available chain does not necessarily <em>just halt</em> during a chain stall. In fact, for a finality gap bound of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> blocks, an adversary could cause <span style="white-space: nowrap">the <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span>-block “tail”</span> of the chain as seen by any given node to “thrash” between different chains. We will call this <span style="white-space: nowrap">a <strong>tail‑thrashing attack</strong>.</span></p>
+<p>If a protocol allowed such attacks then it would be a regression relative to the security we would normally expect from an otherwise similar PoW-based protocol. It only occurs during a finalization stall, but note that we cannot exclude the possibility of an adversary being able to provoke a finalization stall.</p>
+<p>Note that in the Snap‑and‑Chat construction, snapshots of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> are used as input to the BFT protocol. That implies that the tail‑thrashing problem could also affect the input to that protocol, which would be bad (not least for security analysis of availability, which seems somewhat intractable in that case).</p>
+<p>Also, when restarting <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>, we would need to take account of the fact that the adversary has had an arbitrary length of time to build long chains from every block that we could potentially restart from. It could be possible to invalidate those chains by requiring blocks after the restart to be dependent on fresh randomness, but that sounds quite tricky (especially given that we want to restart <em>without</em> manual intervention if possible), and there may be other attacks we haven’t thought of. This motivates using approach a) instead.</p>
+<p>Note that we have still glossed over precisely how consensus rules would change to enforce a). This will be covered later in <a href="design/crosslink/./construction.html">The Crosslink 2 Construction</a>, but first we will discuss other issues with Snap-and-Chat.</p>
+<div style="break-before: page; page-break-before: always;"></div><link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.12.0/dist/katex.min.css" integrity="sha384-AfEj0r4/OFrOo5t7NnNe46zW/tFgW6x/bCJG8FqQCEo3+Aro6EYUG4+cU+KJWu/X" crossorigin="anonymous">
+<h1 id="notes-on-snapandchat"><a class="header" href="#notes-on-snapandchat">Notes on Snap‑and‑Chat</a></h1>
+<p>The discussion in <a href="design/crosslink/./the-arguments-for-bounded-availability-and-finality-overrides.html">The Argument for Bounded Availability and Finality Overrides</a> is at an abstract level, applying to any Ebb‑and‑Flow-like protocol.</p>
+<p>This document considers specifics of the Snap‑and‑Chat construction proposed in <a href="https://eprint.iacr.org/2020/1091.pdf">[NTT2020]</a> (<a href="https://arxiv.org/pdf/2009.04987.pdf">arXiv version</a>).</p>
+<div id="admonition-terminology" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Terminology</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/notes-on-snap-and-chat.html#admonition-terminology"></a></p>
+</div>
+<div>
+<p>We are trying to be precise in this document about use of the terms “Ebb‑and‑Flow”, which is the security model and goal introduced in <a href="https://eprint.iacr.org/2020/1091.pdf">[NTT2020]</a>, vs “Snap‑and‑Chat”, which is the construction proposed in the same paper to achieve that goal. There are other ways to design an Ebb‑and‑Flow protocol that don’t run into the difficulties described in this section (or that run into different difficulties).</p>
+</div>
+</div>
+<h3 id="effect-on-consensus"><a class="header" href="#effect-on-consensus">Effect on consensus</a></h3>
+<p>A general problem with the Snap‑and‑Chat construction is that it does not follow, from enforcement of the original consensus rules on blocks produced in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>, that the properties they are intended to enforce hold for the <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span> or <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">da</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span> ledgers. Less obviously, the converse also does not follow: enforcing unmodified consensus rules on <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> blocks is <strong>both</strong> too lax and too strict.</p>
+<p>Recall from the paper how <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">da</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span> are constructed (starting at the end of page 8 of <a href="https://eprint.iacr.org/2020/1091.pdf">[NTT2020]</a>):</p>
+<blockquote>
+<ol start="3">
+<li><em>Ledger extraction:</em> Finally, how honest nodes compute <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">da</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span> from <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">Ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span> is illustrated in Figure 6. Recall that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">Ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span> is an ordering of snapshots, i.e., a chain of chains of LC blocks. First, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">Ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span> is flattened, i.e. the chains of blocks are concatenated as ordered to arrive at a single sequence of LC blocks. Then, all but the first occurrence of each block are removed (sanitized) to arrive at the finalized ledger <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span> of LC blocks. To form the available ledger <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">da</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span>, which is a sequence of LC blocks, is appended to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span> and the result again sanitized.</li>
+</ol>
+</blockquote>
+<p>This says that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">da</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span> are sequences of transactions, <em>not</em> sequences of blocks. Therefore, <a href="https://zips.z.cash/protocol/protocol.pdf#blockheader">consensus rules defined at the block level</a> are not applicable.</p>
+<div id="admonition-zcash-specific" class="admonition admonish-warning">
+<div class="admonition-title">
+<p>Zcash-specific</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/notes-on-snap-and-chat.html#admonition-zcash-specific"></a></p>
+</div>
+<div>
+<p>Most of these rules are Proof‑of‑Work‑related checks that can be safely ignored at this level. Some are related to the <code>hashBlockCommitments</code> field intended for use by the FlyClient protocol. It is not at all clear how to make FlyClient (or other uses of this commitment) work with the Snap‑and‑Chat construction. In particular, the <code>hashEarliest{Sapling,Orchard}Root</code>, <code>hashLatest{Sapling,Orchard}Root</code>, and <code>n{Sapling,Orchard}TxCount</code> fields don’t make sense in this context since they could only reflect the values in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span>, which have no relation in general to those for any subrange of transactions in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">da</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span>. This problem occurs as a result of sanitization and so will be avoided by Crosslink 2, which does not need sanitization.</p>
+</div>
+</div>
+<p>Since <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">da</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span> does not have blocks, it is not well-defined whether it has “coinbase‑only blocks” when in Stalled Mode. That by itself is not so much of a problem because it would be sufficient for it to have only coinbase transactions in that mode.</p>
+<h3 id="effect-on-issuance"><a class="header" href="#effect-on-issuance">Effect on issuance</a></h3>
+<p>The issuance schedule of Zcash was designed under the assumption that blocks only come from a single chain, and that the difficulty adjustment algorithm keeps the rate of block mining roughly constant over time.</p>
+<p>For Snap‑and‑Chat, if there is a rollback longer than <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> blocks in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>, additional coinbase transactions from the rolled-back chain will be included in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</p>
+<p>We can argue that this will happen rarely enough not to cause any significant problem for the overall issuance schedule. However, it does mean that issuance is less predictable, because the block subsidies will be computed according to their depth in the <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> chain on which they were mined. So it would no longer be the case that coinbase transactions issue a deterministic, non-increasing sequence of block subsidies. (Again, this problem will be avoided by Crosslink 2.)</p>
+<h3 id="effect-on-transaction-ordering"><a class="header" href="#effect-on-transaction-ordering">Effect on transaction ordering</a></h3>
+<p>The order of transactions in any particular <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span> is <em>not</em> in general preserved in either <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> or <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">da</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>. This <em>is</em> considered in the paper (middle of the left column on page 10) but it is very easy to miss it:</p>
+<blockquote>
+<p>Thus, snapshots taken by different nodes or at different times can conflict. However, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> is still safe and thus orders these snapshots linearly. Any transactions invalidated by conflicts are sanitized during ledger extraction.</p>
+</blockquote>
+<p>That is, a transaction from one snapshot might double-spend an output already spent in a different transaction of a different snaphot earlier in the flattening order. If it is omitted, then later transactions could depend on the outputs of the omitted one. The paper is saying that each transaction is only included if (in Bitcoin and Zcash terminology) it satisfies contextual checks for double-spending and existence of inputs at the point in the ledger where it would be added.</p>
+<p>Since nullifiers for shielded spends are public, it is possible to do this even for shielded transactions. Each node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> will construct commitment trees in the order given by <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">da</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mord">.</span></span></span></span></p>
+<div id="admonition-info" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Info</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/notes-on-snap-and-chat.html#admonition-info"></a></p>
+</div>
+<div>
+<p>This means that if <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> is extended by a block that is not the next block in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">da</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> after the finalization point (and that has different note commitments), then <em>all</em> shielded transactions from that point onward in the previous <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">da</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> will be invalidated. It could be possible to do better at the expense of a more complicated note commitment tree structure. In any case, this situation is expected to be rare, because it can only occur if there is a rollback of more than <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> blocks in the <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> consensus chain or a failure of BFT safety.</p>
+</div>
+</div>
+<h3 id="subtlety-in-the-definition-of-sanitization"><a class="header" href="#subtlety-in-the-definition-of-sanitization">Subtlety in the definition of sanitization</a></h3>
+<p>There are two possible ways to interpret how <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0496em;vertical-align:-0.3552em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.5198em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">{</span><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mtight"><span class="mord mathrm mtight">da</span></span><span class="mclose mtight">}</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3552em;"><span></span></span></span></span></span></span></span></span></span> are constructed in Snap‑and‑Chat:</p>
+<ol>
+<li>Concatenate the transactions from each final BFT block snapshot of an LC chain, and sanitize the resulting transaction sequence by including each transaction iff it is contextually valid.</li>
+<li>Concatenate the <em>blocks</em> from each final BFT block snapshot of an LC chain, remove duplicate <em>blocks</em>, and only then sanitize the resulting transaction sequence by including each transaction iff it is contextually valid.</li>
+</ol>
+<p>These are equivalent in the setting considered by <a href="https://eprint.iacr.org/2020/1091.pdf">[NTT2020]</a>, but the argument for their equivalence is not obvious. We definitely want them to be equivalent: in practice there will be many duplicate blocks from chain prefixes in the input to sanitization, and so a literal implementation of the first variant would have to recheck all duplicate transactions for contextual validity. That would have at least <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> complexity (more likely <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span>) in the length <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> of the block chain, because the length of each final snapshot grows with <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span>.</p>
+<p>Suppose that, in a particular <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>, the only reasons for a transaction to be contextually invalid are double-spends and missing inputs. In that case the argument for equivalence is:</p>
+<ul>
+<li>If a transaction is omitted due to a double-spend, then any subsequent time it is checked, that input will still have been double-spent.</li>
+<li>If a transaction is omitted due to a missing input, this can only be because an earlier transaction in the input to sanitization was omitted. So the structure of omitted transactions forms a DAG in which parent links must be to <em>earlier</em> omitted transactions. The roots of the DAG are at double-spending transactions, which cannot be reinstated. A child cannot be reinstated until its parents have been reinstated. Therefore, no transactions are reinstated.</li>
+</ul>
+<p>Note that any <em>other</em> reason for transactions to be contextually invalid might interfere with this argument. Therefore, strictly speaking Snap‑and‑Chat should require of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> that there is no such other reason. This does not seem to be explicitly stated anywhere in <a href="https://eprint.iacr.org/2020/1091.pdf">[NTT2020]</a>.</p>
+<div id="admonition-zcash-specific-1" class="admonition admonish-warning">
+<div class="admonition-title">
+<p>Zcash-specific</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/notes-on-snap-and-chat.html#admonition-zcash-specific-1"></a></p>
+</div>
+<div>
+<p>In Zcash a transaction can also be contextually invalid because it has expired, or because it has a missing anchor. Expiry can be handled by extending the above argument as follows:</p>
+<ul>
+<li>If a transaction is omitted due to having expired, then any subsequent time it is checked, it will still be expired.</li>
+</ul>
+<p>It is not obvious how to extend it to handle missing anchors, because it <em>is</em> technically possible for a duplicate transaction that was invalid because of a missing anchor to <em>become</em> valid in a subsequent block. That situation would require careful manipulation of the commitment trees, but there does not seem to be anything preventing it from being provoked intentionally. The argument that was used above for missing inputs does not work here, because there is no corresponding DAG formed by the transactions with missing anchors: the same commitment treestate can be produced by two unrelated transactions.</p>
+</div>
+</div>
+<h3 id="spending-finalized-outputs"><a class="header" href="#spending-finalized-outputs">Spending finalized outputs</a></h3>
+<p>Transactions in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span> need to be able to spend outputs that are not necessarily from any previous transaction in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span>. This is because, from the point of view of a user of node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> at time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>, the block chain includes all transactions in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">da</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span>. All of the transactions after the finalization point are guaranteed to also be in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span>, but the ones before the finalization point (i.e. in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span>) are not, because they could be from some other <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1318em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span> for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal">u</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span><span class="mrel">=</span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> (intuitively, from some long chain fork that was once considered confirmed by enough nodes).</p>
+<div id="admonition-info-1" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Info</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/notes-on-snap-and-chat.html#admonition-info-1"></a></p>
+</div>
+<div>
+<p>Honest nodes only ever vote for confirmed snapshots, that is, prefixes of their best <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> chain truncated by the confirmation depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>. Obviously the whole point of having the BFT protocol is that chain forks longer than <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> <strong>can</strong> occur in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> — otherwise we'd just use <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> and have done. So it is not that we expect this case to be common, but if it happens then it will never fix itself: the consensus chain in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> will continue on without ever including the transactions from <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span> that were obtained from a snapshot of another fork.</p>
+</div>
+</div>
+<p>A user must be able to spend outputs for which they hold the spending key from <strong>any</strong> finalized transaction, otherwise there would be no point to the finalization.</p>
+<p>The authors of <a href="https://eprint.iacr.org/2020/1091.pdf">[NTT2020]</a> probably just missed this: the paper only has evidence that they simulated their construction, rather than implementing it for Bitcoin or any other concrete block chain as <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>. Let’s try to repair it.</p>
+<p>Suppose that node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> is trying to determine whether <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span> is a consensus-valid chain, which is necessary for deterministic consensus in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>. It cannot decide whether to allow transactions in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span> to spend outputs not in the history of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span> on the basis of its <em>own</em> finalized view <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mpunct">,</span></span></span></span> because <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span> are not in general the same.</p>
+<p>Of course, we hope that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span> are consistent, i.e. one is a prefix of the other. But even if they are consistent, they are not necessarily the same length. In particular, if <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span> is <em>shorter</em> than <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mpunct">,</span></span></span></span> then node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> does not have enough information to fill in the gap — and so it may incorrectly view a transaction in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span> as spending an output that does not exist, when actually it does exist in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∖</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mord">.</span></span></span></span> Conversely if <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span> were longer and node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> were to allow spending an output in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∖</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mpunct">,</span></span></span></span> that would be using information that is not necessarily available to other nodes, and so node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> could diverge from consensus.</p>
+<p>Consensus validity of the block at the tip of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span> can only be a deterministic function of the block itself and its ancestors in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span>. It is crucial to be able to <em>eventually</em> spend outputs from the finalized chain. We are forced to conclude that the chain <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span> <strong>must include</strong> the information needed to calculate <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1318em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">s</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span> for some <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">s</span></span></span></span> not too far behind <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>. That is, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> must be modified to ensure that this is the case. This leads us to strengthen the required properties of an Ebb‑and‑Flow protocol to include another property, “finalization availability”.</p>
+<h2 id="finalization-availability"><a class="header" href="#finalization-availability">Finalization Availability</a></h2>
+<p>In the absence of security flaws and under the security assumptions required by the finality layer, the finalization point will not be seen by any honest node to roll back. However, that does not imply that all nodes will see the same finalized height — which is impossible given network delays and unreliable messaging.</p>
+<p>Both in order to optimize the availability of applications that require finality, and in order to solve the technical issue of spending finalized outputs described in the previous section, we need to consider availability of the information needed to finalize the chain up to a particular point.</p>
+<p>Note that in Bitcoin-like consensus protocols, we don’t generally consider it to be an availability flaw that a block header only <em>commits</em> to the previous block hash and to the Merkle tree of transactions in the block, rather than including them directly. These commitments allow nodes to check that they have the correct information, which can then be requested separately.</p>
+<p>Suppose, then, that each block header in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> commits to the <strong>L</strong>ast <strong>F</strong>inal BFT block known by the <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> block producer. For an LC block chain with block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> at its tip, we will refer to this commitment as <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span></span></span>. We refer to the parent block of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> as <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0972em;vertical-align:-0.2831em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span> (this is a special case of a notation that will be defined in <a href="design/crosslink/./construction.html">The Crosslink 2 Construction</a>).</p>
+<div id="admonition-consensus-rule-extension" class="admonition admonish-success">
+<div class="admonition-title">
+<p>Consensus rule: Extension</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/notes-on-snap-and-chat.html#admonition-consensus-rule-extension"></a></p>
+</div>
+<div>
+<p>We require, as a consensus rule, that if <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> is not the genesis block header, then this BFT block either descends from or is the same as the final BFT block committed to by the <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> block’s parent. <span style="white-space: nowrap">That is, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0972em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span></span></span>.</span></p>
+</div>
+</div>
+<p>This Extension rule will be preserved into Crosslink 2.</p>
+<p>The Extension rule does not prevent the BFT chain from rolling back, if the security assumptions of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> were violated. However, it means that if <span style="white-space: nowrap">a node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span></span> does not observe a rollback in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> at confirmation <span style="white-space: nowrap">depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>,</span> then it will also not observe any instability <span style="white-space: nowrap">in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>,</span> <em>even if</em> the security assumptions of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> are violated. This property holds <strong>by construction</strong>, and in fact regardless <span style="white-space: nowrap">of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</span></p>
+<div id="admonition-info-2" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Info</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/notes-on-snap-and-chat.html#admonition-info-2"></a></p>
+</div>
+<div>
+<p>In the Snap‑and‑Chat construction, we also have BFT block proposals committing to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> snapshots (top of right column of [<a href="https://eprint.iacr.org/2020/1091.pdf">NTT2020</a>, page 7]):</p>
+<blockquote>
+<p>In addition, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span> is used as side information in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> to boycott the finalization of invalid snapshots proposed by the adversary.</p>
+</blockquote>
+<p>This does not cause any circularity, because each protocol only commits to <em>earlier</em> blocks of the other. In fact, BFT validators have to listen to transmission of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> block headers anyway, so that <em>could</em> be also the protocol over which they get the information needed to make and broadcast their own signatures or proposals. (A possible reason not to broadcast individual signatures to all nodes is that with large numbers of validators, the proof that a sufficient proportion of validators/stake has signed can use an aggregate signature, which could be much smaller. Also, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> nodes only need to know about <em>successful</em> BFT block proposals.)</p>
+</div>
+</div>
+<p>Now suppose that, in a Snap‑and‑Chat protocol, the BFT consensus finalizes a <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> snapshot that does not extend the snapshot in the previous block (which can happen if either <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> is unsafe, or <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> suffers a rollback longer than <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> blocks). In that case we will initially not be able to spend outputs from the old snapshot in the new chain. But eventually for some node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> that sees the header <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> at the tip of its best chain at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>,</span> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span></span></span> will be such that from then on (i.e. at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal">u</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>),</span> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1318em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span> includes the output that we want to spend. This assumes <em>liveness</em> of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> and <em>safety</em> <span style="white-space: nowrap">of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</span></p>
+<p>That is, including a reference to a recent final BFT block in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> block headers both incentivizes nodes to propagate this information, <em>and</em> can be used to solve the “spending finalized outputs” problem.</p>
+<p>Optionally, we could incentivize the block producer to include the latest information it has, for example by burning part of the block reward or by giving the producer some limited mining advantage that depends on how many <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> blocks back the finalization information is.</p>
+<p>This raises the question of how we measure how far ahead a given block is relative to the finalization information it provides. As we said before, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span> is a sequence of transactions, not blocks. The transactions will in general be in a different order, and also some transactions from <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span> may have been omitted from <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span> (and even <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">da</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span>) because they were not contextually valid.</p>
+<div id="admonition-info-3" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Info</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/notes-on-snap-and-chat.html#admonition-info-3"></a></p>
+</div>
+<div>
+<p>In <a href="design/crosslink/./construction.html">Crosslink 2</a>, we will sidestep this problem by avoiding the need for sanitization — that is, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span> will correspond exactly to a chain of blocks that is a prefix of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span>. Actually we use the notation <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span> to reflect the fact that it is a bc‑chain, not a sequence of bc‑transactions. This invariant is maintained statefully on each node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>: any rollback past <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span> will be ignored. If a new <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span> would conflict with the old one, the node will refuse to use it. This allows each node to straightforwardly measure how many blocks <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span> is ahead of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span> as the difference in heights. Since this document is intended to explain the development of Crosslink from Snap‑and‑Chat, here we describe the more complicated approach that we originally came up with for Crosslink 1 — which also serves to motivate the simplification in Crosslink 2.</p>
+</div>
+</div>
+<p>Assume that a block unambiguously specifies its ancestor chain. For a block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span>, define:<span class="katex-display fleqn"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">tailhead</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">last-common-ancestor</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.1em;"></span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">finality-depth</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">height</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">height</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">tailhead</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">))</span></span></span></span></span></p>
+<p>Here <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span></span></span> is the BFT block we are providing information for, and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.1em;"></span><span class="mclose">)</span></span></span></span> is the corresponding <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> snapshot. For a node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> that sees <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span></span></span> as the most recent final BFT block at time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span> will definitely contain transactions from blocks up to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">tailhead</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span></span></span>, but usually will not contain subsequent transactions on <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span>’s fork.</p>
+<div id="admonition-info-4" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Info</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/notes-on-snap-and-chat.html#admonition-info-4"></a></p>
+</div>
+<div>
+<p>Strictly speaking, it is possible that a previous BFT block took a snapshot <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> that is between <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">tailhead</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span>. This can only happen if there have been at least two rollbacks longer than <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> blocks (i.e. we went more than <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> blocks down <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span>’s fork from <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">tailhead</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span></span></span>, then reorged to more than <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> blocks down <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.1em;"></span><span class="mclose">)</span></span></span></span>’s fork, then reorged again to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span>’s fork). In that case, the finalized ledger would already have the non-conflicting transactions from blocks between <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">tailhead</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> — and it could be argued that the correct definition of finality depth in such cases is the depth of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> relative to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span>, not of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">tailhead</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span></span></span> relative to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span>.</p>
+<p>However,</p>
+<ul>
+<li>The definition above is simpler and easier to compute.</li>
+<li>The effect of overestimating the finality depth in such corner cases would only cause us to enforce Stalled Mode slightly sooner, which seems fine (and even desirable) in any situation where there have been at least two rollbacks longer than <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> blocks.</li>
+</ul>
+<p>By the way, the “tailhead” of a tailed animal is the area where the posterior of the tail joins the rump (also called the “dock” in some animals).</p>
+</div>
+</div>
+<p>We could alternatively just rely on the fact that some proportion of block producers are honest and will include the latest information they have. However, it turns out that having a definition of finality depth will also be useful to enforce going into Stalled Mode.</p>
+<p>Specifically, if we accept the above definition of finality depth, then the security property we want is</p>
+<div id="admonition-definition" class="admonition admonish-success">
+<div class="admonition-title">
+<p>Definition</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/notes-on-snap-and-chat.html#admonition-definition"></a></p>
+</div>
+<div>
+<p><strong>Bounded hazard-freeness</strong> for a finality gap bound of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span> blocks: There is never, for any node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> at time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>, observed to be a more-available ledger <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">da</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span> with a hazardous transaction that comes from block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span> such that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">finality-depth</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span>.</p>
+</div>
+</div>
+<div id="admonition-info-5" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Info</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/notes-on-snap-and-chat.html#admonition-info-5"></a></p>
+</div>
+<div>
+<p>This assumes that transactions in the non-finalized suffix  <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">da</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∖</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span> come from blocks in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span>. In Snap‑and‑Chat they do by definition, but ideally we wouldn’t depend on that. The difficulty in finding a more general security definition is due to the ledgers in an Ebb‑and‑Flow protocol being specified as sequences of transactions, so that a depth in the ledger would have only a very indirect correspondence to time. We could instead base a definition on timestamps, but that could run into difficulties in ensuring timestamp accuracy.</p>
+<p>Another possibility would be to count the number of coinbase transactions in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">da</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∖</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span> before the hazardous transaction. This would still be somewhat ad hoc (it depends on the fact that coinbase transactions happen once per block and cannot conflict with any other distinct transaction).</p>
+<p>In any case, if <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord textsf">finality-depth</span></span></span></span></span> sometimes overestimates the depth, that cannot weaken this security definition.</p>
+</div>
+</div>
+<p>Note that a node that is validating a chain <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span> must fetch all the chains referenced by BFT blocks reachable from it (back to an ancestor that it has seen before). In theory, there could be a partition that causes there to be multiple disjoint snapshots that get added to the BFT chain in quick succession. However, in practice we expect such long rollbacks to be rare if <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> is meeting its security goals.</p>
+<p>Going into Stalled Mode if there is a long finalization stall helps to reduce the cost of validation when the stall resolves. That is, if there is a partition and nodes build on several long chains, then in unmodified Snap‑and‑Chat, it could be necessary to validate an arbitrary number of transactions on each chain when the stall resolves. Having only coinbase transactions after a certain point in each chain would significantly reduce the concrete validation costs in this situation.</p>
+<p>Nodes should not simply trust that the BFT blocks are correct; they should check validator signatures (or aggregate signatures) and finalization rules. Similarly, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> snapshots should not be trusted just because they are referenced by BFT blocks; they should be fully validated, including the proofs-of-work.</p>
+<p>It is also possible for a snapshot reference to include the subsequent <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> block headers, which are guaranteed to be available for a confirmed snapshot. Having all nodes validate the proofs-of-work in these headers is likely to significantly increase the work that an attacker would need to perform to cause disruption under a partial failure of either <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> or <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>’s security properties.</p>
+<div id="admonition-info-6" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Info</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/notes-on-snap-and-chat.html#admonition-info-6"></a></p>
+</div>
+<div>
+<p>Note that <a href="https://eprint.iacr.org/2020/1091.pdf">[NTT2020]</a> (bottom of right column, page 9) makes a safety assumption about <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> in order to prove the consistency of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> with the output of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>:</p>
+<blockquote>
+<p>As indicated by Algorithm 1, a snapshot of the output of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> becomes final as part of a BFT block only if that snapshot is seen as confirmed by at least one honest node. However, since <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> is safe [i.e., does not roll back further than the confirmation depth <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>],</span> the fact that one honest node sees that snapshot as confirmed implies that every honest node sees the same snapshot as confirmed.</p>
+</blockquote>
+<p>We claim that, while this may be a reasonable assumption to make for parts of the security analysis, in practice we should always require any adversary to do the relevant amount of Proof‑of‑Work to construct block headers that are plausibly confirmed. This is useful even though we cannot require, for every possible attack, that it had those headers at the time they should originally have appeared.</p>
+</div>
+</div>
+<h2 id="enforcing-finalization-availability-and-stalled-mode"><a class="header" href="#enforcing-finalization-availability-and-stalled-mode">Enforcing Finalization Availability and Stalled Mode</a></h2>
+<p>The following idea for enforcing finalization availability <em>and</em> a bound on the finality gap was originally conceived before we had switched to advocating the Stalled Mode approach. It’s simpler to explain first in that variant.</p>
+<p>Suppose that for an <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span>-block availability bound,</span> we required each block header to include the information necessary for a node to finalize to <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span> blocks back.</span> This would automatically enforce a chain stall after the availability bound without any further explicit check, because it would be impossible to produce a block after the bound.</p>
+<p>Note that if full nodes have access to the BFT chain, knowing <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span></span></span> is sufficient to tell whether the correct version of any given BFT block in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span></span></span>’s ancestor chain has been obtained.</p>
+<p>Suppose that the finality gap bound is <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span> blocks.</span> Having already defined <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord textsf">finality-depth</span></span></span></span></span>, the necessary <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> consensus rule is attractively simple:</p>
+<div id="admonition-consensus-rule" class="admonition admonish-success">
+<div class="admonition-title">
+<p>Consensus rule</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/notes-on-snap-and-chat.html#admonition-consensus-rule"></a></p>
+</div>
+<div>
+<p>For every <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">finality-depth</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span>.</p>
+</div>
+</div>
+<p>To adapt this approach to enforce Stalled Mode instead of stalling the chain, we can allow the alternative of producing a block that follows the Stalled Mode restrictions:</p>
+<div id="admonition-consensus-rule-1" class="admonition admonish-success">
+<div class="admonition-title">
+<p>Consensus rule</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/notes-on-snap-and-chat.html#admonition-consensus-rule-1"></a></p>
+</div>
+<div>
+<p>For every <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span>, <font color="#4040f0">either</font> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">finality-depth</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span><font color="#4040f0">, or <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> follows the Stalled Mode restrictions</font>.</p>
+</div>
+</div>
+<p>Note that Stalled Mode will be exited automatically as soon as the finalization point catches up to within <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span> blocks (if it does without an intentional rollback). Typically, after recovery from whatever was causing the finalization stall, the validators will be able to obtain consensus on the same chain as <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">da</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>, and so there will be no rollback (or at least not a long one) of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">da</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</p>
+<div id="admonition-info-7" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Info</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/notes-on-snap-and-chat.html#admonition-info-7"></a></p>
+</div>
+<div>
+<p>An earlier iteration of this idea required the finalization information to be included in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> block headers. This is not necessary when we assume that full nodes have access to the BFT chain and can obtain arbitrary BFT blocks. This also sidesteps any need to relax the rule in order to bound the size of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> block headers. <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> block producers are still incentivized to make the relevant BFT blocks available, because without them the above consensus rule cannot be checked, and so their <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> blocks would not be accepted.</p>
+<p>There is, however, a potential denial-of-service attack by claiming the existence of a BFT block that is very far ahead of the actual BFT chain tip. This attack is not very serious as long as nodes limit the number of BFT blocks they will attempt to obtain in parallel before having checked validator signatures.</p>
+</div>
+</div>
+<h3 id="comment-on-security-assumptions"><a class="header" href="#comment-on-security-assumptions">Comment on security assumptions</a></h3>
+<p>Consider Lemma 5:</p>
+<blockquote>
+<p>Moreover, for a BFT block to become final in the view of an honest node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> under <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathcal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-2.4519em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2481em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.07944em;">Z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0794em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>, at least one vote from an honest node is required, and honest nodes only vote for a BFT block if they view the referenced LC block as confirmed.</p>
+</blockquote>
+<p>The stated assumptions are:</p>
+<blockquote>
+<p><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathcal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.07944em;">Z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0794em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> formalizes the model of <strong>P2</strong>, a synchronous network under dynamic participation, with respect to a bound <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span></span></span></span> on the fraction of awake nodes that are adversarial:</p>
+<ul>
+<li>At all times, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> is required to deliver all messages sent between honest nodes in at most <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Δ</span></span></span></span> slots.</li>
+<li>At all times, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> determines which honest nodes are awake/asleep and when, subject to the constraint that at all times at most fraction <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span></span></span></span> of awake nodes are adversarial and at least one honest node is awake.</li>
+</ul>
+</blockquote>
+<p><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathcal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-2.4519em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2481em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.07944em;">Z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0794em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> is defined as <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathcal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.07944em;">Z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0794em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">))</span></span></span></span>.</p>
+<p>Now consider this statement and figure:</p>
+<blockquote>
+<p>Even if <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> is unsafe (Figure 9c), finalization of a snapshot requires at least one honest vote, and thus only valid snapshots become finalized.</p>
+<p><img src="design/crosslink/./NTT2020-Figure-9.png" alt="Figure 9 of [NTT2020]" /></p>
+</blockquote>
+<p>This argument is technically correct but has to be interpreted with care. It only applies when the number of malicious nodes <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span> is such that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">n</span><span class="mord">/3</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">n</span><span class="mord">/2</span></span></span></span>. What we are trying to do with Crosslink is to ensure that a similar conclusion holds even if <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> is completely subverted, i.e. the adversary has 100% of validators (but only &lt; 50% of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> hash rate).</p>
+<div style="break-before: page; page-break-before: always;"></div><link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.12.0/dist/katex.min.css" integrity="sha384-AfEj0r4/OFrOo5t7NnNe46zW/tFgW6x/bCJG8FqQCEo3+Aro6EYUG4+cU+KJWu/X" crossorigin="anonymous">
+<h1 id="the-crosslink-2-construction"><a class="header" href="#the-crosslink-2-construction">The Crosslink 2 Construction</a></h1>
+<p>We are now ready to give a description of a protocol that takes into account the issues described in <a href="design/crosslink/./notes-on-snap-and-chat.html">Notes on Snap‑and‑Chat</a>, and that implements <a href="design/crosslink/./the-arguments-for-bounded-availability-and-finality-overrides.html#what-is-bounded-availability">bounded availability</a>. We call this the “Crosslink” construction; more precisely the version described here is “Crosslink 2”.</p>
+<p>This description will attempt to be self-contained, but <a href="https://eprint.iacr.org/2020/1091.pdf">[NTT2020]</a> (<a href="https://arxiv.org/pdf/2009.04987.pdf">arXiv version</a>) is useful background on the general model of Ebb-and-Flow protocols.</p>
+<h2 id="conventions"><a class="header" href="#conventions">Conventions</a></h2>
+<p>“<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>” is a metavariable for the name of a protocol. We also use it as a wildcard in protocol names of a particular type, for example <span style="white-space: nowrap">“<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bc”</span> for the name of some best‑chain protocol.</p>
+<p>Protocols are referred to as <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1757em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∗</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> for a <span style="white-space: nowrap">name “<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>”.</span> Where it is useful to avoid ambiguity, when referring to a concept defined by <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1757em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∗</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> we prefix it with <span style="white-space: nowrap">“<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>‑”.</span></p>
+<p>We do not take synchrony or partial synchrony as an implicit assumption of the communication model; that is, <em>unless otherwise specified</em>, messages between protocol participants can be arbitrarily delayed or dropped. A given message is received at most once, and messages are nonmalleably authenticated as originating from a given sender whenever needed by the applicable protocol. Particular subprotocols may require a stronger model.</p>
+<div id="admonition-background" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Background</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/construction.html#admonition-background"></a></p>
+</div>
+<div>
+<p>For an overview of communication models used to analyze distributed protocols, see <a href="https://decentralizedthoughts.github.io/2019-06-01-2019-5-31-models/">this blog post by Ittai Abraham</a>.</p>
+</div>
+</div>
+<details id="admonition-discussion-of-incorrect-applications-of-the-gst-formalization-of-partial-synchrony-to-continuously-operating-protocols" class="admonition admonish-info">
+<summary class="admonition-title">
+<p>Discussion of incorrect applications of the GST formalization of partial synchrony to continuously operating protocols.</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/construction.html#admonition-discussion-of-incorrect-applications-of-the-gst-formalization-of-partial-synchrony-to-continuously-operating-protocols"></a></p>
+</summary>
+<div>
+<p>The original context for the definition of the partially synchronous model in <a href="https://groups.csail.mit.edu/tds/papers/Lynch/jacm88.pdf">[DLS1988]</a> was for “one‑shot” Byzantine Agreement — called “the consensus problem” in that paper. The following argument is used to justify assuming that all messages from the Global Stabilization Time onward are delivered within the <span style="white-space: nowrap">upper time bound <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Δ</span></span></span></span>:</span></p>
+<blockquote>
+<p>Therefore, we impose an additional constraint: For each execution there is a global stabilization time (GST), unknown to the processors, such that the message system respects the upper bound <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Δ</span></span></span></span> from time GST onward.</p>
+<p>This constraint might at first seem too strong: In realistic situations, the upper bound cannot reasonably be expected to hold forever after GST, but perhaps only for a limited time. However, any good solution to the consensus problem in this model would have an <span style="white-space: nowrap">upper bound <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span></span> on the amount of time after GST required for consensus to be reached; in this case it is not really necessary that the bound <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Δ</span></span></span></span> hold forever after time GST, but only up to <span style="white-space: nowrap">time GST <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord">+</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">L</span></span></span></span>.</span> We find it technically convenient to avoid explicit mention of the interval length <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span> in the model, but will instead present the appropriate upper bounds on time for each of our algorithms.</p>
+</blockquote>
+<p>Several subsequent authors applying the partially synchronous model to block chains appear to have forgotten or neglected this context. In particular, the argument depends on the protocol completing soon after GST. Obviously a block‑chain protocol does not satisfy this assumption; it is not a “one‑shot” consensus problem.</p>
+<p>This assumption could be removed, but some authors of papers about block‑chain protocols have taken it to be an essential aspect of modelling partial synchrony. I believe this is contrary to the intent of <a href="https://groups.csail.mit.edu/tds/papers/Lynch/jacm88.pdf">[DLS1988]</a>:</p>
+<blockquote>
+<p>Instead of requiring that the consensus problem be solvable in the GST model, we might think of separating the correctness conditions into safety and termination properties. The safety conditions are that no two correct processors should ever reach disagreement, and that no correct processor should ever make a decision that is contrary to the specified validity conditions. The termination property is just that each correct processor should eventually make a decision. Then we might require an algorithm to satisfy the safety conditions no matter how asynchronously the message system behaves, that is, even if <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Δ</span></span></span></span> does not hold eventually. On the other hand, we might only require termination in case <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Δ</span></span></span></span> holds eventually. It is easy to see that these safety and termination conditions are [for the consensus problem] equivalent to our GST condition: If an algorithm solves the consensus problem when <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Δ</span></span></span></span> holds from time GST onward, then that algorithm cannot possibly violate a safety property even if the message system is completely asynchronous. This is because safety violations must occur at some finite point in time, and there would be some continuation of the violating execution in which <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Δ</span></span></span></span> eventually holds.</p>
+</blockquote>
+<p>This argument is correct as stated, i.e. for the one‑shot consensus problem. Subtly, essentially the same argument can be adapted to protocols with <em>safety</em> properties that need to be satisfied continuously. However, it cannot correctly be applied to <em>liveness</em> properties of non‑terminating protocols. The authors (Cynthia Dwork, Nancy Lynch, and Larry Stockmeyer) would certainly have known this: notice how they carefully distinguish “the GST model” from “partial synchrony”. They cannot plausibly have intended this GST formalization to be applied unmodified to analyze liveness in such protocols, which seems to be common in the block‑chain literature, including in the Ebb-and-Flow paper <a href="https://eprint.iacr.org/2020/1091.pdf">[NTT2020]</a> and the Streamlet paper <a href="https://eprint.iacr.org/2020/088.pdf">[CS2020]</a>.</p>
+<p>The Ebb-and-Flow paper acknowledges the issue by saying “Although in reality, multiple such periods of (a‑)synchrony could alternate, we follow the long‑standing practice in the BFT literature and study only a single such transition.” This is not adequate: “long‑standing practice” notwithstanding, it is not valid in general to infer that properties holding for the first transition to synchrony also apply to subsequent transitions (where the protocol can be in states that would not occur initially), and it is plausible that this inference could fail for real protocols. The Streamlet paper also refers to “periods of synchrony” which indicates awareness of the issue, but then it uses the unmodified GST model in the proofs.</p>
+<p>Informally, to solve this issue it is necessary to also prove that existing progress is maintained during periods of asynchrony, and that during such periods the protocol remains in states where it will be able to take advantage of a future period of synchrony to make further progress.</p>
+<p>This provides further motivation to avoid taking the GST formalization of partial synchrony as a basic assumption.</p>
+<p>Note that the recent result <a href="https://arxiv.org/pdf/2405.10249">[CGSW2024]</a> does not contradict anything we say here. Although the GST and Unknown Latency models are “equally demanding” in the sense of existence of protocols that satisfy a given goal, this result does not show that the models are equivalent for any specific protocol. In particular the requirements of the “clock‑slowing” technique fail in practice for any protocol involving Proof‑of‑Work.</p>
+</div>
+</details>
+<p><span style="white-space: nowrap">A <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>‑execution</span> is the complete set of events (message sends/receives and decisions by protocol participants) that occur in a particular <span style="white-space: nowrap">run of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1757em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∗</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span> from its initiation up to a given time. A prefix of a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>‑execution</span> is also a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>‑execution.</span> Since executions always start from protocol initiation, a strict suffix of a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>‑execution</span> is not a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>‑execution.</span></p>
+<p>Times are modelled as values of a totally ordered type <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathsf">Time</span></span></span></span></span> with minimum value <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>. For convenience, we consider all protocol executions to start at time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>.</p>
+<div id="admonition-remark" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Remark</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/construction.html#admonition-remark"></a></p>
+</div>
+<div>
+<p>Although protocols may be nondeterministic, an execution fixes the events that occur and times at which they occur, for the purpose of modeling.</p>
+<p>For simplicity, we assume that all events occur at global times in a total ordering. This assumption is not realistic in an asynchronous communication model, but it is not essential to the design or analysis and could be removed: we could use a partial happens-before ordering on events in place of a total ordering on times.</p>
+</div>
+</div>
+<p><span style="white-space: nowrap">A “<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>‑node”</span> is a participant in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1757em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∗</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> (the protocol may be implicit). <span style="white-space: nowrap">A <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>‑node</span> is <span style="white-space: nowrap">“honest at time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>”</span> in a given execution iff it has followed the protocol up to and including <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span></span> in that execution.</p>
+<p>A time series on <span style="white-space: nowrap">type <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span></span></span></span></span> is a function <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.1293em;"><span style="top:-2.6293em;"><span class="pstrut" style="height:2.5em;"></span><span class="mord"><span class="mord sizing reset-size6 size1">⦂</span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathsf">Time</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span></span></span></span> assigning a <span style="white-space: nowrap">value of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span></span></span></span></span> to each time in an execution. By convention, we will write the <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span></span> as a superscript: <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7936em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span>.</span></p>
+<p><span style="white-space: nowrap">A <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>‑chain</span> is a nonempty sequence of <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>‑blocks</span>, starting at the <span style="white-space: nowrap">“genesis block” <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1757em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∗</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>,</span> in which each subsequent block refers to its preceding or “parent block” by a collision‑resistant hash. The “tip” of <span style="white-space: nowrap">a <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>‑chain</span> is its last element.</p>
+<p>For convenience, we conflate <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>‑blocks</span> with <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>‑chains;</span> that is, we identify a chain with the block at its tip. This is justified because, assuming that the hash function used for parent links is collision‑resistant, there is exactly one <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>‑chain</span> corresponding to a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>‑block;</span> and conversely there is exactly one <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>‑block</span> at the tip of a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>‑chain.</span></p>
+<p>If <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span> is a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>‑chain,</span> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0991em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∗</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span> means <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span> with the last <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> blocks pruned, except that <span style="white-space: nowrap">if <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">len</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span>,</span> the result is the <span style="white-space: nowrap">genesis <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>‑chain</span> consisting only <span style="white-space: nowrap">of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1757em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∗</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</span></p>
+<p>The block at <span style="white-space: nowrap">depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">N</span></span></span></span></span> in a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>‑chain <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span></span> is defined to be the <span style="white-space: nowrap">tip of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0991em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∗</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span>.</span> Thus the block at <span style="white-space: nowrap">depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span></span> in a chain is the last one that cannot be affected by a rollback of <span style="white-space: nowrap">length <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span></span> (this also applies when <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">len</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> because the <span style="white-space: nowrap">genesis <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>‑chain</span> cannot roll back).</p>
+<div id="admonition-terminology" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Terminology</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/construction.html#admonition-terminology"></a></p>
+</div>
+<div>
+<p>Our usage of “depth” is different from <a href="https://eprint.iacr.org/2020/1091.pdf">[NTT2020]</a>, which uses “depth” to refer to what Bitcoin and Zcash call “height”. It also differs by <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> from the convention for confirmation depths in <code>zcashd</code>, where the tip is considered to be at <span style="white-space: nowrap">depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>,</span> rather <span style="white-space: nowrap">than <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>.</span></p>
+</div>
+</div>
+<p><span id="notation"></span>
+For <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>‑blocks <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span>:</span></p>
+<ul>
+<li>The notation <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1757em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∗</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span> means that the <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>‑chain</span> with <span style="white-space: nowrap">tip <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span></span> is a prefix of the one with <span style="white-space: nowrap">tip <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span>.</span> This includes the <span style="white-space: nowrap">case <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span>.</span></li>
+<li>The notation <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8193em;vertical-align:-0.136em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:-0.5em;"></span></span><span class="base"><span class="strut" style="height:0.786em;vertical-align:-0.15em;"></span><span class="mrel"><span class="mrel">⪰</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1757em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∗</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span> means that <span style="white-space: nowrap">either <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1757em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∗</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span> or <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1757em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∗</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span>.</span> That is, <span style="white-space: nowrap">“one of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span></span> is a prefix of the other”. This also includes the <span style="white-space: nowrap">case <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span>.</span></li>
+<li>The notation <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.5em;"></span><span class="mord"><span class="delimsizing size1">/</span></span><span class="mspace" style="margin-right:-1.06em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:-0.5em;"></span></span><span class="base"><span class="strut" style="height:0.786em;vertical-align:-0.15em;"></span><span class="mrel"><span class="mrel">⪰</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1757em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∗</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span> means that <span style="white-space: nowrap">both <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span></span><span class="base"><span class="strut" style="height:0.786em;vertical-align:-0.15em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1757em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∗</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span></span><span class="base"><span class="strut" style="height:0.786em;vertical-align:-0.15em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1757em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∗</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span>.</span> That is, <span style="white-space: nowrap">“neither of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span></span> is a prefix of the other”.</li>
+</ul>
+<p>A function <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.1293em;"><span style="top:-2.6293em;"><span class="pstrut" style="height:2.5em;"></span><span class="mord"><span class="mord sizing reset-size6 size1">⦂</span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">∗</span><span class="mord text"><span class="mord textsf">-block</span></span></span></span></span> is <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>‑linear</span> iff <span style="white-space: nowrap">for every <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8095em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">t</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">u</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.1293em;"><span style="top:-2.6293em;"><span class="pstrut" style="height:2.5em;"></span><span class="mord"><span class="mord sizing reset-size6 size1">⦂</span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span></span></span></span> where <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span></span> we have <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1757em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∗</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mopen">(</span><span class="mord mathnormal">u</span><span class="mclose">)</span></span></span></span>.</span> (This definition can be applied to time series <span style="white-space: nowrap">where <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathsf">Time</span></span></span></span></span>,</span> or to sequences of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>‑blocks where values of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span></span></span></span> are indices.)</p>
+<p><span id="linear-prefix-lemma"></span></p>
+<div id="admonition-lemma-linear-prefix" class="admonition admonish-success">
+<div class="admonition-title">
+<p>Lemma: Linear prefix</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/construction.html#admonition-lemma-linear-prefix"></a></p>
+</div>
+<div>
+<p>If <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1757em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∗</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1757em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∗</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span> then <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8193em;vertical-align:-0.136em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:-0.5em;"></span></span><span class="base"><span class="strut" style="height:0.786em;vertical-align:-0.15em;"></span><span class="mrel"><span class="mrel">⪰</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1757em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∗</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span>.</p>
+<p>Proof: The chain of ancestors of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span> is <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>-linear, and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> are both on that chain.</p>
+</div>
+</div>
+<p>The notation <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span><span class="mord text"><span class="mord"> for </span></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1757em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∗</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">]</span></span></span></span> means the sequence of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span></span></span> for each <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>‑block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span></span> in chain order from genesis up to and <span style="white-space: nowrap">including <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span>.</span> <span style="white-space: nowrap">(<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> is</span> a bound variable within this construct.)</p>
+<p><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="enclosing" style="background-color:yellow;color:black;"><span class="mord text"><span class="mord textsf">TODO:</span></span></span></span></span></span> remove this if not used:</p>
+<p>We use <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">log</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0307em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">log</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8362em;"><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> (without a subscript <span style="white-space: nowrap">on <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">⪯</span></span></span></span>)</span> to mean that the transaction ledger <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">log</span></span></span></span></span> is a prefix <span style="white-space: nowrap">of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0307em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">log</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8362em;"><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span>.</span> Similarly to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:-0.5em;"></span></span><span class="base"><span class="strut" style="height:0.786em;vertical-align:-0.15em;"></span><span class="mrel"><span class="mrel">⪰</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1757em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∗</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> above, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">log</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:-0.5em;"></span></span><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">⪰</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0307em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">log</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8362em;"><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> means that either <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">log</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0307em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">log</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8362em;"><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> or <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0307em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">log</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8362em;"><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">log</span></span></span></span></span>;</span> that is, <span style="white-space: nowrap">“one of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">log</span></span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0307em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">log</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8362em;"><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span> is a prefix of the other”.</p>
+<h2 id="views"><a class="header" href="#views">Views</a></h2>
+<p>In the simplest case, a block‑chain protocol <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1757em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∗</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> provides a single “view” that, <span style="white-space: nowrap">for a given <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>‑execution,</span> provides <span style="white-space: nowrap">each <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>‑node</span> with a time series <span style="white-space: nowrap">on <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>‑chains.</span> More generally a protocol may define several “views” that provide <span style="white-space: nowrap">each <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>‑node</span> with time series on potentially different chain types.</p>
+<p>We model a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>‑view</span> as a function <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.1293em;"><span style="top:-2.6293em;"><span class="pstrut" style="height:2.5em;"></span><span class="mord"><span class="mord sizing reset-size6 size1">⦂</span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathsf">Node</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathsf">Time</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">∗</span><span class="mord text"><span class="mord textsf">-chain</span></span></span></span></span>.</span> By convention, we will write the <span style="white-space: nowrap">node index <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span></span> as a subscript and the <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span></span> as a superscript: <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mopen">(</span><span class="mord mathnormal">i</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span>.</span></p>
+<p><span id="agreement-on-a-view"></span></p>
+<div id="admonition-definition-agreement-on-a-view" class="admonition admonish-success">
+<div class="admonition-title">
+<p>Definition: Agreement on a view</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/construction.html#admonition-definition-agreement-on-a-view"></a></p>
+</div>
+<div>
+<p>An execution of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Π</span></span></span></span> has <strong>Agreement</strong> on the view <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.1293em;"><span style="top:-2.6293em;"><span class="pstrut" style="height:2.5em;"></span><span class="mord"><span class="mord sizing reset-size6 size1">⦂</span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathsf">Node</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathsf">Time</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">∗</span><span class="mord text"><span class="mord textsf">-chain</span></span></span></span></span> iff <span style="white-space: nowrap">for all times <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span></span> and <span style="white-space: nowrap">all <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Π</span></span></span></span> nodes <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span></span> (potentially the same) such that <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span></span> and <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span>,</span> we have <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:-0.5em;"></span></span><span class="base"><span class="strut" style="height:0.786em;vertical-align:-0.15em;"></span><span class="mrel"><span class="mrel">⪰</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1757em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∗</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0781em;vertical-align:-0.3948em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4413em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3948em;"><span></span></span></span></span></span></span></span></span></span>.</span></p>
+</div>
+</div>
+<h2 id="subprotocols-1"><a class="header" href="#subprotocols-1">Subprotocols</a></h2>
+<p>As in Snap‑and‑Chat, we depend on a <span style="white-space: nowrap">BFT protocol <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>,</span> and a <span style="white-space: nowrap">best‑chain protocol <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>.</span></p>
+<div id="admonition-info" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Info</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/construction.html#admonition-info"></a></p>
+</div>
+<div>
+<p>See <a href="design/crosslink/./the-arguments-for-bounded-availability-and-finality-overrides.html#terminology-note">this terminology note</a> for why we do not call <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> a “longest‑chain” protocol.</p>
+</div>
+</div>
+<p>We modify <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> <span style="white-space: nowrap">(resp. <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>)</span> to give <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> <span style="white-space: nowrap">(resp. <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>)</span> by adding structural elements, changing validity rules, and changing the specified behaviour of honest nodes.</p>
+<p>A Crosslink 2 node must participate in <span style="white-space: nowrap">both <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>;</span> that is, it must maintain a view of the state of each protocol. Acting in more specific roles such as bft‑proposer, bft‑validator, or bc‑block‑producer is optional, but we assume that all such actors are Crosslink 2 nodes.</p>
+<h3 id="model-for-bft-protocols-Πorigbftbft"><a class="header" href="#model-for-bft-protocols-Πorigbftbft">Model for BFT protocols (Π<sub>{origbft,bft}</sub>)</a></h3>
+<p><span style="white-space: nowrap">A <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑node’s view</span> includes a set of <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑block chains</span> each rooted at a fixed genesis <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</span> There is a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑block‑validity</span> rule (specified below), which depends only on the content of the block and its ancestors. A non‑genesis block can only be <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑block‑valid</span> if its parent is <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑block‑valid. <span style="white-space: nowrap">A <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑valid‑chain</span> is a chain of <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑block‑valid</span> blocks.</p>
+<p>Execution proceeds in a sequence of epochs. In each epoch, an honest proposer for that epoch may make a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑proposal</span>.</p>
+<p>A <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑proposal</span> refers to a parent <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑block,</span> and specifies the proposal’s epoch. The content of a proposal is signed by the proposer using a strongly unforgeable signature scheme. We consider the proposal to include this signature. There is a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑proposal‑validity</span> rule, depending only on the content of the proposal and its parent block, and the validity of the proposer’s signature.</p>
+<p>We extend the <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0991em;vertical-align:-0.25em;"></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span></span></span></span> notation to <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑proposals</span> in the obvious way: <span style="white-space: nowrap">if <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8304em;vertical-align:-0.136em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>,</span> <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span> is a <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑proposal</span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> its parent <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑block,</span> then <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1322em;vertical-align:-0.2831em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mrel">:=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1905em;vertical-align:-0.3013em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8892em;"><span style="top:-2.3987em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span><span style="top:-3.1031em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3013em;"><span></span></span></span></span></span></span></span></span></span>.</span></p>
+<div id="admonition-terminology-1" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Terminology</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/construction.html#admonition-terminology-1"></a></p>
+</div>
+<div>
+<p>We will shorten <span style="white-space: nowrap">“<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑block‑valid <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑block”</span> to <span style="white-space: nowrap">“<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑valid‑block”,</span> and <span style="white-space: nowrap">“<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑proposal‑valid <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑proposal”</span> to <span style="white-space: nowrap">“<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑valid‑proposal”.</span></p>
+</div>
+</div>
+<p>For each epoch, there is a fixed number of voting units distributed between the <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑nodes,</span> which they use to vote for a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑proposal.</span> We say that a voting unit has been cast for a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑proposal <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span></span> at a given time in a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑execution,</span> <span style="white-space: nowrap">if and only if</span> <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span> is <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑proposal‑valid</span> and a ballot <span style="white-space: nowrap">for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span></span> authenticated by the holder of the voting unit exists at that time.</p>
+<p>Using knowledge of ballots cast for a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑proposal <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span></span> that collectively satisfy a notarization rule at a given time in a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑execution,</span> and only with such knowledge, it is possible to obtain a valid <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑notarization‑proof <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9386em;vertical-align:-0.2441em;"></span><span class="mord"><span class="mord"><span class="mord mathsf" style="margin-right:0.06944em;">proof</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2342em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">P</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span></span></span></span>.</span> The notarization rule must require at least a two‑thirds absolute supermajority of voting units <span style="white-space: nowrap">in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span>’s epoch</span> to have been cast <span style="white-space: nowrap">for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span>.</span> It may also require other conditions.</p>
+<p>A voting unit is cast non‑honestly for an epoch’s proposal iff:</p>
+<ul>
+<li>it is cast other than by the holder of the unit (due to key compromise or any flaw in the voting protocol, for example); or</li>
+<li>it is double‑cast (i.e. there are at least two ballots casting it for distinct proposals); or</li>
+<li>the holder of the unit following the conditions for honest voting <span style="white-space: nowrap">in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>,</span> according to its view, should not have cast that vote.</li>
+</ul>
+<p>Note that a unit should be considered to be cast non-honestly in the case of key compromise, because it is then effectively under the control of an adversary. The key compromise may or may not be attributable to another flaw in the protocol, but such a flaw would not be a break of the consensus mechanism per se.</p>
+<p><span id="one-third-bound"></span></p>
+<div id="admonition-definition-onethird-bound-on-nonhonest-voting" class="admonition admonish-success">
+<div class="admonition-title">
+<p>Definition: One‑third bound on non‑honest voting</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/construction.html#admonition-definition-onethird-bound-on-nonhonest-voting"></a></p>
+</div>
+<div>
+<p>An execution of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has the <strong>one‑third bound on non‑honest voting</strong> property iff for every epoch, <em>strictly</em> fewer than one third of the total voting units for that epoch are ever cast non‑honestly.</p>
+</div>
+</div>
+<div id="admonition-info-1" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Info</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/construction.html#admonition-info-1"></a></p>
+</div>
+<div>
+<p>It may be the case that a ballot cast for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span> is not <a href="design/crosslink/construction.html#in-honest-view">in honest view</a> when it is used to create a notarization proof for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span>. Since we are not assuming synchrony, it may also be the case that such a ballot is in honest view but that any given node has not received it (and perhaps will never receive it).</p>
+<p>There may be multiple distinct ballots or distinct ballot messages attempting to cast a given voting unit for the same proposal; this is undesirable for bandwidth usage, but it is not necessary to consider it to be non‑honest behaviour for the purpose of security analysis, as long as such ballots are not double‑counted toward the two‑thirds threshold.</p>
+</div>
+</div>
+<p><span id="one-third-bound-caveat"></span></p>
+<div id="admonition-security-caveat" class="admonition admonish-warning">
+<div class="admonition-title">
+<p>Security caveat</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/construction.html#admonition-security-caveat"></a></p>
+</div>
+<div>
+<p>The <strong>one‑third bound on non‑honest voting</strong> property considers all ballots cast in the entire execution. In particular, it is possible that a validator’s key is compromised and then used to cast its voting units for a proposal of an epoch long finished. If the number of voting units cast non-honestly for any epoch <em>ever</em> reaches one third of the total voting units for that epoch during an execution, then the <strong>one‑third bound on non‑honest voting</strong> property is violated for that execution.</p>
+<p>Therefore, validator keys of honest nodes must remain secret indefinitely. Whenever a key is rotated, the old key must be securely deleted. For further discussion and potential improvements, see <a href="https://github.com/Electric-Coin-Company/tfl-book/issues/140">tfl-book issue #140</a>.</p>
+</div>
+</div>
+<p>A <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑block</span> consists <span style="white-space: nowrap">of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord mathsf" style="margin-right:0.06944em;">proof</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2342em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">P</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span> re‑signed by the same proposer using a strongly unforgeable signature scheme. It is <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑block‑valid</span> iff:</p>
+<ul>
+<li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span> is <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑proposal‑valid</span>; and</li>
+<li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9386em;vertical-align:-0.2441em;"></span><span class="mord"><span class="mord"><span class="mord mathsf" style="margin-right:0.06944em;">proof</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2342em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">P</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span></span></span></span> is a valid proof that some subset of ballots cast for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span> are sufficient to satisfy the notarization rule; and</li>
+<li>the proposer’s outer signature <span style="white-space: nowrap">on <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord mathsf" style="margin-right:0.06944em;">proof</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2342em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">P</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span> is valid.</li>
+</ul>
+<p><span style="white-space: nowrap">A <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑proposal’s</span> parent reference hashes the entire <span style="white-space: nowrap">parent <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑block,</span> i.e. proposal, proof, and outer signature.</p>
+<div id="admonition-info-2" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Info</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/construction.html#admonition-info-2"></a></p>
+</div>
+<div>
+<p>Neither <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9386em;vertical-align:-0.2441em;"></span><span class="mord"><span class="mord"><span class="mord mathsf" style="margin-right:0.06944em;">proof</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2342em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">P</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span></span></span></span> nor the proposer’s outer signature is unique for a <span style="white-space: nowrap">given <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span>.</span> The proposer’s outer signature is however third‑party nonmalleable, by definition of a strongly unforgeable signature scheme. An <span style="white-space: nowrap">“honest <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑proposal”</span> is a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑proposal</span> made for a given epoch by a proposer who is honest in that epoch. Such a proposer will only create one proposal and only sign at most once for each epoch, and so there will be at most one <span style="white-space: nowrap">“honestly submitted”</span> <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑block</span> for each epoch.</p>
+<p>It is possible for there to be multiple <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑valid‑blocks</span> for the same proposal, with different notarization proofs and/or outer signatures, if the proposer is not honest. However, the property that there will be at most one <span style="white-space: nowrap">“honestly submitted”</span> <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑block</span> for each epoch is important for liveness, even though we cannot guarantee that any particular proposer for an epoch is honest.</p>
+<p><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="enclosing" style="background-color:yellow;color:black;"><span class="mord text"><span class="mord textsf">TODO:</span></span></span></span></span></span> check that we are correctly using this in the liveness analysis.</p>
+</div>
+</div>
+<p>There is an efficiently computable function <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">∗</span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.1293em;"><span style="top:-2.6293em;"><span class="pstrut" style="height:2.5em;"></span><span class="mord"><span class="mord sizing reset-size6 size1">⦂</span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">∗</span><span class="mord text"><span class="mord textsf">bft-block</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">∗</span><span class="mord text"><span class="mord textsf">bft-block</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∪</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord">⊥</span><span class="mclose">}</span></span></span></span>.</span> <span style="white-space: nowrap">For a <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑block‑valid</span> input <span style="white-space: nowrap">block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span>,</span> this function outputs the last ancestor of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span> that is final in the <span style="white-space: nowrap">context of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span>.</span></p>
+<div id="admonition-info-3" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Info</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/construction.html#admonition-info-3"></a></p>
+</div>
+<div>
+<p>The chain of ancestors is unambiguously determined because a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑proposal’s</span> parent reference hashes the entire parent <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑block;</span> each <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑block</span> commits to a proposal; and the parent hashes are collision‑resistant. This holds despite the caveat mentioned above that there may be multiple <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑valid‑blocks</span> for the same proposal.</p>
+</div>
+</div>
+<p><span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">∗</span><span class="mord text"><span class="mord textsf">bft-last-final</span></span></span></span></span></span> must satisfy all of the following:</p>
+<ul>
+<li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∗</span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7184em;vertical-align:-0.024em;"></span><span class="mord">⊥</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟺</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span> is not <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑block‑valid.</li>
+<li>If <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span> is <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑block‑valid, then:
+<ul>
+<li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∗</span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span> (and therefore it must also be <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑block‑valid);</span></li>
+<li>for all <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑valid‑blocks</span> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span> such that <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span>,</span> <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∗</span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∗</span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mclose">)</span></span></span></span>.</span></li>
+</ul>
+</li>
+<li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∗</span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</li>
+</ul>
+<div id="admonition-info-4" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Info</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/construction.html#admonition-info-4"></a></p>
+</div>
+<div>
+<p>It is correct to talk about the “last final block” of a given chain (that is, each <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑valid-block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span></span> unambiguously determines a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑valid-block</span> <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∗</span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mclose">)</span></span></span></span>)</span>, but it is not correct to refer to a given <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑block</span> as objectively <span style="white-space: nowrap">“<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑final”.</span></p>
+</div>
+</div>
+<p>A particular BFT protocol might need adaptations to fit it into this model <span style="white-space: nowrap">for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>,</span> <em>before</em> we apply the Crosslink 2 modifications to <span style="white-space: nowrap">obtain <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</span> Any such adaptions are necessarily protocol-specific. In particular:</p>
+<ul>
+<li>origbft‑proposal‑validity should correspond to the strongest property of an origbft‑proposal that is objectively and feasibly verifiable from the content of the proposal and its parent origbft‑block at the time the proposal is made. It must include verification of the proposer’s signature.</li>
+<li>origbft‑block‑validity should correspond to the strongest property of an origbft‑block that is objectively and feasibly verifiable from the content of the block and its ancestors at the time the block is added to an origbft‑chain. It should typically include all of the relevant checks from origbft‑proposal‑validity that apply to the created block (or equivalent checks). It must also include verification of the notarization proof and the proposer’s outer signature.</li>
+<li>If a node observes an origbft‑valid block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span>, then it should be infeasible for an adversary to cause a rollback in that node’s view <span style="white-space: nowrap">past <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">origbft-last-final</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mclose">)</span></span></span></span>,</span> and the view of the chain <span style="white-space: nowrap">up to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">origbft-last-final</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mclose">)</span></span></span></span></span> should agree with that of all other honest nodes. This is formalized in the next section.</li>
+</ul>
+<h4 id="safety-of-Πbft"><a class="header" href="#safety-of-Πbft">Safety of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></a></h4>
+<p>The intuition behind the following safety property is that:</p>
+<ul>
+<li>For <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> to be safe, it should never be the case that two honest nodes observe (at any time) <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑blocks <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span> respectively that they each consider final in some context, but <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8193em;vertical-align:-0.136em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:-0.5em;"></span></span><span class="base"><span class="strut" style="height:0.786em;vertical-align:-0.15em;"></span><span class="mrel"><span class="mrel">⪰</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span> does not hold.</li>
+<li>By definition, an honest node observes a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑block</span> to be final in the context of another <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span>,</span> <span style="white-space: nowrap">iff <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∗</span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mclose">)</span></span></span></span>.</span></li>
+</ul>
+<p><span id="in-honest-view"></span>
+We say that a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑block</span> is “in honest view” if a party observes it at some time at which that party is honest.</p>
+<p><span id="final-agreement"></span></p>
+<div id="admonition-definition-final-agreement" class="admonition admonish-success">
+<div class="admonition-title">
+<p>Definition: Final Agreement</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/construction.html#admonition-definition-final-agreement"></a></p>
+</div>
+<div>
+<p>An execution of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has <strong>Final Agreement</strong> iff for all <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑valid blocks <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span></span> in honest view at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span></span> and <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> in honest view</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span>,</span> we have <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∗</span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:-0.5em;"></span></span><span class="base"><span class="strut" style="height:0.786em;vertical-align:-0.15em;"></span><span class="mrel"><span class="mrel">⪰</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord">∗</span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>.</span></p>
+</div>
+</div>
+<p>Note that it is possible for this property to hold for an execution of a BFT protocol in an asynchronous communication model. As <a href="design/crosslink/construction.html#one-third-bound-caveat">previously mentioned</a>, if the <a href="design/crosslink/construction.html#one-third-bound"><strong>one‑third bound on non‑honest voting</strong></a> property is <em>ever</em> broken at any time in an execution, then it may not be possible to maintain <strong>Final Agreement</strong> from that point on.</p>
+<details id="admonition-adapting-the-streamlet-bft-protocol" class="admonition admonish-info">
+<summary class="admonition-title">
+<p>Adapting the Streamlet BFT protocol.</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/construction.html#admonition-adapting-the-streamlet-bft-protocol"></a></p>
+</summary>
+<div>
+<p>Streamlet as described in <a href="https://eprint.iacr.org/2020/088.pdf">[CS2020]</a> has three possible states of a block in a player’s view:</p>
+<ul>
+<li>“valid” (but not notarized or final);</li>
+<li>“notarized” (but not final);</li>
+<li>“final”.</li>
+</ul>
+<p>By “valid” the Streamlet paper means just that it satisfies the structural property of being part of a block chain with parent hashes. The role of <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑block‑validity</span> in our model corresponds roughly to Streamlet’s “notarized”. It turns out that with some straightforward changes relative to Streamlet, we can identify “origbft‑block‑valid” with “notarized” and consider an origbft‑valid‑chain to only consist of notarized blocks. This is not obvious, but is a useful simplification.</p>
+<p>Here is how the paper defines “notarized”:</p>
+<blockquote>
+<p>When a block gains votes from at least <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mord mathnormal">n</span><span class="mord">/3</span></span></span></span> distinct players, it becomes notarized. A chain is notarized if its constituent blocks are all notarized.</p>
+</blockquote>
+<p>This implies that blocks can be added to chains independently of notarization. However, the paper also says that an honest leader always proposes a block extending from a notarized chain. Therefore, only <em>notarized</em> chains really matter in the protocol.</p>
+<p>In unmodified Streamlet, the order in which a player sees signatures might cause it to view blocks as notarized out of order. Streamlet’s security analysis is in a synchronous model, and assumes for liveness that any vote will have been received by all players (Streamlet nodes) within two epochs.</p>
+<p>In Crosslink 2, however, we need origbft‑block‑validity to be an objectively and feasibly verifiable property. We also would prefer reliable message delivery within bounded time not to be a basic assumption of our communication model. (This does not dictate what assumptions about message delivery are made for particular security analyses.) If we did not make a modification to the protocol to take this into account, then some Crosslink 2 nodes might receive a two‑thirds absolute supermajority of voting messages and consider a BFT block to be notarized, while others might never receive enough of those messages.</p>
+<p>Obviously a <em>proposal</em> cannot include signatures on itself — but the block formed from it can include proofs about the proposal and signatures. We can therefore say that when a proposal gains a two‑thirds absolute supermajority of signatures, a block is created from it that contains a proof (such as an aggregate signature) that it had such a supermajority. For example, we can have the proposer itself make this proof once it has enough votes, sign the <span style="white-space: nowrap">resulting <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord mathsf" style="margin-right:0.06944em;">proof</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2342em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">P</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span> to create a block, then <em>submit</em> that block in a separate message. (The proposer has most incentive to do this in order to gain whatever reward attaches to a successful proposal; it can outsource the proving task if needed.) Then the origbft‑block‑validity rule can require a valid supermajority proof, which is objectively and feasibly verifiable. Players that see an origbft‑valid‑block can immediately consider it notarized.</p>
+<p>Note that for the liveness analysis to be unaffected, we need to assume that the combined latency of messages, of collecting and aggregating signatures, and of block submission is such that all adapted‑Streamlet nodes will receive a notarized block corresponding to a given proposal (rather than just all of the votes for the proposal) within two epochs. Alternatively we could re‑do the timing analysis.</p>
+<p>With this change, “origbft‑block‑valid” and “notarized” do not need to be distinguished.</p>
+<p>Streamlet’s finality rule is:</p>
+<blockquote>
+<p>If in any notarized chain, there are three adjacent blocks with consecutive epoch numbers, the prefix of the chain up to the second of the three blocks is considered final. When a block becomes final, all of its prefix must be final too.</p>
+</blockquote>
+<p>We can straightforwardly express this as an <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord textsf">origbft-last-final</span></span></span></span></span> function of a context block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span>, as required by the model:</p>
+<p>For an origbft‑valid‑block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">origbft-last-final</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mclose">)</span></span></span></span> is the last origbft‑valid‑block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span> such that either <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> or <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> is the second block of a group of three adjacent blocks with consecutive epoch numbers.</p>
+<p>Note that “When a block becomes final, all of its prefix must be final too.” is implicit in the model.</p>
+</div>
+</details>
+<h3 id="model-for-best-chain-protocols-Πorigbcbc"><a class="header" href="#model-for-best-chain-protocols-Πorigbcbc">Model for best-chain protocols (Π<sub>{origbc,bc}</sub>)</a></h3>
+<p>A node’s view in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> includes a set of <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bc‑block chains</span> each rooted at a fixed <span style="white-space: nowrap">genesis <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bc‑block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</span> There is a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bc‑block‑validity rule</span> (often described as a collection of <span style="white-space: nowrap">“consensus rules”),</span> depending only on the content of the block and its ancestors. A non‑genesis block can only be <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bc‑block‑valid</span> if its parent is <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bc‑block‑valid.</span> <span style="white-space: nowrap">By “<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bc‑valid‑chain”</span> we mean a chain of <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bc‑block‑valid blocks.</span></p>
+<div id="admonition-terminology-2" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Terminology</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/construction.html#admonition-terminology-2"></a></p>
+</div>
+<div>
+<p>The terminology commonly used in the block‑chain community does not distinguish between rules that are part of the consensus protocol proper, and rules required for validity of the economic computation supported by the block chain. Where it is necessary to distinguish, the former can be called “L0” consensus rules, and the latter “L1” consensus rules.</p>
+</div>
+</div>
+<p>The definition of <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bc‑block‑validity</span> is such that it is hard for a block producer to extend a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bc‑valid‑chain</span> unless they are selected by a random process that chooses a block producer in proportion to their resources with an approximately known and consistent time distribution, subject to some assumption about the total proportion of resources held by honest nodes.</p>
+<p>There is a function <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord"><span class="mord mathsf">score</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.1293em;"><span style="top:-2.6293em;"><span class="pstrut" style="height:2.5em;"></span><span class="mord"><span class="mord sizing reset-size6 size1">⦂</span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">∗</span><span class="mord text"><span class="mord textsf">bc-valid-chain</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathsf">Score</span></span></span></span></span>, with <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">&lt;</span></span></span></span> a <a href="https://proofwiki.org/wiki/Definition:Strict_Total_Ordering">strict total ordering</a></span> <span style="white-space: nowrap">on <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathsf">Score</span></span></span></span></span>.</span> An honest node will choose one of the <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bc‑valid‑chains</span> with highest score as the <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bc‑best‑chain</span> in its view. Any rule can be specified for breaking ties.</p>
+<p>The <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord"><span class="mord mathsf">score</span></span></span></span></span> function is required to satisfy <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0972em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord mathsf">score</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mspace mtight" style="margin-right:-0.1952em;"></span><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">score</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span></span></span> for any non‑genesis <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bc‑valid‑chain <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span>.</span></p>
+<div id="admonition-info-5" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Info</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/construction.html#admonition-info-5"></a></p>
+</div>
+<div>
+<p>For a Proof‑of‑Work protocol, the score of a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bc‑chain</span> should be its accumulated work.</p>
+</div>
+</div>
+<p>Unless an adversary is able to censor knowledge of other chains from a node’s view, it should be difficult to cause the node to switch to a chain with a last common ancestor more than <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> blocks</span> back from the tip of its previous <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bc‑best‑chain.</span></p>
+<p>Let <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathsf">ch</span></span></span></span></span> be a <a href="design/crosslink/construction.html#views">view</a> such that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span> is <span style="white-space: nowrap">node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>’s <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bc‑best‑chain</span> at time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>. (This matches the notation used in <a href="https://eprint.iacr.org/2020/1091.pdf">[NTT2020]</a>.) We define <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1454em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8984em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span> to be <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</p>
+<p>A <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bc‑valid‑block</span> is assumed to commit to a collection (usually, a sequence) of <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bc‑transactions</span>. Unlike in Crosslink 1 or Snap-and-Chat, we do not need to explicitly model <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bc‑transaction</span> validity or impose any additional constraints on it. The consensus rules applying to <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bc‑transactions</span> are entirely unchanged, including any rules that depend on <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bc‑block</span> height or previous <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bc‑blocks.</span> This is because Crosslink 2 never reorders or selectively “sanitizes” transactions as Snap-and-Chat does. If a bc‑block is included in a Crosslink 2 block chain then its entire parent bc‑block chain is included just as it would have been in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> (only modified by the <a href="design/crosslink/construction.html#structural-additions">structural additions</a> described later), so block heights are also preserved.</p>
+<p>A “coinbase transaction” is a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bc‑transaction</span> that only distributes newly issued funds and has no inputs.</p>
+<p>Define <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord textsf">is-coinbase-only-block</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.1293em;"><span style="top:-2.6293em;"><span class="pstrut" style="height:2.5em;"></span><span class="mord"><span class="mord sizing reset-size6 size1">⦂</span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">∗</span><span class="mord text"><span class="mord textsf">bc-block</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathsf">boolean</span></span></span></span></span> so that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">is-coinbase-only-block</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.5714em;"></span><span class="mord"><span class="mord mathsf">true</span></span></span></span></span> iff <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> has exactly one transaction that is a coinbase transaction.</p>
+<p>Each <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bc‑block</span> is summarized by a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bc‑header</span> that commits to the block. There is a notion of <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bc‑header‑validity</span> that is necessary, but not sufficient, for validity of the block. We will only make the distinction between <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bc‑headers</span> and <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bc‑blocks</span> when it is necessary to avoid ambiguity.</p>
+<details id="admonition-header-validity-for-proofofwork-protocols" class="admonition admonish-info">
+<summary class="admonition-title">
+<p>Header validity for Proof‑of‑Work protocols.</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/construction.html#admonition-header-validity-for-proofofwork-protocols"></a></p>
+</summary>
+<div>
+<p>In a Proof‑of‑Work protocol, it is normally possible to check the Proof‑of‑Work of a block using only the header. There is a difficulty adjustment function that determines the target difficulty for a block based on its parent chain. So, checking that the correct difficulty target has been used relies on knowing that the header’s parent chain is valid.</p>
+<p>Checking header validity before expending further resources on a purported block can be relevant to mitigating denial‑of‑service attacks that attempt to inflate validation cost.</p>
+</div>
+</details>
+<p>Typically, Bitcoin‑derived best chain protocols do not need much adaptation to fit into this model. The model still omits some details that would be important to implementing Crosslink 2, but distracting for this level of abstraction.</p>
+<h4 id="safety-of-Πbc"><a class="header" href="#safety-of-Πbc">Safety of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></a></h4>
+<p>We make an assumption on executions of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> that we will call <strong>Prefix Consistency</strong> (introduced in [<a href="https://eprint.iacr.org/2016/454.pdf">PSS2016</a>, section 3.3] as just “consistency”):</p>
+<p><span id="prefix-consistency"></span></p>
+<div id="admonition-definition-prefix-consistency" class="admonition admonish-success">
+<div class="admonition-title">
+<p>Definition: Prefix Consistency</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/construction.html#admonition-definition-prefix-consistency"></a></p>
+</div>
+<div>
+<p>An execution of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has <strong>Prefix Consistency</strong> at confirmation depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>, iff <span style="white-space: nowrap">for all times <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span></span> and <span style="white-space: nowrap">all nodes <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span></span> (potentially the same) such that <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span></span> and <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span>,</span> we have that <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.161em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mspace mtight" style="margin-right:-0.1952em;"></span><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1318em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span>.</span></p>
+</div>
+</div>
+<details id="admonition-explain-the-confusion-in-the-literature-about-what-variants-of-this-property-are-called" class="admonition admonish-info">
+<summary class="admonition-title">
+<p>Explain the confusion in the literature about what variants of this property are called.</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/construction.html#admonition-explain-the-confusion-in-the-literature-about-what-variants-of-this-property-are-called"></a></p>
+</summary>
+<div>
+<p>The literature uses the same name, <span style="white-space: nowrap">“common‑prefix property”,</span> for two different properties of very different strength.</p>
+<p>[<a href="https://eprint.iacr.org/2016/454.pdf">PSS2016</a>, section 3.3] introduced the stronger variant. That paper first describes the weaker variant, calling it the “common‑prefix property by Garay et al <a href="https://link.springer.com/chapter/10.1007/978-3-662-46803-6_10">[GKL2015]</a>.” Then it explains what is essentially a bug in that variant, and describes the stronger variant which it just calls “consistency”:</p>
+<blockquote>
+<p>The common‑prefix property by Garay et al <a href="https://link.springer.com/chapter/10.1007/978-3-662-46803-6_10">[GKL2015]</a>, which was already considered and studied by Nakamoto <a href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.221.9986">[Nakamoto2008]</a>, requires that in any round <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span></span></span></span>, the record chains of any two honest players <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> agree on all, but potentially the last <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span></span></span></span>, records. We note that this property (even in combination with the other two desiderata [of Chain Growth and Chain Quality]) provides quite weak guarantees: even if any two honest parties perfectly agree on the chains, the chain could be completely different on, say, even rounds and odd rounds. We here consider a stronger notion of consistency which additionally stipulates players should be consistent with their “future selves”.</p>
+<p>Let <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1606em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">consistent</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9106em;"><span style="top:-3.1323em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">view</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> iff for <span style="white-space: nowrap">all rounds <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span>,</span> and <span style="white-space: nowrap">all players <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span></span> (potentially the same) such that <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> is honest at <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7337em;"></span><span class="mord"><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">view</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7337em;"><span style="top:-3.1323em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span></span></span></span></span></span></span></span></span> and <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> is honest at <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0118em;"></span><span class="mord"><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">view</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0118em;"><span style="top:-3.1323em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8278em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span>,</span> we have that the prefixes of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0087em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.05834em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4413em;margin-left:-0.0583em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">view</span></span><span class="mclose">)</span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3373em;vertical-align:-0.3948em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.05834em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9425em;"><span style="top:-2.4413em;margin-left:-0.0583em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8278em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3948em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">view</span></span><span class="mclose">)</span></span></span></span> consisting of the first <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">ℓ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0087em;vertical-align:-0.2587em;"></span><span class="mord">∣</span><span class="mord"><span class="mord mathcal" style="margin-right:0.05834em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4413em;margin-left:-0.0583em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">view</span></span><span class="mclose">)</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span></span></span></span> records are identical.</p>
+</blockquote>
+<p>Unfortunately, <a href="https://eprint.iacr.org/2014/765.pdf">[GKL2020]</a>, which is a revised version of <a href="https://link.springer.com/chapter/10.1007/978-3-662-46803-6_10">[GKL2015]</a>, switches to the stronger variant <em>without changing the name</em>.</p>
+<p>(The <a href="https://eprint.iacr.org/archive/versions/2014/765">eprint version history</a> may be useful; the change was made in <a href="https://eprint.iacr.org/archive/2014/765/20181013:200033">version 20181013:200033</a>, page 17.)</p>
+<p>Note that [GKL2020] uses an adaptive‑corruption model, “meaning that the adversary is allowed to take control of parties on the fly”, and so their wording in Definition 3:</p>
+<blockquote>
+<p>... for any pair of <span style="white-space: nowrap">honest players <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span> adopting the <span style="white-space: nowrap">chains <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.05834em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0583em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.05834em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0583em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span> at rounds <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.786em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> in <span style="white-space: nowrap"><font style="font-variant: small-caps;">view</font><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3411em;vertical-align:-0.4296em;"></span><span class="mord"><span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9115em;"><span style="top:-2.4065em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">Π</span><span class="mpunct mtight">,</span><span class="mord mathcal mtight">A</span><span class="mpunct mtight">,</span><span class="mord mathcal mtight" style="margin-right:0.07944em;">Z</span></span></span></span><span style="top:-3.1809em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4296em;"><span></span></span></span></span></span></span></span></span></span></span> respectively, it holds that <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3111em;vertical-align:-0.2663em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.05834em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em;"><span style="top:-2.4337em;margin-left:-0.0583em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span><span style="top:-3.2198em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">⌈</span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2663em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.05834em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0583em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</span></p>
+</blockquote>
+<p>is intended to mean the same as our</p>
+<blockquote>
+<p>... <span style="white-space: nowrap">for all times <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span></span> and <span style="white-space: nowrap">all nodes <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span></span> (potentially the same) such that <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span></span> and <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span>,</span> we have that <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.161em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mspace mtight" style="margin-right:-0.1952em;"></span><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1318em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span>.</span></p>
+</blockquote>
+<p>The latter is closer to <a href="https://eprint.iacr.org/2016/454.pdf">[PSS2016]</a>.</p>
+<p>Incidentally, this property does <em>not</em> seem to be mentioned in <a href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.221.9986">[Nakamoto2008]</a>, contrary to the [PSS2016] authors’ assertion. Maybe implicitly, but it’s a stretch.</p>
+</div>
+</details>
+<details id="admonition-discussion-of-gkl2020s-communication-model-and-network-partition" class="admonition admonish-info">
+<summary class="admonition-title">
+<p>Discussion of [GKL2020]’s communication model and network partition.</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/construction.html#admonition-discussion-of-gkl2020s-communication-model-and-network-partition"></a></p>
+</summary>
+<div>
+<p>When <strong>Prefix Consistency</strong> is taken to hold of typical PoW-based block‑chain protocols like Bitcoin (as it often is), this implies that, in the relevant executions, the network of honest nodes is never <span style="white-space: nowrap">partitioned —</span> unless any partition lasts only for a short length of time relative to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> block times. <span style="white-space: nowrap">If node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span></span> is on one side of a full partition and <span style="white-space: nowrap">node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span></span> on the other, then after <span style="white-space: nowrap">node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>’s</span> best chain has been extended by more than <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> blocks,</span> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.161em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mspace mtight" style="margin-right:-0.1952em;"></span><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span> will contain information that has no way to get to <span style="white-space: nowrap">node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span>.</span> And even if the partition is incomplete, we cannot guarantee that the Prefix Consistency property will hold for any given pair of nodes.</p>
+<p>It might be possible to maintain Prefix Consistency if the honest nodes on one side of the partition knew that they should not continue building on their chain until the partition has healed, but it is unclear how that would be done in general without resorting to a BFT protocol (as opposed to in specific cases like a single node being unable to connect to the rest of the network). Certainly there is no mechanism to explicitly detect and respond to partitions in protocols derived from Bitcoin.</p>
+<p>And yet, <a href="https://eprint.iacr.org/2014/765.pdf">[GKL2020]</a> claims to <em>prove</em> Prefix Consistency from other assumptions. So we know that those assumptions must also rule out a long partition between honest nodes. In fact the required assumption is <em>implicit</em> in the communication model:</p>
+<ul>
+<li>A synchronous network cannot be partitioned.</li>
+<li>A partially synchronous network <span style="white-space: nowrap">—that is,</span> providing reliable delivery with bounded but <span style="white-space: nowrap">unknown delay—</span> cannot be partitioned for longer than the delay.</li>
+</ul>
+<p>We might be concerned that these implicit assumptions are stronger than we would like. In practice, the peer‑to‑peer network protocol of Bitcoin and Zcash attempts to flood blocks to all nodes. This protocol might have weaknesses, but it is not intended to (and plausibly does not) depend on all messages being received. (Incidentally, Streamlet also implicitly floods messages to all nodes.)</p>
+<p>Also, Streamlet and many other BFT protocols do <em>not</em> assume <em>for safety</em> that the network is not partitioned. That is, BFT protocols can be safe in a fully asynchronous communication model with unreliable messaging. That is why we avoid taking synchrony or partial synchrony as an implicit assumption of the communication model, or else we could end up with a protocol with weaker safety properties than <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> alone.</p>
+<p>This leaves the question of whether the <strong>Prefix Consistency</strong> property is still too strong, even if we do not rely on it for the analysis of safety when <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has not been subverted. In particular, if a particular <span style="white-space: nowrap">node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">h</span></span></span></span></span> is not well-connected to the rest of the network, then that will inevitably affect <span style="white-space: nowrap">node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">h</span></span></span></span>’s</span> security, but should not affect other honest nodes’ security.</p>
+<p>Fortunately, it is not the case that disconnecting a single <span style="white-space: nowrap">node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">h</span></span></span></span></span> from the network causes the security assumption to be voided. The solution is to view <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">h</span></span></span></span> as not honest in that case (even though it would follow the protocol if it could). This achieves the desired effect within the model, because other nodes can no longer rely on <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">h</span></span></span></span>’s honest input.</span> Although viewing <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">h</span></span></span></span> as potentially adversarial might seem conservative from the point of view of other nodes, bear in mind that an adversary could censor an arbitrary subset of incoming and outgoing messages from the node, and this may be best modelled by considering it to be effectively controlled by the adversary.</p>
+</div>
+</details>
+<p>Prefix Consistency compares the <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>-truncated chain</span> of <span style="white-space: nowrap">some node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span></span> with the <em>untruncated</em> chain of <span style="white-space: nowrap">node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span>.</span> For our analysis of safety of the derived ledgers, we will also need to make an assumption on executions of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> that at <span style="white-space: nowrap">any given time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>,</span> any two <span style="white-space: nowrap">honest nodes <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span></span> <strong>agree</strong> on their confirmed <span style="white-space: nowrap">prefixes —</span> with only the caveat that one may have observed more of the chain than the other. <span style="white-space: nowrap">That is:</span></p>
+<p><span id="prefix-agreement"></span></p>
+<div id="admonition-definition-prefix-agreement" class="admonition admonish-success">
+<div class="admonition-title">
+<p>Definition: Prefix Agreement</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/construction.html#admonition-definition-prefix-agreement"></a></p>
+</div>
+<div>
+<p>An execution of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has <strong>Prefix Agreement</strong> at confirmation <span style="white-space: nowrap">depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span></span> iff it has <a href="design/crosslink/construction.html#agreement-on-a-view"><strong>Agreement</strong></a> on the <span style="white-space: nowrap">view <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">i</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">↦</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.161em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mspace mtight" style="margin-right:-0.1952em;"></span><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span>.</span></p>
+</div>
+</div>
+<details id="admonition-why-are-this-property-and-prefix-consistency-above-stated-as-unconditional-properties-of-protocol-executions-rather-than-as-probabilistic-assumptions" class="admonition admonish-info">
+<summary class="admonition-title">
+<p>Why are this property, and Prefix Consistency above, stated as unconditional properties of protocol <em>executions</em>, rather than as probabilistic assumptions?</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/construction.html#admonition-why-are-this-property-and-prefix-consistency-above-stated-as-unconditional-properties-of-protocol-executions-rather-than-as-probabilistic-assumptions"></a></p>
+</summary>
+<div>
+<p>Our security arguments that depend on these properties will all be of the form “in an execution where ⟨safety properties⟩ are not violated, ⟨undesirable thing⟩ cannot happen”.</p>
+<p>It is not necessary to involve probability in arguments of this form. Any probabilistic reasoning can be done separately.</p>
+<p>In particular, if a statement of this form holds, and ⟨safety properties⟩ are violated with probability <span style="white-space: nowrap">at most <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span></span></span></span></span> under certain conditions, then it immediately follows that under those conditions ⟨undesirable thing⟩ happens with probability <span style="white-space: nowrap">at most <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span></span></span></span>.</span> Furthermore, ⟨undesirable thing⟩ can only happen <em>after</em> ⟨safety properties⟩ have been violated, because the execution up to that point has been an execution in which ⟨safety properties⟩ are <em>not</em> violated.</p>
+<p>With few exceptions, involving probability in a security argument is best done only to account for nondeterministic choices in the protocol itself. This is opinionated advice, but a lot of security proofs would likely be simpler if inherently probabilistic arguments were more distinctly separated from unconditional ones.</p>
+<p>In the case of the Prefix Agreement property, an alternative approach would be to prove that Prefix Agreement holds with some probability given Prefix Consistency and some other chain properties. This is what <a href="https://eprint.iacr.org/2020/1091.pdf">[NTT2020]</a> does in its <span style="white-space: nowrap">Theorem 2,</span> which essentially says that under certain conditions Prefix Agreement holds except with <span style="white-space: nowrap">probability <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9271em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9271em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">Ω</span><span class="mopen mtight">(</span><span class="mord sqrt mtight"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8059em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mtight" style="padding-left:0.833em;"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span><span style="top:-2.7659em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail mtight" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z'/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2341em;"><span></span></span></span></span></span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span></span></span></span>.</span></p>
+<p>The conclusions that can be obtained from this approach are necessarily probabilistic, and depending on the techniques used, the proof may not be tight; that is, the proof may obtain a bound on the probability of failure that is (either asymptotically or concretely) higher than needed. This is the case for <span>[<a href="https://eprint.iacr.org/2020/1091.pdf">NTT2020</a>, Theorem 2];</span> <span style="white-space: nowrap">footnote 10</span> in that paper points out that the expression for the probability can be asymptotically improved:</p>
+<blockquote>
+<p>Using the recursive bootstrapping argument developed in <span>[<a href="https://eprint.iacr.org/2020/601.pdf">DKT+2020</a>, Section 4.2],</span> it is possible to bring the error <span style="white-space: nowrap">probability <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9271em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9271em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">Ω</span><span class="mopen mtight">(</span><span class="mord sqrt mtight"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8059em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mtight" style="padding-left:0.833em;"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span><span style="top:-2.7659em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail mtight" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z'/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2341em;"><span></span></span></span></span></span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span></span></span></span></span> as close to an exponential decay as possible. In this context, <span style="white-space: nowrap">for any <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">ϵ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>,</span> it is possible to find <span style="white-space: nowrap">constants <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">ϵ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">ϵ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span> such that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">p</span><span class="mclose">)</span></span></span></span> is secure after <span style="white-space: nowrap">
+<em>C</em><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mop">max</span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">GST</span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathsf">GAT</span></span><span class="mclose">))</span></span></span></span></span> with confirmation time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> except with <span style="white-space: nowrap">probability <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1369em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">ϵ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9869em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight"><span class="mord mathnormal mtight">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1645em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">ϵ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mspace mtight" style="margin-right:0.1952em;"></span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight">ϵ</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span>.</span></p>
+</blockquote>
+<p>(Here <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span></span></span></span> is the probability that any given node gets to produce a block in any given time slot.)</p>
+<p>In fact none of the proofs of security properties for Snap‑and‑Chat depend on the particular expression <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9271em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9271em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">Ω</span><span class="mopen mtight">(</span><span class="mord sqrt mtight"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8059em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mtight" style="padding-left:0.833em;"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span><span style="top:-2.7659em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail mtight" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z'/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2341em;"><span></span></span></span></span></span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span></span></span></span>;</span> for example in the proofs of <span style="white-space: nowrap">Lemma 5</span> and <span style="white-space: nowrap">Theorem 1,</span> this probability just “passes through” the proof from the premisses to the conclusion, because the argument is not probabilistic. The same will be true of our safety arguments.</p>
+<p>Talking about what is possible in particular executions has further advantages:</p>
+<ul>
+<li>It sidesteps the issue of how to interpret results in the GST model of partial synchrony, when we do not know what <span style="white-space: nowrap">
+<em>C</em><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mop">max</span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">GST</span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathsf">GAT</span></span><span class="mclose">))</span></span></span></span> is.</span> See also the critique of applying the GST model to block‑chain protocols under <a href="design/crosslink/construction.html#admonition-discussion-of-gkl2020s-communication-model-and-network-partition">“Discussion of [GKL2020]’s communication model and network partition”</a> above. (This is not an inherent problem with analyzing the protocol in the partially synchronous setting, but only with inappropriate use of the GST model of that setting.)</li>
+<li>We do not require <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> to be a Nakamoto‑style Proof‑of‑Work block chain protocol. Some other kind of protocol could potentially satisfy Prefix Consistency and Prefix Agreement.</li>
+<li>It is not clear whether a <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9271em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9271em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">Ω</span><span class="mopen mtight">(</span><span class="mord sqrt mtight"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8059em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mtight" style="padding-left:0.833em;"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span><span style="top:-2.7659em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail mtight" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z'/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2341em;"><span></span></span></span></span></span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span></span></span></span> probability of failure would be concretely adequate. That would depend on the value of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> and the constant hidden by the <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Ω</span></span></span></span> notation. The asymptotic property using <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Ω</span></span></span></span> tells us whether a sufficiently large <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> <em>could</em> be chosen, but we are more interested in what needs to be assumed for a given concrete choice of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>.</li>
+<li>If a violation of a required safety property occurs <em>in a given execution</em>, then the safety argument for Crosslink that depended on the property fails for that execution, regardless of what the probability of that occurrence was. This approach therefore more precisely models the consequences of such violations.</li>
+</ul>
+</div>
+</details>
+<details id="admonition-why-intuitively-should-we-believe-that-prefix-agreement-and-prefix-consistency-for-a-large-enough-confirmation-depth-hold-with-high-probability-for-executions-of-a-powbased-bestchain-protocol" class="admonition admonish-info">
+<summary class="admonition-title">
+<p>Why, intuitively, should we believe that Prefix Agreement and Prefix Consistency for a large enough confirmation depth hold with high probability for executions of a PoW‑based best‑chain protocol?</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/construction.html#admonition-why-intuitively-should-we-believe-that-prefix-agreement-and-prefix-consistency-for-a-large-enough-confirmation-depth-hold-with-high-probability-for-executions-of-a-powbased-bestchain-protocol"></a></p>
+</summary>
+<div>
+<p>Roughly speaking, the intuition behind both properties is as follows:</p>
+<p>Honest nodes are collectively able to find blocks faster than an adversary, and communication between honest nodes is sufficiently reliable that they act as a combined network racing against that adversary. Then by the argument in <a href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.221.9986">[Nakamoto2008]</a>, modified by <a href="https://arxiv.org/pdf/1702.02867.pdf">[GP2020]</a> to correct an error in the concrete analysis, a private mining attack that attempts to cause a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>‑block</span> rollback will, with high probability, fail for <span style="white-space: nowrap">large enough <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>.</span> A private mining attack is optimal by the argument in <a href="https://arxiv.org/pdf/2005.10484.pdf">[DKT+2020]</a>.</p>
+<p>Any further analysis of the conditions under which these properties hold should be done in the context of a <span style="white-space: nowrap">particular <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</span></p>
+</div>
+</details>
+<details id="admonition-why-is-the-quantification-in-prefix-agreement-over-two-different-times-t-and-t" class="admonition admonish-info">
+<summary class="admonition-title">
+<p>Why is the quantification in Prefix Agreement over two <em>different</em> times <em>t</em> and <em>t′</em>?</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/construction.html#admonition-why-is-the-quantification-in-prefix-agreement-over-two-different-times-t-and-t"></a></p>
+</summary>
+<div>
+<p>This strengthens the security property, relative to quantifying over a single time. The question can then be split into several parts:</p>
+<ol>
+<li><em><strong>What does the strengthened property mean, intuitively?</strong></em>
+Consider the full tree of <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bc‑valid-blocks</span> that honest nodes have considered to be part of their <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bc‑best-chain</span> at any times during the execution. This property holds iff, when we strip off all branches of length up to and including <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> blocks,</span> the resulting tree <span style="white-space: nowrap">is <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bc‑linear.</span></li>
+<li><em><strong>Why is the strengthening needed?</strong></em>
+Suppose that time were split into periods such that honest nodes agreed on one chain in odd periods, and a completely different chain in even periods. This would obviously not satisfy the intent, but it would satisfy a version of the property that did not quantify over different <span style="white-space: nowrap">times <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span>.</span></li>
+<li><em><strong>Why should we expect the strengthened property to hold?</strong></em>
+If <span style="white-space: nowrap">node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span></span> were far ahead, <span style="white-space: nowrap">i.e. <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">u</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≫</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>,</span> then it is obvious that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.161em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mspace mtight" style="margin-right:-0.1952em;"></span><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1331em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mspace mtight" style="margin-right:-0.1952em;"></span><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span> should hold. Conversely, if node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> were far ahead then it is obvious that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1331em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mspace mtight" style="margin-right:-0.1952em;"></span><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.161em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mspace mtight" style="margin-right:-0.1952em;"></span><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span> should hold. The case where <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span> is the same as quantifying over a single time. By considering intermediate cases where <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span></span> converge from the extremes or where they diverge from being equal, you should be able to convince yourself that the property holds for any relative values of <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span>,</span> in executions of a reasonable best‑chain protocol.</li>
+</ol>
+</div>
+</details>
+<h2 id="definition-of-crosslink-2"><a class="header" href="#definition-of-crosslink-2">Definition of Crosslink 2</a></h2>
+<h3 id="parameters"><a class="header" href="#parameters">Parameters</a></h3>
+<p>Crosslink 2 is parameterized by a <span style="white-space: nowrap">bc‑confirmation‑depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7713em;"></span><span class="mord"><span class="mord mathbb">N</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7713em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">+</span></span></span></span></span></span></span></span></span></span></span></span> (as in Snap‑and‑Chat), and also a finalization gap bound <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7224em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">L</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7713em;"></span><span class="mord"><span class="mord mathbb">N</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7713em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">+</span></span></span></span></span></span></span></span></span></span></span> with <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span> significantly greater <span style="white-space: nowrap">than <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>.</span></p>
+<p>Each node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> always uses the fixed confirmation <span style="white-space: nowrap">depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span></span> to obtain its view of the finalized chain <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.1293em;"><span style="top:-2.6293em;"><span class="pstrut" style="height:2.5em;"></span><span class="mord"><span class="mord sizing reset-size6 size1">⦂</span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord textsf">bc-valid-chain</span></span></span></span></span>.</span> Unlike in Snap‑and‑Chat or Crosslink 1, this is just a block chain; because we do not need sanitization, there is no need to express it as a log of transactions rather than blocks.</p>
+<p>Each node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> chooses a potentially different <span style="white-space: nowrap">bc‑confirmation‑depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">N</span></span></span></span> where <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6835em;vertical-align:-0.0391em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8304em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span></span> to obtain its view of the bounded‑available ledger at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span></span>, <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0797em;vertical-align:-0.2861em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ba</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.1293em;"><span style="top:-2.6293em;"><span class="pstrut" style="height:2.5em;"></span><span class="mord"><span class="mord sizing reset-size6 size1">⦂</span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord textsf">bc-valid-chain</span></span></span></span></span>.</span> (We make the restriction <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8304em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> because there is no reason to choose a <span style="white-space: nowrap">larger <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span></span></span></span>.)</span></p>
+<div id="admonition-security-caveat-1" class="admonition admonish-warning">
+<div class="admonition-title">
+<p>Security caveat</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/construction.html#admonition-security-caveat-1"></a></p>
+</div>
+<div>
+<p>Choosing <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span></span> is at the node’s own risk and may increase the risk of rollback attacks against <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0797em;vertical-align:-0.2861em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ba</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span> (it does not <span style="white-space: nowrap">affect <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span>).</span> Using small values of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span></span></span></span> is not recommended. The default should <span style="white-space: nowrap">be <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>.</span></p>
+</div>
+</div>
+<h3 id="stalled-mode"><a class="header" href="#stalled-mode">Stalled Mode</a></h3>
+<p>Consider, roughly speaking, the number of bc‑blocks that are not yet finalized at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span></span> (a more precise definition will be given in the section on <a href="design/crosslink/construction.html#%CE%A0bc-changes-from-%CE%A0origbc"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> changes from <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span></a>). We call this the “finality gap” at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>.</span> Under an assumption about the distribution of bc‑block intervals, if this gap stays roughly constant then it corresponds to the approximate time that transactions take to be finalized after being included in a bc‑block (if they are finalized at all) just prior to <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>.</span></p>
+<p>As explained in detail by <a href="design/crosslink/./the-arguments-for-bounded-availability-and-finality-overrides.html">The Arguments for Bounded Availability and Finality Overrides</a>, if this bound exceeds a threshold <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span>, then it likely signals an exceptional or emergency condition, in which it is undesirable to keep accepting user transactions that spend funds into new bc‑blocks. In practice, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span> should be <em>at least</em> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>.</p>
+<p>The condition that the network enters in such cases will be called “Stalled Mode”. For a given higher‑level transaction protocol, we can define a policy for which bc‑blocks will be accepted in Stalled Mode. This will be modelled by a predicate <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord textsf">is-stalled-block</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.1293em;"><span style="top:-2.6293em;"><span class="pstrut" style="height:2.5em;"></span><span class="mord"><span class="mord sizing reset-size6 size1">⦂</span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord textsf">bc-block</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathsf">boolean</span></span></span></span></span>.</span> A bc‑block for which <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord textsf">is-stalled-block</span></span></span></span></span> returns <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5714em;"></span><span class="mord"><span class="mord mathsf">true</span></span></span></span></span> is called a “stalled block”.</p>
+<div id="admonition-caution" class="admonition admonish-warning">
+<div class="admonition-title">
+<p>Caution</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/construction.html#admonition-caution"></a></p>
+</div>
+<div>
+<p>A bc‑block producer is only constrained to produce stalled blocks while, roughly speaking, its view of the finalization point is not advancing. In particular an adversary that has subverted the BFT protocol in a way that does <em>not</em> keep the finalization point from advancing, can always avoid being constrained by Stalled Mode.</p>
+</div>
+</div>
+<p>The desired properties of stalled blocks and a possible Stalled Mode policy for Zcash are discussed in the <a href="design/crosslink/./the-arguments-for-bounded-availability-and-finality-overrides.html#how-to-block-hazards">How to block hazards</a> section of <a href="design/crosslink/./the-arguments-for-bounded-availability-and-finality-overrides.html">The Arguments for Bounded Availability and Finality Overrides</a>.</p>
+<p>In practice a node's view of the finalized chain, <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span>,</span> is likely to lag only a few blocks behind <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ba</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span> (depending on the latency overhead imposed <span style="white-space: nowrap">by <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>),</span> <em>unless</em> the chain has entered Stalled Mode. <span style="white-space: nowrap">So when <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>,</span> the main factor influencing the choice of a given application to use <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span> or <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0797em;vertical-align:-0.2861em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ba</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span> is not the average latency, but rather the desired behaviour in the case of a finalization stall: i.e. stall immediately, or keep processing user transactions until <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span> blocks</span> have passed.</p>
+<h3 id="structural-additions"><a class="header" href="#structural-additions">Structural additions</a></h3>
+<ol>
+<li>Each bc‑header has, in addition to origbc‑header fields, a <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">context_bft</span></span></span></span></span> field that commits to a bft‑block.</li>
+<li>Each bft‑proposal has, in addition to origbft‑proposal fields, a <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">headers_bc</span></span></span></span></span> field containing a sequence of exactly <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> bc‑headers</span> (zero‑indexed, deepest first).</li>
+<li>Each non‑genesis bft‑block has, in addition to origbft‑block fields, a <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">headers_bc</span></span></span></span></span> field containing a sequence of exactly <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> bc‑headers</span> (zero-indexed, deepest first). The genesis bft‑block has <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">headers_bc</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6633em;vertical-align:-0.0817em;"></span><span class="mord amsrm">∅</span></span></span></span>.</span></li>
+</ol>
+<p>For a bft‑block or bft‑proposal <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span>, define <span class="katex-display fleqn"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.75em;"><span class="pstrut" style="height:3.75em;"></span><span class="mord"><span class="mord"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mord" style="color:transparent;"><span class="mord" style="color:transparent;"><span class="mord mathsf" style="color:transparent;">LF</span></span><span class="mopen" style="color:transparent;">(</span><span class="mord mathnormal" style="margin-right:0.08125em;color:transparent;">H</span><span class="mclose" style="color:transparent;">)</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.75em;"><span class="pstrut" style="height:3.75em;"></span><span class="mord"><span class="mspace" style="margin-right:-0.1667em;"></span><span class="mspace" style="margin-right:-0.1667em;"></span><span class="mspace" style="margin-right:-0.1667em;"></span><span class="mspace" style="margin-right:-0.1667em;"></span><span class="mrel">:=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">headers_bc</span></span><span class="mopen">[</span><span class="mord">0</span><span class="mclose">]</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mpunct">,</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord">if </span></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">headers_bc</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord amsrm">∅</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord">otherwise.</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span></span></span></span></span>
+For a bc‑block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span>, define <span class="katex-display fleqn"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mord" style="color:transparent;"><span class="mord" style="color:transparent;"><span class="mord mathsf" style="color:transparent;">snapshot</span></span><span class="mopen" style="color:transparent;">(</span><span class="mord mathnormal" style="margin-right:0.05017em;color:transparent;">B</span><span class="mclose" style="color:transparent;">)</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">candidate</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mspace" style="margin-right:-0.1667em;"></span><span class="mspace" style="margin-right:-0.1667em;"></span><span class="mspace" style="margin-right:-0.1667em;"></span><span class="mspace" style="margin-right:-0.1667em;"></span><span class="mrel">:=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">context_bft</span></span><span class="mclose">)</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mspace" style="margin-right:-0.1667em;"></span><span class="mspace" style="margin-right:-0.1667em;"></span><span class="mspace" style="margin-right:-0.1667em;"></span><span class="mspace" style="margin-right:-0.1667em;"></span><span class="mrel">:=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord text"><span class="mord textsf">last-common-ancestor</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.1em;"></span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span></span></span></span></span></p>
+<p>When <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> is the tip of a node’s bc‑best‑chain, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">candidate</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span></span></span> will give the candidate finalization point, subject to a condition described below that prevents local rollbacks.</p>
+<details id="admonition-use-of-the-headers_bc-field-and-its-relation-to-the-ch-field-in-snapandchat" class="admonition admonish-info">
+<summary class="admonition-title">
+<p>Use of the headers_bc field, and its relation to the ch field in Snap‑and‑Chat.</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/construction.html#admonition-use-of-the-headers_bc-field-and-its-relation-to-the-ch-field-in-snapandchat"></a></p>
+</summary>
+<div>
+<p>For a bft‑proposal or <span style="white-space: nowrap">bft‑block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span>,</span> the role of the bc‑chain snapshot referenced by <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1641em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">headers_bc</span></span><span class="mopen">[</span><span class="mord">0</span><span class="mclose">]</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span> is comparable to the <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">lc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> snapshot</span> referenced by <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">ch</span></span></span></span></span> in the Snap‑and‑Chat construction from <a href="https://eprint.iacr.org/2020/1091.pdf">[NTT2020]</a>. The motivation for the additional headers is to demonstrate, to any party that sees a bft‑proposal (resp. bft‑block), that the snapshot had been confirmed when the proposal (resp. the block’s proposal) was made.</p>
+<p>Typically, a node that is validating an honest bft‑proposal or bft‑block will have seen at least the snapshotted bc‑block (and possibly some of the subsequent bc‑blocks in the <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">headers_bc</span></span></span></span></span> chain) before. For this not to be the case, the validator’s bc‑best‑chain would have to be more than <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> bc‑blocks</span> behind the honest proposer’s bc‑best‑chain at a given time, which would violate the <strong>Prefix Consistency</strong> property <span style="white-space: nowrap">of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</span></p>
+<p>If the headers do not connect to any bc‑valid‑chain known to the validator, then the validator should be suspicious that the proposer might not be honest. It can assign a lower priority to validating the proposal in this case, or simply drop it. The latter option could drop a valid proposal, but this does not in practice cause a problem as long as a sufficient number of validators are properly synced (so that <strong>Prefix Consistency</strong> holds for them).</p>
+<p>If the headers <em>do</em> connect to a known bc‑valid‑chain, it could still be the case that the whole header chain up to and including <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">headers_bc</span></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">]</span></span></span></span> is not a bc‑valid‑chain. Therefore, to limit denial‑of‑service attacks the validator should first check the Proofs‑of‑Work and difficulty adjustment <span style="white-space: nowrap">—which it can do locally using only the headers—</span> before attempting to download and validate any bc‑blocks that it has not already seen. This is why we include the full headers rather than just the block hashes. Nodes may “trim” (i.e. not explicitly store) headers in a bft‑block that overlap with those referred to by its <span style="white-space: nowrap">ancestor bft‑block(s).</span></p>
+</div>
+</details>
+<details id="admonition-why-is-a-distinguished-value-needed-for-the-headers_bc-field-in-the-genesis-bftblock" class="admonition admonish-info">
+<summary class="admonition-title">
+<p>Why is a distinguished value needed for the headers_bc field in the genesis bft‑block?</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/construction.html#admonition-why-is-a-distinguished-value-needed-for-the-headers_bc-field-in-the-genesis-bftblock"></a></p>
+</summary>
+<div>
+<p>It would be conceptually nice for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0972em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">headers</span></span><span class="mopen">[</span><span class="mord">0</span><span class="mclose">]</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span> to refer to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>, as well as <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">context_bft</span></span></span></span></span> being <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> so that <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1em;vertical-align:-0.35em;"></span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">context_bft</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</span> That reflects the fact that we know <span style="white-space: nowrap">“from the start”</span> that neither genesis block can be rolled back.</p>
+<p>This is not literally implementable using block hashes because it would involve a hash cycle, but we achieve the same effect by defining a <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span></span></span></span> function that allows us to “patch” <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> <span style="white-space: nowrap">to be <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</span> We do it this way around rather than “patching” the link from a bc‑block to a bft‑block, because the genesis bft‑block already needs a special case since there are not <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> bc‑headers</span> available.</p>
+</div>
+</details>
+<details id="admonition-why-is-the-context_bft-field-needed-why-not-use-a-final_bft-field-to-refer-directly-to-the-last-final-bftblock-before-the-context-block" class="admonition admonish-info">
+<summary class="admonition-title">
+<p>Why is the context_bft field needed? Why not use a final_bft field to refer directly to the last final bft‑block before the context block?</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/construction.html#admonition-why-is-the-context_bft-field-needed-why-not-use-a-final_bft-field-to-refer-directly-to-the-last-final-bftblock-before-the-context-block"></a></p>
+</summary>
+<div>
+<p>The finality of some bft‑block is only defined in the context of another bft‑block. One possible design would be for a bc‑block to have both <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">final_bft</span></span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">context_bft</span></span></span></span></span> fields, so that the finality of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">final_bft</span></span></span></span></span> could be checked objectively in the context <span style="white-space: nowrap">of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">context_bft</span></span></span></span></span>.</span></p>
+<p>However, specifying just the context block is sufficient information to determine its last final ancestor. There would never be any need to give a context block and a final ancestor that is not the last one. The <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord textsf">bft-last-final</span></span></span></span></span> function can be computed efficiently for typical BFT protocols. Therefore, having just the <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">context_bft</span></span></span></span></span> field is sufficient.</p>
+</div>
+</details>
+<h3 id="locally-finalized-chain"><a class="header" href="#locally-finalized-chain">Locally finalized chain</a></h3>
+<p>Each node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> keeps track of a “locally finalized” <span style="white-space: nowrap">bc‑chain <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span></span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>.</span> Each node’s locally finalized bc‑chain starts <span style="white-space: nowrap">at <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0728em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</span> However, this chain state should not be exposed to clients of the node until it <a href="design/crosslink/construction.html#syncing-and-checkpoints">has synced</a>.</p>
+<p><span id="local-finalization-linearity"></span></p>
+<div id="admonition-definition-local-finalization-linearity" class="admonition admonish-success">
+<div class="admonition-title">
+<p>Definition: Local finalization linearity</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/construction.html#admonition-definition-local-finalization-linearity"></a></p>
+</div>
+<div>
+<p>Node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> has <strong>Local finalization linearity</strong> up to time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span> iff the time series of <span style="white-space: nowrap">bc‑blocks <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.162em;vertical-align:-0.2769em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8852em;"><span style="top:-2.4231em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.14em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span><span class="mspace mtight" style="margin-right:0.1952em;"></span><span class="mrel mtight">≤</span><span class="mspace mtight" style="margin-right:0.1952em;"></span><span class="mord mathnormal mtight">t</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2769em;"><span></span></span></span></span></span></span></span></span></span></span> is <span style="white-space: nowrap">
+<a href="design/crosslink/construction.html#notation">bc‑linear</a>.</span></p>
+</div>
+</div>
+<p>When <span style="white-space: nowrap">node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>’s bc‑best‑chain view</span> is updated <span style="white-space: nowrap">from <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9957em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">s</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span> to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span>,</span> <span style="white-space: nowrap">the node’s <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span></span> will become <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1279em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">candidate</span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> if and only if this is a descendant <span style="white-space: nowrap">of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9531em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">s</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span>.</span> <span style="white-space: nowrap">Otherwise <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span> will stay at <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9531em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">s</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span>.</span> This guarantees <strong>Local finalization linearity</strong> by construction.</p>
+<p>If when making this update, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2279em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">candidate</span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.5em;"></span><span class="mord"><span class="delimsizing size1">/</span></span><span class="mspace" style="margin-right:-1.06em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:-0.5em;"></span></span><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">⪰</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9531em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">s</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span> (i.e. <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1279em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">candidate</span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9531em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">s</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span> are on different forks), then the node should record a finalization safety hazard. This can only happen if global safety assumptions are violated. Note that <strong>Local finalization linearity</strong> on each node is <em>not</em> sufficient for <a href="design/crosslink/construction.html#assured-finality"><strong>Assured Finality</strong></a>, but it is necessary.</p>
+<p>This can be expressed by the following state update algorithm, where <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">s</span></span></span></span> is the time of the last update and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span> is the time of the current update:</p>
+<p><span class="katex-display fleqn"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1279em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">update-fin</span></span><span class="mopen">(</span><span class="mord text"><span class="mord">mut </span></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">s</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mspace" style="margin-right:2em;"></span><span class="mord text"><span class="mord">let </span></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1279em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">candidate</span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.8304em;vertical-align:-0.136em;"></span><span class="mspace" style="margin-right:2em;"></span><span class="mord text"><span class="mord">if </span></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪰</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9614em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">s</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord text"><span class="mord">:</span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:1.0906em;vertical-align:-0.247em;"></span><span class="mspace" style="margin-right:2em;"></span><span class="mspace" style="margin-right:2em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8436em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">←</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mspace" style="margin-right:2em;"></span><span class="mord text"><span class="mord">else:</span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:1.0906em;vertical-align:-0.247em;"></span><span class="mspace" style="margin-right:2em;"></span><span class="mspace" style="margin-right:2em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8436em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">←</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9614em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">s</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mspace" style="margin-right:2em;"></span><span class="mspace" style="margin-right:2em;"></span><span class="mord text"><span class="mord">if </span></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span></span><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9614em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">s</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord text"><span class="mord">:</span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mspace" style="margin-right:2em;"></span><span class="mspace" style="margin-right:2em;"></span><span class="mspace" style="margin-right:2em;"></span><span class="mord text"><span class="mord">record finalization safety hazard</span></span></span></span></span></span></p>
+<p>A safety hazard record should include <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span> and the history of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> updates including and since the last one that was an ancestor of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span></span></span></span>.</p>
+<p><span id="local-fin-depth"></span></p>
+<div id="admonition-lemma-local-findepth" class="admonition admonish-success">
+<div class="admonition-title">
+<p>Lemma: Local fin‑depth</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/construction.html#admonition-lemma-local-findepth"></a></p>
+</div>
+<div>
+<p>In any execution of Crosslink 2, for any <span style="white-space: nowrap">node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span></span> that is honest at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>,</span> there exists a <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span></span> <span style="white-space: nowrap">such that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0331em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span>.</span></p>
+<p>Proof: By the definition of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathsf">candidate</span></span></span></span></span> we have <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">candidate</span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0331em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span> for all <span style="white-space: nowrap">times <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span>.</span> <span style="white-space: nowrap">Let <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span></span> be the last time at which <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span> changed,</span> or the <span style="white-space: nowrap">genesis time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span> if it has never changed. <span style="white-space: nowrap">Then for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span> we have <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9531em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">candidate</span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0331em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span>,</span> and <span style="white-space: nowrap">for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span> we have <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9531em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0331em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span></span> <span style="white-space: nowrap">(because <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1454em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8984em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>,</span> and <span style="white-space: nowrap">truncating <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span> always <span style="white-space: nowrap">yields <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>).</span></p>
+</div>
+</div>
+<details id="admonition-why-does-fini-need-to-be-maintained-using-local-state" class="admonition admonish-info">
+<summary class="admonition-title">
+<p>Why does fin<sub>i</sub> need to be maintained using local state?</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/construction.html#admonition-why-does-fini-need-to-be-maintained-using-local-state"></a></p>
+</summary>
+<div>
+<p>When a node’s view of the bc‑best‑chain reorgs to a different fork (even if the reorg is shorter than <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> blocks), it may be the case that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1279em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">candidate</span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> rolls back. If <a href="design/crosslink/construction.html#final-agreement"><strong>Final Agreement</strong></a> of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> holds up to time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>, the new snapshot should in that case be an ancestor of the old one. If all is well then this snapshot will subsequently roll forward along the same path. However, we do not want applications using the node to see the temporary rollback.</p>
+</div>
+</details>
+<p><span id="assured-finality"></span>
+<a href="design/crosslink/../../terminology.html#definition-assured-finality">Assured Finality</a> is our main safety goal for Crosslink 2. It is essentially the same goal as <strong>Final Agreement</strong> but applied to nodes’ locally finalized bc‑chains; intuitively it means that honest nodes never see conflicting locally finalized chains. We intend to prove that this goal holds under reasonable assumptions about <em>either</em> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> <em>or</em> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>.</p>
+<div id="admonition-definition-assured-finality" class="admonition admonish-success">
+<div class="admonition-title">
+<p>Definition: Assured Finality</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/construction.html#admonition-definition-assured-finality"></a></p>
+</div>
+<div>
+<p>An execution of Crosslink 2 has <strong>Assured Finality</strong> iff <span style="white-space: nowrap">for all times <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span></span> and <span style="white-space: nowrap">all nodes <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span></span> (potentially the same) such that <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span></span> and <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span>,</span> we have <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:-0.5em;"></span></span><span class="base"><span class="strut" style="height:0.786em;vertical-align:-0.15em;"></span><span class="mrel"><span class="mrel">⪰</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0892em;vertical-align:-0.3948em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3948em;"><span></span></span></span></span></span></span></span></span></span>.</span></p>
+</div>
+</div>
+<p>Note that if an execution of Crosslink 2 has <strong>Assured Finality</strong>, then all nodes that are honest for that execution have <a href="design/crosslink/construction.html#local-finalization-linearity"><strong>Local finalization linearity</strong></a>. That is because the restriction of <strong>Assured Finality</strong> to the <span style="white-space: nowrap">case <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span></span> is equivalent to <strong>Local finalization linearity</strong> <span style="white-space: nowrap">for node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span></span> up to any time at which <span style="white-space: nowrap">node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> is honest.</span></p>
+<details id="admonition-why-do-we-need-to-use-candidateh-rather-than-snapshotlfh" class="admonition admonish-info">
+<summary class="admonition-title">
+<p>Why do we need to use candidate(H) rather than snapshot(LF(H))?</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/construction.html#admonition-why-do-we-need-to-use-candidateh-rather-than-snapshotlfh"></a></p>
+</summary>
+<div>
+<p>This ensures that the candidate is at least <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>‑confirmed.</span></p>
+<p>In practice <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">candidate</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span></span></span> will rarely differ from <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.1em;"></span><span class="mclose">)</span></span></span></span>, but using the former patches over a potential gap in the safety proof. The <a href="design/crosslink/construction.html#last-final-snapshot-rule"><strong>Last Final Snapshot rule</strong></a> specified later will guarantee that <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.1em;"></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span>,</span> and this ensures that <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.1em;"></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:-0.5em;"></span></span><span class="base"><span class="strut" style="height:0.786em;vertical-align:-0.15em;"></span><span class="mrel"><span class="mrel">⪰</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0331em;vertical-align:-0.2831em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span>.</span> However, the depth of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.1em;"></span><span class="mclose">)</span></span></span></span> relative to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> is not guaranteed to be <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>. For the proof we will need <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">candidate</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0331em;vertical-align:-0.2831em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span>,</span> so that we can use the <a href="design/crosslink/construction.html#local-fin-depth"><strong>Local fin‑depth lemma</strong></a> together with <a href="design/crosslink/construction.html#prefix-agreement"><strong>Prefix Agreement</strong></a> of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> at confirmation depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> to prove <a href="design/crosslink/construction.html#assured-finality"><strong>Assured Finality</strong></a>.</p>
+<p>An alternative would be to change the <a href="design/crosslink/construction.html#last-final-snapshot-rule"><strong>Last Final Snapshot rule</strong></a> to directly require <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.1em;"></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0331em;vertical-align:-0.2831em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span>.</span></p>
+<p><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="enclosing" style="background-color:yellow;color:black;"><span class="mord text"><span class="mord textsf">TODO:</span></span></span></span></span></span> Choose between these options based on what works well for the security proofs and finalization latency.</p>
+</div>
+</details>
+<h3 id="locally-boundedavailable-chain"><a class="header" href="#locally-boundedavailable-chain">Locally bounded‑available chain</a></h3>
+<p>Define the locally bounded‑available chain on node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> for bc‑confirmation‑depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span></span></span></span>, as <span class="katex-display fleqn"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1297em;vertical-align:-0.2861em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ba</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8436em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em;"><span style="top:-2.3987em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.1809em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3013em;"><span></span></span></span></span></span></span><span class="mpunct">,</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mpunct">,</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord">if </span></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em;"><span style="top:-2.3987em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.1809em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3013em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord">otherwise.</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>
+<p>Like the locally finalized bc‑chain, this chain state should not be exposed to clients of the node until it <a href="design/crosslink/construction.html#syncing-and-checkpoints">has synced</a>.</p>
+<p><span id="ledger-prefix-property"></span></p>
+<div id="admonition-theorem-ledger-prefix-property" class="admonition admonish-success">
+<div class="admonition-title">
+<p>Theorem: Ledger prefix property</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/construction.html#admonition-theorem-ledger-prefix-property"></a></p>
+</div>
+<div>
+<p>For any node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> that is honest at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>,</span> and any confirmation <span style="white-space: nowrap">depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span></span></span></span>,</span> <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0797em;vertical-align:-0.2861em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ba</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span>.</span></p>
+<p>Proof: By construction of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0797em;vertical-align:-0.2861em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ba</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span>.</span></p>
+</div>
+</div>
+<p><span id="local-ba-depth"></span></p>
+<div id="admonition-lemma-local-badepth" class="admonition admonish-success">
+<div class="admonition-title">
+<p>Lemma: Local ba‑depth</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/construction.html#admonition-lemma-local-badepth"></a></p>
+</div>
+<div>
+<p>In any execution of Crosslink 2, for any <span style="white-space: nowrap">confirmation depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8304em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span></span> and any <span style="white-space: nowrap">node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span></span> that is honest at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>,</span> there exists a <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span></span> <span style="white-space: nowrap">such that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0797em;vertical-align:-0.2861em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ba</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0836em;vertical-align:-0.3013em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em;"><span style="top:-2.3987em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.1809em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3013em;"><span></span></span></span></span></span></span></span></span></span>.</span></p>
+<p>Proof: Either <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0797em;vertical-align:-0.2861em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ba</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span>, in which case the result follows by the <a href="design/crosslink/construction.html#local-fin-depth"><strong>Local fin‑depth lemma</strong></a> since <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8304em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>, or <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0797em;vertical-align:-0.2861em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ba</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1792em;vertical-align:-0.3013em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em;"><span style="top:-2.3987em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.1809em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3013em;"><span></span></span></span></span></span></span></span></span></span> in which case it follows trivially with <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>.</p>
+</div>
+</div>
+<p>Our security goal for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ba</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> will be <a href="design/crosslink/construction.html#agreement-on-a-view"><strong>Agreement</strong></a> on <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ba</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">μ</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span></span> as already defined.</p>
+<details id="admonition-why-is-the-otherwise-case-in-the-definition-of-baμit-necessary" class="admonition admonish-info">
+<summary class="admonition-title">
+<p>Why is the ‘otherwise’ case in the definition of (ba<sub>μ</sub>)<sub>i</sub><sup>t</sup> necessary?</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/construction.html#admonition-why-is-the-otherwise-case-in-the-definition-of-baμit-necessary"></a></p>
+</summary>
+<div>
+<p>Assume for this discussion that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> uses PoW.</p>
+<p>Depending on the value of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>, the timestamps of bc‑blocks, and the difficulty adjustment rule, it can be the case that if <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> switches to a different fork, the difficulty on that fork is greater than on the chain of the previous snapshot. Then, the new bc‑chain could reach a higher score than the previous chain in fewer than <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> blocks</span> from the fork point, and so <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1792em;vertical-align:-0.3013em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em;"><span style="top:-2.3987em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.1809em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3013em;"><span></span></span></span></span></span></span></span></span></span> might not be a descendant of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span> (which is more likely <span style="white-space: nowrap">if <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>).</span> This can occur even when all safety assumptions are satisfied.</p>
+<p>For <a href="https://zips.z.cash/protocol/protocol.pdf#diffadjustment">Zcash’s difficulty adjustment algorithm</a>, the difficulty of each block is adjusted based on the median timestamps and difficulty target thresholds over a range of <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">PoWAveragingWindow</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">17</span></span></span></span> blocks,</span> where each median is taken over <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathsf">PoWMedianBlockSpan</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">11</span></span></span></span> blocks.</span> Other damping factors and clamps are applied in order to prevent instability and to reduce the influence that adversarially chosen timestamps can have on difficulty adjustment. This makes it unlikely that an adversary could gain a significant advantage by manipulating the difficulty adjustment. So it is safe to use <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span> in this case: even though it does not have <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span></span></span></span> confirmations</span> relative to <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span>,</span> it does have at least the required amount of work “on top” of it.</p>
+<p>Defining <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0797em;vertical-align:-0.2861em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ba</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span> this way also has the advantage of making the proof of the <strong>Ledger prefix property</strong> trivial.</p>
+</div>
+</details>
+<h3 id="syncing-and-checkpoints"><a class="header" href="#syncing-and-checkpoints">Syncing and checkpoints</a></h3>
+<p>It is recommended that node implementations “bake in” a checkpointed <span style="white-space: nowrap">bft‑block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathsf mtight">checkpoint</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span></span> to each released version, and that node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> should only expose <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0797em;vertical-align:-0.2861em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ba</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span></span> to its clients once it is “synced”, that is:</p>
+<ul>
+<li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathsf mtight">checkpoint</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1279em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>;  and</li>
+<li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathsf mtight">checkpoint</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span>;  and</li>
+<li>the timestamp of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span> is within some threshold of the current time.</li>
+</ul>
+<h3 id="Πbft-changes-from-Πorigbft"><a class="header" href="#Πbft-changes-from-Πorigbft">Π<sub>bft</sub> changes from Π<sub>origbft</sub></a></h3>
+<h4 id="Πbft-proposal-and-block-validity"><a class="header" href="#Πbft-proposal-and-block-validity">Π<sub>bft</sub> proposal and block validity</a></h4>
+<p><span id="genesis-bft-block-rule"></span>
+<strong>Genesis bft‑block rule:</strong> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> is bft‑block‑valid.</p>
+<p>A bft‑proposal (resp. non‑genesis bft‑block) <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> is bft‑proposal‑valid (resp. bft‑block‑valid) iff all of the following hold:</p>
+<ul>
+<li><span id="inherited-origbft-rules"></span>
+<strong>Inherited origbft rules:</strong> The corresponding origbft‑proposal‑validity (resp. origbft‑block‑validity) rules hold for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span>.</li>
+<li><span id="linearity-rule"></span>
+<strong>Linearity rule:</strong> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0972em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span>.</li>
+<li><span id="tail-confirmation-rule"></span>
+<strong>Tail Confirmation rule:</strong> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">headers_bc</span></span></span></span></span> form the <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>‑block tail of a bc‑valid‑chain.</li>
+</ul>
+<p><span id="parent-rule"></span>
+The “corresponding validity rules” are assumed to include the <strong>Parent rule</strong> that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span>’s parent is bft‑valid.</p>
+<p>Note: origbft‑block‑validity rules may be different to origbft‑proposal‑validity rules. For example, in adapted Streamlet, a origbft‑block needs evidence that it was voted for by a supermajority, and an origbft‑proposal doesn’t. Such differences also apply to bft‑block‑validity vs bft‑proposal‑validity.</p>
+<details id="admonition-why-have-validity-rules-been-separated-from-the-honest-voting-condition-below" class="admonition admonish-info">
+<summary class="admonition-title">
+<p>Why have validity rules been separated from the honest voting condition below?</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/construction.html#admonition-why-have-validity-rules-been-separated-from-the-honest-voting-condition-below"></a></p>
+</summary>
+<div>
+<p>The reason to separate the validity rules from the honest voting condition, is that the validity rules are objective: they don’t depend on an observer’s view of the bc‑best‑chain. Therefore, they can be checked independently of validator signatures. Even a proposal voted for by 100% of validators will not be considered bft‑proposal‑valid by other nodes unless it satisfies the above rules. If more than two thirds of voting units are cast for an <strong>invalid</strong> proposal, something is seriously <em>and visibly</em> wrong; in any case, the block will not be accepted as a bft‑valid‑block. Importantly, a purportedly valid bft‑block will not be recognized as such by <em>any</em> honest Crosslink 2 node even if it includes a valid notarization proof, if it does not meet other bft‑block‑validity rules.</p>
+<p>This is essential to making the finalized chain safe against a flaw in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> or its security assumptions (even, say, a complete break of the validator signature algorithm), as long as <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> remains safe.</p>
+</div>
+</details>
+<details id="admonition-what-does-the-linearity-rule-do" class="admonition admonish-info">
+<summary class="admonition-title">
+<p>What does the <strong>Linearity rule</strong> do?</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/construction.html#admonition-what-does-the-linearity-rule-do"></a></p>
+</summary>
+<div>
+<p>This rule is key to combining simplicity with strong security properties in Crosslink 2. It essentially says that, in a given bft‑valid‑chain, the snapshots pointed to by blocks in that chain cannot roll back.</p>
+<p>This allows the informal safety argument for Crosslink 2 to be rather intuitive.</p>
+<p>Informally, if <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has <a href="design/crosslink/construction.html#final-agreement"><strong>Final Agreement</strong></a>, then all nodes see only one consistent <a href="design/crosslink/construction.html#notation">bft‑linear</a> chain (restricting to bft‑blocks that are final in the context of some bft‑block <a href="design/crosslink/construction.html#in-honest-view">in honest view</a>). Within such a bft‑chain, the <a href="design/crosslink/construction.html#linearity-rule"><strong>Linearity rule</strong></a> ensures <em>by construction</em> that the sequence of referenced bc‑chain snapshots is <a href="design/crosslink/construction.html#notation">bc‑linear</a>. This implies <a href="design/crosslink/construction.html#assured-finality"><strong>Assured Finality</strong></a>, without needing to assume any safety property of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</p>
+<p>We will <em>also</em> be able to prove safety of the finalized snapshots based only on safety of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> (for a confirmation depth of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>), without needing to assume any safety property of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>. Informally, that is because each node sees each candidate final snapshot at a given time as a <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>-confirmed prefix of its bc‑best‑chain at that time (this can be proven based on the <a href="design/crosslink/construction.html#%CE%A0bc-changes-from-%CE%A0origbc"><strong>Last Final Snapshot rule</strong></a> and the fact that a snapshot includes <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> subsequent headers), and <a href="design/crosslink/construction.html#prefix-agreement"><strong>Prefix Agreement</strong></a> implies that honest nodes agree on this prefix. We will leave a more detailed argument until after we have presented the <span style="white-space: nowrap">
+<a href="design/crosslink/construction.html#%CE%A0bc-changes-from-%CE%A0origbc"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> changes from <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span></a>.</span></p>
+<p>The <a href="design/crosslink/construction.html#linearity-rule"><strong>Linearity rule</strong></a> replaces the “Increasing Score rule” used in Crosslink 1. The Increasing Score rule required that each snapshot in a bft‑valid‑chain either be the same snapshot, or a higher-scoring snapshot to that of its parent block. Since scores strictly increase within a bc‑valid‑chain, the <a href="design/crosslink/construction.html#linearity-rule"><strong>Linearity rule</strong></a> <em>implies</em> the Increasing Score rule. It retains the same or stronger positive effects:</p>
+<ul>
+<li>It prevents potential attacks that rely on proposing a bc‑valid‑chain that forks from a much earlier block. This is necessary because the difficulty (or stake threshold) at that point could have been much lower.</li>
+<li>It limits the extent of disruption an adversary can feasibly cause to the bounded‑available <span style="white-space: nowrap">chain <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>,</span> <em>even if</em> it has <span style="white-space: nowrap">subverted <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</span> Informally, because the finalized chain <em>is</em> a <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> chain, its safety is no worse than <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> alone for a rollback of <em>any</em> depth.</li>
+<li>It ensures that either progress is made (the snapshot advances relative to that of the parent bft‑block), <em>or</em> there is no further validation that needs to be done for the snapshot because it was already validated.</li>
+</ul>
+<p>Note that the adversary could take advantage of an “accidental” fork and start its attack from the base of that fork, so that not all of this work is done by it alone. This is also possible in the case of a standard “private mining” attack, and is not so much of a problem in practice because accidental forks are expected to be short. In any case, <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> should</span> be chosen to take it into account.</p>
+<p>The <a href="design/crosslink/construction.html#linearity-rule"><strong>Linearity rule</strong></a> is also critical to removing the need for one of the most complex elements of Snap‑and‑Chat and Crosslink 1, “sanitization”. In those protocols, because bc‑chain snapshots could be unrelated to each other, it was necessary to sanitize the chain formed from these snapshots to remove transactions that were contextually invalid (e.g. because they double‑spend). The negative consequences of this are described in <a href="design/crosslink/./notes-on-snap-and-chat.html">Notes on Snap‑and‑Chat</a>; avoiding it is much simpler.</p>
+<p>The linearity property is intentionally always relative to the snapshot of the parent bft‑block, even if it is not final in the context of the current bft‑block. This is because the rule needs to hold if and when it becomes final in the context of some descendant bft‑block.</p>
+<p><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="enclosing" style="background-color:yellow;color:black;"><span class="mord text"><span class="mord textsf">TODO:</span></span></span></span></span></span> PoS Desideratum: we want leader selection with good security / performance properties that will be relevant to this rule. (Suggested: <a href="https://arxiv.org/pdf/2010.08154.pdf">PoSAT</a>.)</p>
+</div>
+</details>
+<details id="admonition-why-does-the-linearity-rule-allow-keeping-the-same-snapshot-as-the-parent" class="admonition admonish-info">
+<summary class="admonition-title">
+<p>Why does the <strong>Linearity rule</strong> allow keeping the same snapshot as the parent?</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/construction.html#admonition-why-does-the-linearity-rule-allow-keeping-the-same-snapshot-as-the-parent"></a></p>
+</summary>
+<div>
+<p>This is necessary in order to preserve liveness of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> relative <span style="white-space: nowrap">to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>.</span> Liveness of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> might require honest proposers to make proposals at a minimum rate. That requirement could be consistently violated if it were not always possible to make a valid proposal. But given that it is allowed to repeat the same snapshot as in the parent bft‑block, neither the <a href="design/crosslink/construction.html#linearity-rule"><strong>Linearity rule</strong></a> nor the <a href="design/crosslink/construction.html#tail-confirmation-rule"><strong>Tail Confirmation rule</strong></a> can prevent making a valid proposal — and all other rules of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> affecting the ability to make valid proposals are the same as <span style="white-space: nowrap">in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>.</span> <span style="white-space: nowrap">(In principle,</span> changes to voting in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> could also affect its liveness; we’ll discuss that in the liveness proof later.)</p>
+<p>For example, Streamlet requires three notarized blocks <em>in consecutive epochs</em> in order to finalize <span style="white-space: nowrap">a block [<a href="https://eprint.iacr.org/2020/088.pdf">CS2020</a>, section 1.1].</span> Its proof of liveness depends on the assumption that in each epoch for which the leader is honest, that leader will make a proposal, and that during a “period of synchrony” this proposal will be received by <span style="white-space: nowrap">every node [<a href="https://eprint.iacr.org/2020/088.pdf">CS2020</a>, section 3.6].</span> This argument can also be extended to adapted‑Streamlet.</p>
+<p>We could alternatively have allowed to always make a “null” proposal, rather than to always make a proposal with the same snapshot as the parent. We prefer the latter because the former would require specifying the rules for null proposals <span style="white-space: nowrap">in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>.</span></p>
+<p>As a clarification, no BFT protocol that uses leader election can <em>require</em> a proposal in each epoch, because the leader might be dishonest. The above issue concerns liveness of the protocol when assumptions about the attacker’s share of bft‑validators or stake are met, so that it can be assumed that sufficiently long periods with enough honest leaders to make progress <span style="white-space: nowrap">(5 consecutive</span> epochs in the case of Streamlet), will occur with significant probability.</p>
+</div>
+</details>
+<h4 id="Πbft-block-finality-in-context"><a class="header" href="#Πbft-block-finality-in-context">Π<sub>bft</sub> block finality in context</a></h4>
+<p>The finality rule for bft‑blocks in a given context is unchanged from origbft‑finality. That is, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord textsf">bft-last-final</span></span></span></span></span> is defined in the same way as <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord textsf">origbft-last-final</span></span></span></span></span> (modulo referring to bft‑block‑validity and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>).</p>
+<h4 id="Πbft-honest-proposal"><a class="header" href="#Πbft-honest-proposal">Π<sub>bft</sub> honest proposal</a></h4>
+<p>An honest proposer of a bft‑proposal <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span> chooses <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">headers_bc</span></span></span></span></span> as the <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>‑block tail of its bc‑best‑chain, provided that it is consistent with the <a href="design/crosslink/construction.html#linearity-rule"><strong>Linearity rule</strong></a>. If it would not be consistent with that rule, it sets <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">headers_bc</span></span></span></span></span> to the same <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">headers_bc</span></span></span></span></span> field as <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span>’s parent bft‑block. It does not make proposals until its bc‑best‑chain is at least <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> blocks long.</p>
+<details id="admonition-why-σ--1" class="admonition admonish-info">
+<summary class="admonition-title">
+<p>Why σ + 1?</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/construction.html#admonition-why-σ--1"></a></p>
+</summary>
+<div>
+<p>If the length were less than <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> blocks, it would be impossible to construct the <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">headers_bc</span></span></span></span></span> field of the proposal.</p>
+<p>Note that when the length of the proposer’s bc‑best‑chain is exactly <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> blocks, the snapshot must be of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">.</span></span></span></span> But this does not violate the <a href="design/crosslink/construction.html#linearity-rule"><strong>Linearity rule</strong></a>, because <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> matches the previous snapshot by <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</p>
+</div>
+</details>
+<details id="admonition-how-is-it-possible-that-the-linearity-rulelinearity-rule-would-not-be-satisfied-by-choosing-headers-from-an-honest-proposers-bcbestchain" class="admonition admonish-info">
+<summary class="admonition-title">
+<p>How is it possible that the <a href="design/crosslink/construction.html#linearity-rule"><strong>Linearity rule</strong></a> would not be satisfied by choosing headers from an honest proposer’s bc‑best‑chain?</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/construction.html#admonition-how-is-it-possible-that-the-linearity-rulelinearity-rule-would-not-be-satisfied-by-choosing-headers-from-an-honest-proposers-bcbestchain"></a></p>
+</summary>
+<div>
+<p>As in the answer to <a href="design/crosslink/construction.html#admonition-why-is-the-otherwise-case-in-the-definition-of-bai%CE%BCt-necessary">Why is the ‘otherwise’ case in the definition of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0797em;vertical-align:-0.2861em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ba</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span> necessary?</a> above, after a reorg on the bc‑chain, the <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>-confirmed block on the new chain might not be a descendant of the <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>-confirmed block on the old chain, which could break the <a href="design/crosslink/construction.html#linearity-rule"><strong>Linearity rule</strong></a>.</p>
+</div>
+</details>
+<h4 id="Πbft-honest-voting"><a class="header" href="#Πbft-honest-voting">Π<sub>bft</sub> honest voting</a></h4>
+<p>An honest validator considering a proposal <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span>, first updates its view of both subprotocols with the bc‑headers given in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">headers_bc</span></span></span></span></span>, downloading bc‑blocks for these headers and checking their bc‑block‑validity.</p>
+<p>For each downloaded bc‑block, the bft‑chain referenced by its <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">context_bft</span></span></span></span></span> field might need to be validated if it has not been seen before.</p>
+<details id="admonition-wait-what-how-much-validation-is-that" class="admonition admonish-info">
+<summary class="admonition-title">
+<p>Wait what, how much validation is that?</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/construction.html#admonition-wait-what-how-much-validation-is-that"></a></p>
+</summary>
+<div>
+<p>In general the entire referenced bft‑chain needs to be validated, not just the referenced block — and for each bft‑block, the bc‑chain in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">headers_bc</span></span></span></span></span> needs to be validated, and so on recursively. If this sounds overwhelming, note that:</p>
+<ul>
+<li>We should check the requirement that a bft‑valid‑block must have been voted for by a two‑thirds absolute supermajority of validators, and any other <em>non‑recursive</em> bft‑validity rules, <em>first</em>.</li>
+<li>Before validating a bc‑chain referenced by a <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">headers_bc</span></span></span></span></span> field, we check that it connects to an already-validated bc‑chain and that the Proofs‑of‑Work are valid. This implies that the amount of bc‑block validation is constrained by how fast the network can find valid Proofs‑of‑Work.</li>
+<li>The <a href="design/crosslink/construction.html#linearity-rule"><strong>Linearity rule</strong></a> reduces the worst‑case validation effort, by ensuring that only one bc‑chain needs to be validated for any bft‑chain. Assuming safety of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> and that the adversary does not have an overwhelming advantage in computing the Proof‑of‑Work, this is effectively only one bc‑chain overall with, at most, short side branches.</li>
+</ul>
+<p>In summary, the order of validation is important to avoid denial‑of‑service — but it already is in Bitcoin and Zcash.</p>
+</div>
+</details>
+<p>After updating its view, the validator will vote for a proposal <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span> <em>only if</em>:</p>
+<ul>
+<li><span id="valid-proposal-criterion"></span>
+<strong>Valid proposal criterion:</strong> it is bft‑proposal‑valid, and</li>
+<li><span id="confirmed-best-chain-criterion"></span>
+<strong>Confirmed best‑chain criterion:</strong> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mclose">)</span></span></span></span> is part of the validator’s bc‑best‑chain at a bc‑confirmation‑depth of at least <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>.</li>
+</ul>
+<details id="admonition-blocks-in-a-bcbestchain-are-by-definition-bcblockvalid-if-were-checking-the-confirmed-bestchain-criterion-why-do-we-need-to-have-separately-checked-that-the-blocks-referenced-by-the-headers-are-bcblockvalid" class="admonition admonish-info">
+<summary class="admonition-title">
+<p>Blocks in a bc‑best‑chain are by definition bc‑block‑valid. If we’re checking the <strong>Confirmed best‑chain criterion</strong>, why do we need to have separately checked that the blocks referenced by the headers are bc‑block‑valid?</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/construction.html#admonition-blocks-in-a-bcbestchain-are-by-definition-bcblockvalid-if-were-checking-the-confirmed-bestchain-criterion-why-do-we-need-to-have-separately-checked-that-the-blocks-referenced-by-the-headers-are-bcblockvalid"></a></p>
+</summary>
+<div>
+<p>The <strong>Confirmed best‑chain criterion</strong> is quite subtle. It ensures that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1641em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">headers_bc</span></span><span class="mopen">[</span><span class="mord">0</span><span class="mclose">]</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span> is bc‑block‑valid and has <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> bc‑block‑valid blocks after it in the validator’s bc‑best‑chain. However, it need not be the case that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">headers_bc</span></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">]</span></span></span></span> is part of the validator’s bc‑best‑chain after it updates its view. That is, the chain could fork after <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mclose">)</span></span></span></span>.</p>
+<p>The bft‑proposal‑validity rule must be objective; it can’t depend on what the validator’s bc‑best‑chain is. The validator’s bc‑best‑chain <em>may</em> have been updated to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">headers_bc</span></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">]</span></span></span></span> (if it has the highest score), but it also may not.</p>
+<p>However, if the validator’s bc‑best‑chain <em>was</em> updated, that makes it more likely that it will be able to vote for the proposal.</p>
+<p>In any case, if the validator does not check that all of the blocks referenced by the headers are bc‑block‑valid, then its vote may be invalid.</p>
+</div>
+</details>
+<details id="admonition-how-does-this-compare-to-snapandchat" class="admonition admonish-info">
+<summary class="admonition-title">
+<p>How does this compare to Snap‑and‑Chat?</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/construction.html#admonition-how-does-this-compare-to-snapandchat"></a></p>
+</summary>
+<div>
+<p>Snap‑and‑Chat already had the voting condition:</p>
+<blockquote>
+<p>An honest node only votes for a proposed BFT block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> if it views <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">ch</span></span></span></span></span> as confirmed.</p>
+</blockquote>
+<p>but it did <em>not</em> give the headers potentially needed to update the validator’s view, and it did not require a proposal to be for an <em>objectively confirmed</em> snapshot as a matter of validity.</p>
+<p>If a Crosslink‑like protocol were to require an objectively confirmed snapshot but without including the bc‑headers in the proposal, then validators would not immediately know which bc‑blocks to download to check its validity. This would increase latency, and would be likely to lead proposers to be more conservative and only propose blocks that they think will <em>already</em> be in at least a two‑thirds absolute supermajority of validators’ best chains.</p>
+<p>That is, showing <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">headers_bc</span></span></span></span></span> to all of the validators is advantageous to the proposer, because the proposer does not have to guess what blocks the validators might have already seen. It is also advantageous for the protocol goals in general, because it improves the trade‑off between finalization latency and security.</p>
+</div>
+</details>
+<h3 id="Πbc-changes-from-Πorigbc"><a class="header" href="#Πbc-changes-from-Πorigbc">Π<sub>bc</sub> changes from Π<sub>origbc</sub></a></h3>
+<h4 id="Πbc-block-validity"><a class="header" href="#Πbc-block-validity">Π<sub>bc</sub> block validity</a></h4>
+<p><span id="genesis-bc-block-rule"></span>
+<strong>Genesis bc‑block rule:</strong> For the genesis bc‑block we must have <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">context_bft</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>,</span> and therefore <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1em;vertical-align:-0.35em;"></span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">context_bft</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</span></p>
+<p>A bc‑block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> is bc‑block‑valid iff all of the following hold:</p>
+<ul>
+<li><span id="inherited-origbc-rules"></span>
+<strong>Inherited origbc rules:</strong> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> satisfies the corresponding origbc‑block‑validity rules.</li>
+<li><span id="valid-context-rule"></span>
+<strong>Valid context rule:</strong> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">context_bft</span></span></span></span></span> is bft‑block‑valid.</li>
+<li><span id="extension-rule"></span>
+<strong>Extension rule:</strong> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0972em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span></span></span>.</li>
+<li><span id="last-final-snapshot-rule"></span>
+<strong>Last Final Snapshot rule:</strong> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.1em;"></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span>.</li>
+<li><span id="finality-depth-rule"></span>
+<strong>Finality depth rule:</strong> Define: <span class="katex-display fleqn"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">finality-depth</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">height</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">height</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.1em;"></span><span class="mclose">))</span></span></span></span></span> Then either <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">finality-depth</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span> or <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">is-stalled-block</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span></span></span>.</span></li>
+</ul>
+<details id="admonition-explain-the-definition-of-finalitydepth" class="admonition admonish-info">
+<summary class="admonition-title">
+<p>Explain the definition of finality‑depth.</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/construction.html#admonition-explain-the-definition-of-finalitydepth"></a></p>
+</summary>
+<div>
+<p>The finality depth must be objectively defined, since it is used in a consensus rule. Therefore it should measure the height of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> relative to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.1em;"></span><span class="mclose">)</span></span></span></span>, which is an objectively defined function of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span>, rather than relative to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span>. (These will only differ for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span> when node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> has just reorged, and only then in corner cases.)</p>
+<p>Note that the <a href="design/crosslink/construction.html#last-final-snapshot-rule"><strong>Last Final Snapshot rule</strong></a> ensures that it is meaningful to simply use the difference in heights, since <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.1em;"></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span>.</p>
+</div>
+</details>
+<h4 id="Πbc-contextual-validity"><a class="header" href="#Πbc-contextual-validity">Π<sub>bc</sub> contextual validity</a></h4>
+<p>The consensus rule changes above are all non-contextual. Modulo these changes, contextual validity <span style="white-space: nowrap">in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span> is the same as <span style="white-space: nowrap">in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>.</span></p>
+<h4 id="Πbc-honest-block-production"><a class="header" href="#Πbc-honest-block-production">Π<sub>bc</sub> honest block production</a></h4>
+<p>An honest producer of a <span style="white-space: nowrap">bc‑block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span></span> must follow the consensus rules under <a href="design/crosslink/construction.html#%CE%A0bc-block-validity"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> block validity</a> above. In particular, it must produce a stalled block if required to do so by the <a href="design/crosslink/construction.html#finality-depth-rule"><strong>Finality depth rule</strong></a>.</p>
+<p>To <span style="white-space: nowrap">choose <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">context_bft</span></span></span></span></span>,</span> the producer considers a subset of the tips of bft‑valid‑chains in its view: <span class="katex-display fleqn"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="mord text"><span class="mord"> is bft‑block‑valid and </span></span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="mclose">)}</span></span></span></span></span> It chooses one of the longest of these chains, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span>, breaking ties by maximizing <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">score</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mclose">)))</span></span></span></span>,</span> and if there is still a tie then by taking <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span> with the smallest hash.</p>
+<p>The honest block producer then sets <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">context_bft</span></span></span></span></span> to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span>.</span></p>
+<div id="admonition-attention" class="admonition admonish-warning">
+<div class="admonition-title">
+<p>Attention</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/construction.html#admonition-attention"></a></p>
+</div>
+<div>
+<p>An honest bc‑block‑producer must not use information from the BFT protocol, other than the specified consensus rules, to decide which bc‑valid‑chain to follow. The specified consensus rules that depend on <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> have been carefully constructed to preserve safety of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> relative <span style="white-space: nowrap">to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>.</span> Imposing any additional constraints could potentially allow an adversary that is able to <span style="white-space: nowrap">subvert <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>,</span> to influence the evolution of the bc‑best‑chain in ways that are not considered in the safety argument.</p>
+</div>
+</div>
+<details id="admonition-why-not-choose-t--such-that-h-1bc--context_bft--bft--bftlastfinalt-" class="admonition admonish-info">
+<summary class="admonition-title">
+<p>Why not choose <em>T</em>  such that <em>H</em> ⌈<sup>1</sup><sub>bc</sub> . context_bft  ⪯<sub>bft</sub>  bft‑last‑final(<em>T</em> )?</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/construction.html#admonition-why-not-choose-t--such-that-h-1bc--context_bft--bft--bftlastfinalt-"></a></p>
+</summary>
+<div>
+<p>The effect of this would be to tend to more often follow the last bft‑block seen by the producer of the parent bc‑block, if there is a choice. It is not always possible to do so, though: the resulting set of candidates <span style="white-space: nowrap">for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span></span> might be empty.</p>
+<p>Also, it is not clear that giving the parent bc‑block‑producer the chance to “guide” what bft‑block should be chosen next is beneficial, since that producer might be adversarial and the resulting incentives are difficult to reason about.</p>
+</div>
+</details>
+<details id="admonition-why-choose-the-longest-c-rather-than-the-longest-bftlastfinalc-" class="admonition admonish-info">
+<summary class="admonition-title">
+<p>Why choose the longest <em>C</em>, rather than the longest bft‑last‑final(<em>C</em> )?</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/construction.html#admonition-why-choose-the-longest-c-rather-than-the-longest-bftlastfinalc-"></a></p>
+</summary>
+<div>
+<p>We could have instead chosen <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span> to maximize the length <span style="white-space: nowrap">of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mclose">)</span></span></span></span>.</span> The rule we chose follows Streamlet, which builds on the longest notarized chain, not the longest finalized chain. This may call for more analysis specific to the chosen BFT protocol.</p>
+</div>
+</details>
+<details id="admonition-why-this-tiebreaking-rule" class="admonition admonish-info">
+<summary class="admonition-title">
+<p>Why this tie‑breaking rule?</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/construction.html#admonition-why-this-tiebreaking-rule"></a></p>
+</summary>
+<div>
+<p>Choosing the bft‑chain that has the last final snapshot with the highest score, tends to inhibit an adversary’s ability to finalize its own chain if it has a lesser score. (If it has a greater score, then it has already won a hash race and we cannot stop the adversary chain from being finalized.)</p>
+</div>
+</details>
+<p>For discussion of potentially unifying the roles of bc‑block producer and bft‑proposer, see <a href="design/crosslink/./potential-changes.html#what-about-making-the-bcblockproducer-the-bftproposer">What about making the bc‑block‑producer the bft‑proposer?</a> in <a href="design/crosslink/./potential-changes.html">Potential changes to Crosslink</a>.</p>
+<p>At this point we have completed the definition of Crosslink 2. In <a href="design/crosslink/./security-analysis.html">Security Analysis of Crosslink 2</a>, we will prove it secure.</p>
+<div style="break-before: page; page-break-before: always;"></div><link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.12.0/dist/katex.min.css" integrity="sha384-AfEj0r4/OFrOo5t7NnNe46zW/tFgW6x/bCJG8FqQCEo3+Aro6EYUG4+cU+KJWu/X" crossorigin="anonymous">
+<h1 id="security-analysis-of-crosslink-2"><a class="header" href="#security-analysis-of-crosslink-2">Security Analysis of Crosslink 2</a></h1>
+<p>This document analyzes the security of Crosslink 2 in terms of liveness and safety. It assumes that you have read <a href="design/crosslink/./construction.html">The Crosslink 2 Construction</a> which defines the protocol.</p>
+<h2 id="liveness-argument"><a class="header" href="#liveness-argument">Liveness argument</a></h2>
+<p>First note that Crosslink 2 <em>intentionally</em> sacrifices <strong>availability</strong> if there is a long finalization stall.</p>
+<div id="admonition-info" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Info</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/security-analysis.html#admonition-info"></a></p>
+</div>
+<div>
+<p>This is technically independent of the other changes; you can omit the <a href="design/crosslink/./construction.html#finality-depth-rule"><strong>Finality depth rule</strong></a> and the protocol would still have security advantages over Snap‑and‑Chat, as well as being much simpler and solving its “spending from finalized outputs” issue. In that case the incentives to “pull” the finalization point forward to include new final bft‑blocks would be weaker, but honest bc‑block‑producers would still do it.</p>
+</div>
+</div>
+<p>It would still be a bug if there were any situation in which <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> failed to be <strong>live</strong>, though, because that would allow <a href="design/crosslink/./the-arguments-for-bounded-availability-and-finality-overrides.html#tail-thrashing-attacks">tail‑thrashing attacks</a>.</p>
+<p>Crosslink 2 involves a bidirectional dependence between <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathsf mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathsf mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>. The Ebb‑and‑Flow paper <a href="https://eprint.iacr.org/2020/1091.pdf">[NTT2020]</a> argues in Appendix E (&quot;Bouncing Attack on Casper FFG&quot;) that it can be more difficult to reason about liveness given a bidirectional dependence between protocols:</p>
+<blockquote>
+<p>To ensure consistency among the two tiers [of Casper FFG], the fork choice rule of the blockchain is modified to always respect ‘the justified checkpoint of the greatest <span style="white-space: nowrap">height [*]’ <a href="https://arxiv.org/abs/1710.09437">[22]</a>.</span> There is thus a bidirectional interaction between the block proposal and the finalization layer: blocks proposed by the blockchain are input to finalization, while justified checkpoints constrain future block proposals. This bidirectional interaction is intricate to reason about and a gateway for liveness attacks.</p>
+</blockquote>
+<div id="admonition-info-1" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Info</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/security-analysis.html#admonition-info-1"></a></p>
+</div>
+<div>
+<p>[*] The quotation changes this to “[depth]”, but our terminology is consistent with Ethereum here and not with [NTT2020]’s idiosyncratic use of “depth” to mean “block height”.</p>
+</div>
+</div>
+<p>The argument is correct as far as it goes. The main reason why this does not present any great difficulty to proving liveness of Crosslink, is due to a fundamental difference from Casper FFG: in Crosslink 2 the fork‑choice rule of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> is not modified.</p>
+<p>Let <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathcal" style="margin-right:0.075em;">S</span></span></span></span> be the subset of bc‑blocks <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">is-coinbase-only-block</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mord text"><span class="mord"> and </span></span><span class="mord text"><span class="mord textsf">is-stalled-block</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)}</span></span></span></span>. Assume that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathcal" style="margin-right:0.075em;">S</span></span></span></span> is such that a bc‑block‑producer can always produce a block in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathcal" style="margin-right:0.075em;">S</span></span></span></span>.</p>
+<p>In that case it is straightforward to convince ourselves that the additional bc‑block‑validity rules are never an obstacle to producing a new bc‑block in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathcal" style="margin-right:0.075em;">S</span></span></span></span>:</p>
+<ul>
+<li>The changes to the definition of contextual validity do not interfere with liveness, since they do not affect coinbase transactions, and therefore do not affect blocks in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathcal" style="margin-right:0.075em;">S</span></span></span></span>. That is, the bc‑block‑producer can always just omit non‑coinbase transactions that are contextually invalid.</li>
+<li>The <a href="design/crosslink/./construction.html#genesis-bc-block-rule"><strong>Genesis bc‑block rule</strong></a> doesn’t apply to new bc‑blocks.</li>
+<li>The <a href="design/crosslink/./construction.html#valid-context-rule"><strong>Valid context rule</strong></a>, <a href="design/crosslink/./construction.html#extension-rule"><strong>Extension rule</strong></a>, and <a href="design/crosslink/./construction.html#last-final-snapshot-rule"><strong>Last Final Snapshot rule</strong></a> are always satisfiable by referencing the same context bft‑block as the parent bc‑block.</li>
+<li>The <a href="design/crosslink/./construction.html#finality-depth-rule"><strong>Finality depth rule</strong></a> always allows the option of producing a stalled block, and therefore does not affect blocks in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathcal" style="margin-right:0.075em;">S</span></span></span></span>.</li>
+</ul>
+<p>The instructions to an honest bc‑block‑producer allow it to produce a block in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathcal" style="margin-right:0.075em;">S</span></span></span></span>. Therefore, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> remains live under the same conditions as <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>.</p>
+<p>The additional bft‑proposal‑validity, bft‑block‑validity, bft‑finality, and honest voting rules are also not an obstacle to making, voting for, or finalizing bft‑proposals:</p>
+<ul>
+<li>Because <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> is live, there will always be some point in time at which a fresh valid bc‑header chain that satisfies both the <a href="design/crosslink/./construction.html#linearity-rule"><strong>Linearity rule</strong></a> and the <a href="design/crosslink/./construction.html#tail-confirmation-rule"><strong>Tail confirmation rule</strong></a> exists for use in the <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord">.</span><span class="mord"><span class="mord mathsf">headers_bc</span></span></span></span></span> field.</li>
+<li>If no fresh valid bc‑header chain is available, the <a href="design/crosslink/./construction.html#linearity-rule"><strong>Linearity rule</strong></a> and <a href="design/crosslink/./construction.html#tail-confirmation-rule"><strong>Tail confirmation rule</strong></a> allow an honest bft‑proposer to choose <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">headers_bc</span></span></span></span></span> to be the same as in the previous bft‑block. So, if liveness of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> depends on an honest proposer always being able to make a proposal (as it does in adapted‑Streamlet for example), then this requirement will not be violated.</li>
+<li>The changes to voting are only requiring a vote to be for a proposal that could have been honestly proposed.</li>
+<li>The bft‑finality rules are unchanged from origbft‑finality.</li>
+</ul>
+<p>Therefore, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> remains live under the same conditions as <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>.</p>
+<p>The only other possibility for a liveness issue in Crosslink 2 would be if the change to the constructions of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> or <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ba</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> could cause either of them to stall, even when <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> are both still live.</p>
+<p>However, liveness of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> and the <a href="design/crosslink/./construction.html#linearity-rule"><strong>Linearity rule</strong></a> together imply that at each point in time, provided there are sufficient honest bft‑proposers/validators, eventually a new bft‑block with a higher-scoring snapshot will become final in the context of the longest bft‑valid‑chain. <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="enclosing" style="background-color:yellow;color:black;"><span class="mord text"><span class="mord textsf">TODO:</span></span></span></span></span></span> make that more precise.</p>
+<p>Because of the <a href="design/crosslink/./construction.html#extension-rule"><strong>Extension rule</strong></a>, this new bft‑block must be a descendent of the previous final bft‑block in the context visible to bc‑block‑producers. Therefore, the new finalized chain will extend the old finalized chain.</p>
+<p>Finally, we need to show that Stalled Mode is only triggered when it should be; that is, when the assumptions needed for liveness of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> are violated. Informally, that is the case because, as long as there are sufficient honest bc‑block‑producers <em>and</em> sufficient honest bft‑proposers/validators, the finalization point implied by the <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">context_bft</span></span></span></span></span> field at the tip of the bc‑best chain in any node’s view will advance fast enough for the finalization gap bound <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span> not to be hit. This depends on the value of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span> relative to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>, the network delay, the hash rate of honest bc‑block‑producers, the number of honest bft‑proposers and the proportion of voting units they hold, and other details of the BFT protocol. <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="enclosing" style="background-color:yellow;color:black;"><span class="mord text"><span class="mord textsf">TODO:</span></span></span></span></span></span> more detailed argument needed, especially for the dependence on <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span>.</p>
+<h2 id="safety-argument"><a class="header" href="#safety-argument">Safety argument</a></h2>
+<p><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="enclosing" style="background-color:yellow;color:black;"><span class="mord text"><span class="mord textsf">TODO:</span></span></span></span></span></span> Not updated for Crosslink 2 below this point.</p>
+<h3 id="discussion"><a class="header" href="#discussion">Discussion</a></h3>
+<p>Recall the definition of <a href="design/crosslink/./construction.html#assured-finality"><strong>Assured Finality</strong></a>.</p>
+<p><span id="assured-finality"></span></p>
+<div id="admonition-definition-assured-finality" class="admonition admonish-success">
+<div class="admonition-title">
+<p>Definition: Assured Finality</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/security-analysis.html#admonition-definition-assured-finality"></a></p>
+</div>
+<div>
+<p>An execution of Crosslink 2 has <strong>Assured Finality</strong> iff <span style="white-space: nowrap">for all times <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span></span> and <span style="white-space: nowrap">all nodes <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span></span> (potentially the same) such that <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span></span> and <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span>,</span> we have <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:-0.5em;"></span></span><span class="base"><span class="strut" style="height:0.786em;vertical-align:-0.15em;"></span><span class="mrel"><span class="mrel">⪰</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0892em;vertical-align:-0.3948em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3948em;"><span></span></span></span></span></span></span></span></span></span>.</span></p>
+</div>
+</div>
+<p>First we prove that <strong>Assured Finality</strong> is implied by <a href="design/crosslink/./construction.html#prefix-agreement"><strong>Prefix Agreement</strong></a> of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</p>
+<p><span id="prefix-agreement"></span></p>
+<div id="admonition-definition-prefix-agreement" class="admonition admonish-success">
+<div class="admonition-title">
+<p>Definition: Prefix Agreement</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/security-analysis.html#admonition-definition-prefix-agreement"></a></p>
+</div>
+<div>
+<p>An execution of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has <strong>Prefix Agreement</strong> at confirmation <span style="white-space: nowrap">depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>,</span> iff <span style="white-space: nowrap">for all times <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span></span> and <span style="white-space: nowrap">all nodes <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span></span> (potentially the same) such that <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span></span> and <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span>,</span> we have <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.161em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mspace mtight" style="margin-right:-0.1952em;"></span><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:-0.5em;"></span></span><span class="base"><span class="strut" style="height:0.786em;vertical-align:-0.15em;"></span><span class="mrel"><span class="mrel">⪰</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1331em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mspace mtight" style="margin-right:-0.1952em;"></span><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span>.</span></p>
+</div>
+</div>
+<p><span id="prefix-agreement-implies-assured-finality"></span></p>
+<div id="admonition-safety-theorem-prefix-agreement-of-Πbc-implies-assured-finality" class="admonition admonish-success">
+<div class="admonition-title">
+<p>Safety Theorem: Prefix Agreement of Π<sub>bc</sub> implies Assured Finality</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/security-analysis.html#admonition-safety-theorem-prefix-agreement-of-Πbc-implies-assured-finality"></a></p>
+</div>
+<div>
+<p>In an execution of Crosslink 2 for which the <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> subprotocol has <strong>Prefix Agreement</strong> at confirmation <span style="white-space: nowrap">depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>,</span> that execution has <strong>Assured Finality</strong>.</p>
+<p>Proof: Suppose that we have times <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span></span> and <span style="white-space: nowrap">nodes <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span></span> (potentially the same) such that <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span></span> and <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span>.</span> Then, by the <a href="design/crosslink/./construction.html#local-fin-depth"><strong>Local fin-depth lemma</strong></a> applied to each of <span style="white-space: nowrap">node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span></span> and <span style="white-space: nowrap">node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span>,</span> there exist <span style="white-space: nowrap">times <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span></span></span></span></span> at which <span style="white-space: nowrap">node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span></span> is honest and <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span> at which <span style="white-space: nowrap">node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span></span> is honest, such that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0331em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span> and <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0892em;vertical-align:-0.3948em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3948em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.4099em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0268em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8278em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span>.</span> By <a href="design/crosslink/./construction.html#prefix-agreement"><strong>Prefix Agreement at confirmation depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">σ</span></span></span></span></span></span></strong></a>, we have <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0331em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:-0.5em;"></span></span><span class="base"><span class="strut" style="height:0.786em;vertical-align:-0.15em;"></span><span class="mrel"><span class="mrel">⪰</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.4099em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0268em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8278em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span>.</span> Wlog due to symmetry, suppose <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0331em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.4099em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0268em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8278em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span>.</span> Then <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.4099em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0268em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8278em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span></span> (by transitivity <span style="white-space: nowrap">of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.786em;vertical-align:-0.15em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>)</span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0892em;vertical-align:-0.3948em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3948em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.4099em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0268em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8278em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span> (as above),</span> so <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:-0.5em;"></span></span><span class="base"><span class="strut" style="height:0.786em;vertical-align:-0.15em;"></span><span class="mrel"><span class="mrel">⪰</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0892em;vertical-align:-0.3948em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3948em;"><span></span></span></span></span></span></span></span></span></span></span> by the <a href="design/crosslink/./construction.html#linear-prefix-lemma"><strong>Linear prefix lemma</strong></a>.</p>
+</div>
+</div>
+<p>Then we prove that <strong>Assured Finality</strong> is also implied by <a href="design/crosslink/./construction.html#final-agreement"><strong>Final Agreement</strong></a> of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</p>
+<p><span id="final-agreement"></span></p>
+<div id="admonition-definition-final-agreement" class="admonition admonish-success">
+<div class="admonition-title">
+<p>Definition: Final Agreement</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/security-analysis.html#admonition-definition-final-agreement"></a></p>
+</div>
+<div>
+<p>An execution of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has <strong>Final Agreement</strong> iff for all <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑valid blocks <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span></span> in honest view at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span></span> and <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> in honest view</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span>,</span> we have <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∗</span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:-0.5em;"></span></span><span class="base"><span class="strut" style="height:0.786em;vertical-align:-0.15em;"></span><span class="mrel"><span class="mrel">⪰</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord">∗</span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>.</span></p>
+</div>
+</div>
+<p><span id="final-agreement-implies-assured-finality"></span></p>
+<div id="admonition-safety-theorem-final-agreement-of-Πbft-implies-assured-finality" class="admonition admonish-success">
+<div class="admonition-title">
+<p>Safety Theorem: Final Agreement of Π<sub>bft</sub> implies Assured Finality</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/security-analysis.html#admonition-safety-theorem-final-agreement-of-Πbft-implies-assured-finality"></a></p>
+</div>
+<div>
+<p>In an execution of Crosslink 2 for which the <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> subprotocol has <strong>Final Agreement</strong>, that execution has <strong>Assured Finality</strong>.</p>
+<p>Proof: Suppose that we have times <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span></span> and <span style="white-space: nowrap">nodes <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span></span> (potentially the same) such that <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span></span> and <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span>.</span> Then, by the <a href="design/crosslink/./construction.html#local-fin-depth"><strong>Local fin-depth lemma</strong></a> applied to each of <span style="white-space: nowrap">node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span></span> and <span style="white-space: nowrap">node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span>,</span> there exist <span style="white-space: nowrap">times <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span></span></span></span></span> at which <span style="white-space: nowrap">node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span></span> is honest and <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span> at which <span style="white-space: nowrap">node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span></span> is honest, such that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0331em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span> and <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0892em;vertical-align:-0.3948em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3948em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.4099em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0268em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8278em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span>.</span> By <a href="design/crosslink/./construction.html#prefix-agreement"><strong>Prefix Agreement at confirmation depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">σ</span></span></span></span></span></span></strong></a>, we have <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0331em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:-0.5em;"></span></span><span class="base"><span class="strut" style="height:0.786em;vertical-align:-0.15em;"></span><span class="mrel"><span class="mrel">⪰</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.4099em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0268em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8278em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span>.</span> Wlog due to symmetry, suppose <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0331em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.4099em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0268em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8278em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span>.</span> Then <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.4099em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0268em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8278em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span></span> (by transitivity <span style="white-space: nowrap">of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.786em;vertical-align:-0.15em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>)</span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0892em;vertical-align:-0.3948em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3948em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.4099em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0268em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8278em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span> (as above),</span> so <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0522em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:-0.5em;"></span></span><span class="base"><span class="strut" style="height:0.786em;vertical-align:-0.15em;"></span><span class="mrel"><span class="mrel">⪰</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0892em;vertical-align:-0.3948em;"></span><span class="mord"><span class="mord mathsf">fin</span></span><span class="mord"><span class="mspace" style="margin-right:0.04em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3948em;"><span></span></span></span></span></span></span></span></span></span></span> by the <a href="design/crosslink/./construction.html#linear-prefix-lemma"><strong>Linear prefix lemma</strong></a>.</p>
+</div>
+</div>
+<p><span id="prefix-consistency"></span></p>
+<div id="admonition-safety-theorem-prefix-consistency-of-Πbc-implies-prefix-consistency-of-ba" class="admonition admonish-success">
+<div class="admonition-title">
+<p>Safety Theorem: Prefix Consistency of Π<sub>bc</sub> implies Prefix Consistency of ba</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/security-analysis.html#admonition-safety-theorem-prefix-consistency-of-Πbc-implies-prefix-consistency-of-ba"></a></p>
+</div>
+<div>
+<p>By the <a href="design/crosslink/./construction.html#local-ba-depth"><strong>Local ba-depth lemma</strong></a>, we have:</p>
+<blockquote>
+<p>In any execution of Crosslink 2, for any <span style="white-space: nowrap">confirmation depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8304em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span></span> and any <span style="white-space: nowrap">node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span></span> that is honest at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>,</span> there exists a <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span></span> <span style="white-space: nowrap">such that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0797em;vertical-align:-0.2861em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ba</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0836em;vertical-align:-0.3013em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em;"><span style="top:-2.3987em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.1809em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3013em;"><span></span></span></span></span></span></span></span></span></span>.</span></p>
+</blockquote>
+<p>Renaming <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span> to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span></span></span></span> in the definition of <a href="design/crosslink/./construction.html#prefix-consistency"><strong>Prefix Consistency</strong></a> gives:</p>
+<blockquote>
+<p>An execution of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has <strong>Prefix Consistency</strong> at confirmation depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span></span></span></span>, iff <span style="white-space: nowrap">for all times <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span></span> and <span style="white-space: nowrap">all nodes <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span></span> (potentially the same) such that <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span></span></span></span></span> and <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span>,</span> we have that <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0836em;vertical-align:-0.3013em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em;"><span style="top:-2.3987em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.1809em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3013em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1318em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span>.</span></p>
+</blockquote>
+<p>Since any node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> that is honest at time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span> is also honest at time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>, and by transitivity of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.786em;vertical-align:-0.15em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>, we therefore have:</p>
+<blockquote>
+<p>In any execution of Crosslink 2 that has <strong>Prefix Consistency</strong> at confirmation depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span></span></span></span>, <span style="white-space: nowrap">for all times <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span></span> and <span style="white-space: nowrap">all nodes <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span></span> (potentially the same) such that <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span></span> and <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span>,</span> we have that <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0797em;vertical-align:-0.2861em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ba</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-2.4413em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1318em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span>.</span></p>
+</blockquote>
+</div>
+</div>
+----
+<p>The <strong>Extension rule</strong> ensures that, informally, if a given node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>’s view of its bc‑best‑chain at a depth of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> blocks does not roll back, then neither does its view of the bft‑final block referenced by its bc‑best‑chain, and therefore neither does its view of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span>.</p>
+<p>This does <em>not</em> by itself imply that all nodes are seeing the “same” confirmed bc‑best‑chain (up to propagation timing), or the same <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span>. If the network is partitioned <em>and</em> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> is subverted, it could be that the nodes on each side of the partition follow a different fork, and the adversary arranges for each node’s view of the last final bft‑block to be consistent with the fork it is on. It can potentially do this if it has more than one third of validators, because if the validators are partitioned in the same way as other nodes, it can vote with an additional one third of them on each side of the fork.</p>
+<p>This is, if you think about it, unavoidable. <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> doesn’t include the mechanisms needed to maintain finality under partition; it takes a different position on the CAP trilemma. In order to maintain finality under partition, we need <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> not to be subverted (and to actually work!)</p>
+<p>So what is the strongest security property we can realistically get? It is stronger than what Snap‑and‑Chat provides. Snap‑and‑Chat is unsafe <em>even without a partition</em> if <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> is subverted. Ideally we would have a protocol with safety that is <em>only</em> limited by attacks “like” the unavoidable attack described above — which also applies to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> used on its own.</p>
+<h3 id="proof-of-safety-for-logfin"><a class="header" href="#proof-of-safety-for-logfin">Proof of safety for LOG<sub>fin</sub></a></h3>
+<p>In order to capture the intuition that it is hard in practice to cause a consistent partition of the kind described in the previous section, we will need to assume that the <a href="design/crosslink/./construction.html#prefix-agreement"><strong>Prefix Agreement</strong></a> safety property holds for the relevant executions of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>. The structural and consensus modifications to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> relative to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> seem unlikely to have any significant effect on this property, given that we proved above that they do not affect liveness. ==TODO: that is a handwave; we should be able to do better, as we do for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> below.== So, to the extent that it is reasonable to assume that <a href="design/crosslink/./construction.html#prefix-agreement"><strong>Prefix Agreement</strong></a> holds for executions of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> under some conditions, it should also be reasonable to assume it holds for executions of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> under the same conditions.</p>
+<p>Recall that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1em;vertical-align:-0.35em;"></span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">context_bft</span></span><span class="mclose">)</span></span></span></span>.</p>
+<p><span id="prefix-lemma"></span></p>
+<div id="admonition-prefix-lemma" class="admonition admonish-success">
+<div class="admonition-title">
+<p>Prefix Lemma</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/security-analysis.html#admonition-prefix-lemma"></a></p>
+</div>
+<div>
+<p>If <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> are bc‑valid blocks with <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>, then <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>.</p>
+<p>Proof: Using the <strong>Extension rule</strong>, by induction on the distance between <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</p>
+</div>
+</div>
+<p>Using the <strong>Prefix Lemma</strong> once for each direction, we can transfer the <a href="design/crosslink/./construction.html#prefix-agreement"><strong>Prefix Agreement</strong></a> property to the referenced bft‑blocks:</p>
+<p><span id="prefix-agreement-lemma"></span></p>
+<div id="admonition-prefix-agreement-lemma" class="admonition admonish-success">
+<div class="admonition-title">
+<p>Prefix Agreement Lemma</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/security-analysis.html#admonition-prefix-agreement-lemma"></a></p>
+</div>
+<div>
+<p>In an execution of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> that has <a href="design/crosslink/./construction.html#prefix-agreement"><strong>Prefix Agreement</strong></a> at confirmation <span style="white-space: nowrap">depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>,</span> <span style="white-space: nowrap">for all times <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span></span> and <span style="white-space: nowrap">all nodes <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span></span> (potentially the same) such that <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span></span> and <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span>,</span> we have <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.161em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:-0.5em;"></span></span><span class="base"><span class="strut" style="height:0.786em;vertical-align:-0.15em;"></span><span class="mrel"><span class="mrel">⪰</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1331em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>.</span></p>
+</div>
+</div>
+<p>Let <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">chain-txns</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">ch</span></span><span class="mclose">)</span></span></span></span> be the sequence of transactions in the given <span style="white-space: nowrap">chain <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathsf">ch</span></span></span></span></span>,</span> starting from genesis.</p>
+<p>Recall that <span class="katex-display fleqn"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:9.722em;vertical-align:-4.611em;"></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:5.111em;"><span style="top:-7.271em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord textsf">san-ctx</span></span></span></span><span style="top:-6.071em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord textsf">san-ctx</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mclose">)</span></span></span><span style="top:-4.871em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord textsf">fin</span></span></span></span><span style="top:-3.671em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">fin</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span></span><span style="top:-2.471em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord textsf">fin-ctx</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span></span><span style="top:-1.271em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord textsf">ba-ctx</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">μ</span><span class="mclose">)</span></span></span><span style="top:-0.0331em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span><span style="top:1.2279em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">ba</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">μ</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:4.611em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:5.111em;"><span style="top:-7.271em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mrel"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.1293em;"><span style="top:-2.6293em;"><span class="pstrut" style="height:2.5em;"></span><span class="mord"><span class="mord sizing reset-size6 size1">⦂</span></span></span></span></span></span></span></span></span><span style="top:-6.071em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mrel">:=</span></span></span><span style="top:-4.871em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mrel"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.1293em;"><span style="top:-2.6293em;"><span class="pstrut" style="height:2.5em;"></span><span class="mord"><span class="mord sizing reset-size6 size1">⦂</span></span></span></span></span></span></span></span></span><span style="top:-3.671em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mrel">:=</span></span></span><span style="top:-2.471em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mrel">:=</span></span></span><span style="top:-1.271em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mrel">:=</span></span></span><span style="top:-0.0331em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mrel">:=</span></span></span><span style="top:1.2279em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mrel">:=</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:4.611em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:5.111em;"><span style="top:-7.271em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">[</span><span class="mord text"><span class="mord textsf">bc-chain</span></span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord text"><span class="mord textsf">bc-context</span></span></span></span><span style="top:-6.071em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">sanitize</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">concat</span></span><span class="mopen">([</span><span class="mord text"><span class="mord textsf">chain-txns</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mclose">)</span><span class="mord text"><span class="mord"> for </span></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord text"><span class="mord"> in </span></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mclose">]))</span></span></span><span style="top:-4.871em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord textsf">bc-block</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">[</span><span class="mord text"><span class="mord textsf">bc-chain</span></span><span class="mclose">]</span></span></span><span style="top:-3.671em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">[</span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mord text"><span class="mord"> for </span></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)]</span></span></span><span style="top:-2.471em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord textsf">san-ctx</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathrm">fin</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">))</span></span></span><span style="top:-1.271em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord textsf">san-ctx</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathrm">fin</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">∣∣</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em;"><span style="top:-2.3987em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.1809em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3013em;"><span></span></span></span></span></span></span><span class="mclose">])</span></span></span><span style="top:-0.0331em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord textsf">context-txns</span></span><span class="mopen">(</span><span class="mord text"><span class="mord textsf">fin-ctx</span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose">))</span></span></span><span style="top:1.2279em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord textsf">context-txns</span></span><span class="mopen">(</span><span class="mord text"><span class="mord textsf">ba-ctx</span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">μ</span><span class="mclose">))</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:4.611em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span></span></span></span></span></p>
+<p>Because <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathrm">fin</span></span></span></span></span> takes the form <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathrm">fin</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span><span class="mord text"><span class="mord"> for </span></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)]</span></span></span></span>,</span> we have that <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟹</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathrm">fin</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathrm">fin</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>.</span> (This would be true for any well‑typed <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span></span></span></span>.)</span></p>
+<p>By this observation and the <strong>Prefix Agreement Lemma</strong>, we also have that, in an execution of Crosslink 2 where <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has the <a href="design/crosslink/./construction.html#prefix-agreement"><strong>Prefix Agreement</strong></a> safety property at confirmation depth <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>,</span> <span style="white-space: nowrap">for all times <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span></span> and <span style="white-space: nowrap">all nodes <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span></span> (potentially the same) such that <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span></span> and <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span>,</span> <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1279em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathrm">fin</span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:-0.5em;"></span></span><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">⪰</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1331em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord mathrm">fin</span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>.</span></p>
+<p>Because <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6794em;"></span><span class="mord"><span class="mord mathsf">sanitize</span></span></span></span></span> only considers previous state, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5714em;"></span><span class="mord text"><span class="mord textsf">context-txns</span></span></span></span></span> ∘ <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5714em;"></span><span class="mord text"><span class="mord textsf">san-ctx</span></span></span></span></span> must be a prefix-preserving map; that is, <span style="white-space: nowrap">if <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> then <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">context-txns</span></span><span class="mopen">(</span><span class="mord text"><span class="mord textsf">san-ctx</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">))</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">context-txns</span></span><span class="mopen">(</span><span class="mord text"><span class="mord textsf">san-ctx</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">))</span></span></span></span>.</span> Therefore:</p>
+<div id="admonition-theorem-logfin-safety-from-prefix-agreement-of-Πbc" class="admonition admonish-success">
+<div class="admonition-title">
+<p>Theorem: LOG<sub>fin</sub> Safety (from Prefix Agreement of Π<sub>bc</sub>)</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/security-analysis.html#admonition-theorem-logfin-safety-from-prefix-agreement-of-Πbc"></a></p>
+</div>
+<div>
+<p>In an execution of Crosslink 2 where <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has <a href="design/crosslink/./construction.html#prefix-agreement"><strong>Prefix Agreement</strong></a> at confirmation depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>, <span style="white-space: nowrap">for all times <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span></span> and <span style="white-space: nowrap">all nodes <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span></span> (potentially the same) such that <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span></span> and <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span>,</span> <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:-0.5em;"></span></span><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">⪰</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1318em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span></span></span></span>.</span></p>
+</div>
+</div>
+<p>Notice that this does not depend on any safety property of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>, and is an elementary proof. ([<a href="https://eprint.iacr.org/2020/1091.pdf">NTT2020</a>, Theorem 2] is a much more complicated proof that takes nearly 3 pages, not counting the reliance on results from <a href="https://eprint.iacr.org/2016/918.pdf">[PS2017]</a>.)</p>
+<p><em>In addition,</em> just as in Snap‑and‑Chat, safety of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> can be inferred from safety of <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>,</span> which follows from safety of <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>.</span> We prove this based on the <a href="design/crosslink/./construction.html#final-agreement"><strong>Final Agreement</strong></a> property for executions of <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>:</span></p>
+<div id="admonition-definition-final-agreement-1" class="admonition admonish-success">
+<div class="admonition-title">
+<p>Definition: Final Agreement</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/security-analysis.html#admonition-definition-final-agreement-1"></a></p>
+</div>
+<div>
+<p>An execution of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> has the <a href="design/crosslink/./construction.html#final-agreement"><strong>Final Agreement</strong></a> safety property iff for all origbft‑valid <span style="white-space: nowrap">blocks <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span></span> <a href="design/crosslink/security-analysis.html#in-honest-view">in honest view</a> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> in honest view at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span>,</span> we have <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">origbft-last-final</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:-0.5em;"></span></span><span class="base"><span class="strut" style="height:0.9221em;vertical-align:-0.2861em;"></span><span class="mrel"><span class="mrel">⪰</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">origbft-last-final</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span></span></span></span>.</span></p>
+</div>
+</div>
+<p>The changes in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> relative to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> only add structural components and tighten bft‑block‑validity and bft‑proposal‑validity rules. So for any legal execution of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> there is a corresponding legal execution of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>, with the structural additions erased and with the same nodes honest at any given time. A safety property, by definition, only asserts that executions not satisfying the property do not occur. Safety properties of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> necessarily do not refer to the added structural components in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>. Therefore, for any safety property of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>, including <a href="design/crosslink/./construction.html#final-agreement"><strong>Final Agreement</strong></a>, the corresponding safety property holds for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</p>
+<p>By the definition of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathrm">fin</span></span></span></span></span> as above, in an execution of Crosslink 2 where <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has <a href="design/crosslink/./construction.html#final-agreement"><strong>Final Agreement</strong></a>, <span style="white-space: nowrap">for all times <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span></span> and <span style="white-space: nowrap">all nodes <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span></span> (potentially the same) such that <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span></span> and <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span>,</span> <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1279em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathrm">fin</span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:-0.5em;"></span></span><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">⪰</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1331em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord mathrm">fin</span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span></span></span></span>.</span> Therefore, by an argument similar to the one above using the fact that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5714em;"></span><span class="mord text"><span class="mord textsf">context-txns</span></span></span></span></span> ∘ <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5714em;"></span><span class="mord text"><span class="mord textsf">san-ctx</span></span></span></span></span> is a prefix-preserving map:</p>
+<div id="admonition-theorem-logfin-safety-from-final-agreement-of-Πbft-or-Πorigbft" class="admonition admonish-success">
+<div class="admonition-title">
+<p>Theorem: LOG<sub>fin</sub> Safety (from Final Agreement of Π<sub>bft</sub> or Π<sub>origbft</sub>)</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/security-analysis.html#admonition-theorem-logfin-safety-from-final-agreement-of-Πbft-or-Πorigbft"></a></p>
+</div>
+<div>
+<p>In an execution of Crosslink 2 where <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has <a href="design/crosslink/./construction.html#final-agreement"><strong>Final Agreement</strong></a>, <span style="white-space: nowrap">for all times <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span></span> and <span style="white-space: nowrap">all nodes <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span></span> (potentially the same) such that <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span></span> and <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span>,</span> <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:-0.5em;"></span></span><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">⪰</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1318em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span></span></span></span>.</span></p>
+</div>
+</div>
+<h3 id="proof-of-safety-for-logba"><a class="header" href="#proof-of-safety-for-logba">Proof of safety for LOG<sub>ba</sub></a></h3>
+<p>From the two <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> Safety theorems and the <a href="design/crosslink/./construction.html#admonition-theorem-ledger-prefix-property"><strong>Ledger prefix property</strong></a>, we immediately have:</p>
+<div id="admonition-theorem-logba-does-not-roll-back-past-the-agreed-logfin" class="admonition admonish-success">
+<div class="admonition-title">
+<p>Theorem: LOG<sub>ba</sub> does not roll back past the agreed LOG<sub>fin</sub></p>
+<p><a class="admonition-anchor-link" href="design/crosslink/security-analysis.html#admonition-theorem-logba-does-not-roll-back-past-the-agreed-logfin"></a></p>
+</div>
+<div>
+<p>Let <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">μ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> be an arbitrary choice of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">ba</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> confirmation depth for each node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>. Consider an execution of Crosslink 2 where either <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has <a href="design/crosslink/./construction.html#prefix-agreement"><strong>Prefix Agreement</strong></a> at confirmation depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> or <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has <a href="design/crosslink/./construction.html#final-agreement"><strong>Final Agreement</strong></a>.</p>
+<p>In such an execution, <span style="white-space: nowrap">for all times <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span></span> and <span style="white-space: nowrap">all nodes <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span></span> (potentially the same) such that <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span></span> and <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span>,</span> either <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1318em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1931em;vertical-align:-0.4443em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">ba</span></span><span class="mpunct mtight">,</span><span class="mord mtight"><span class="mord mathnormal mtight">μ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2819em;"><span></span></span></span></span></span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4443em;"><span></span></span></span></span></span></span></span></span></span> or <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1318em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">ba</span></span><span class="mpunct mtight">,</span><span class="mord mtight"><span class="mord mathnormal mtight">μ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span>.</p>
+<p><strong>Corollary:</strong> Under the same conditions, if wlog <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1318em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span>, then <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.3222em;vertical-align:-0.4443em;"></span><span class="mopen">{</span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">ba</span></span><span class="mpunct mtight">,</span><span class="mord mtight"><span class="mord mathnormal mtight">μ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">ba</span></span><span class="mpunct mtight">,</span><span class="mord mtight"><span class="mord mathnormal mtight">μ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2819em;"><span></span></span></span></span></span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4443em;"><span></span></span></span></span></span></span><span class="mclose">}</span></span></span></span>.</p>
+</div>
+</div>
+<p>The above property is not as strong as we would like for practical uses of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">ba</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>, because it does not say anything about rollbacks up to the finalization point, and such rollbacks may be of unbounded length. (Loosely speaking, the number of non‑Stalled Mode bc‑blocks after the consensus finalization point is bounded by <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span>,</span> but we have also not proven that so far.)</p>
+<p>As documented in the <a href="design/crosslink/./construction.html#model-for-bft-protocols-%CE%A0origbftbft">Model for BFT protocols</a> section of <a href="design/crosslink/./construction.html">The Crosslink 2 Construction</a>):</p>
+<blockquote>
+<p>For each epoch, there is a fixed number of voting units distributed between the players, which they use to vote for a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑proposal.</span> We say that a voting unit has been cast for a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑proposal <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span></span> at a given time in a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑execution,</span> <span style="white-space: nowrap">if and only if</span> <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span> is <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑proposal‑valid</span> and a ballot <span style="white-space: nowrap">for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span></span> authenticated by the holder of the voting unit exists at that time.</p>
+<p>Using knowledge of ballots cast for a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑proposal <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span></span> that collectively satisfy a notarization rule at a given time in a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑execution,</span> and only with such knowledge, it is possible to obtain a valid <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>bft‑notarization‑proof <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9386em;vertical-align:-0.2441em;"></span><span class="mord"><span class="mord"><span class="mord mathsf" style="margin-right:0.06944em;">proof</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2342em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">P</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span></span></span></span>.</span> The notarization rule must require at least a two‑thirds absolute supermajority of voting units <span style="white-space: nowrap">in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span>’s epoch</span> to have been cast <span style="white-space: nowrap">for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span>.</span> It may also require other conditions.</p>
+<p>A voting unit is cast non‑honestly for an epoch’s proposal iff:</p>
+<ul>
+<li>it is cast other than by the holder of the unit (due to key compromise or any flaw in the voting protocol, for example); or</li>
+<li>it is double‑cast (i.e. there are two ballots casting it for distinct proposals); or</li>
+<li>the holder of the unit following the conditions for honest voting <span style="white-space: nowrap">in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>,</span> according to its view, should not have cast that vote.</li>
+</ul>
+</blockquote>
+<p><span id="one-third-bound"></span></p>
+<div id="admonition-definition-onethird-bound-on-nonhonest-voting" class="admonition admonish-success">
+<div class="admonition-title">
+<p>Definition: One‑third bound on non‑honest voting</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/security-analysis.html#admonition-definition-onethird-bound-on-nonhonest-voting"></a></p>
+</div>
+<div>
+<p>An execution of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has the <strong>one‑third bound on non‑honest voting</strong> property iff for every epoch, <em>strictly</em> fewer than one third of the total voting units for that epoch are ever cast non‑honestly.</p>
+</div>
+</div>
+<p><span id="theorem-on-bft-valid-blocks"></span></p>
+<div id="admonition-theorem-on-bftvalid-blocks-for-a-given-epoch-in-honest-view" class="admonition admonish-success">
+<div class="admonition-title">
+<p>Theorem: On bft‑valid blocks for a given epoch in honest view</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/security-analysis.html#admonition-theorem-on-bftvalid-blocks-for-a-given-epoch-in-honest-view"></a></p>
+</div>
+<div>
+<p>By a well known argument often used to prove safety of BFT protocols, in an execution of Crosslink 2 where <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has the <strong>one‑third bound on non‑honest voting</strong> property (and assuming soundness of notarization proofs), any bft‑valid block for a given epoch <a href="design/crosslink/security-analysis.html#in-honest-view">in honest view</a> must commit to the same proposal.</p>
+<p>Proof (adapted from [<a href="https://eprint.iacr.org/2020/088.pdf">CS2020</a>, Lemma 1]): Suppose there are two bft‑proposals <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span>, both for <span style="white-space: nowrap">epoch <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">e</span></span></span></span>,</span> such that <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span> is</span> committed to by some bft‑block‑valid <span style="white-space: nowrap">block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span>,</span> and <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> is</span> committed to by some bft‑block‑valid <span style="white-space: nowrap">block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span>.</span> This implies that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> have valid notarization proofs. Let the number of voting units for <span style="white-space: nowrap">epoch <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">e</span></span></span></span></span> <span style="white-space: nowrap">be <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</span> Assuming soundness of the notarization proofs, it must be that at least <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">/3</span></span></span></span> voting units for <span style="white-space: nowrap">epoch <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">e</span></span></span></span>,</span> denoted as the <span style="white-space: nowrap">set <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span></span></span></span>,</span> were cast <span style="white-space: nowrap">for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span>,</span> and at least <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">/3</span></span></span></span> voting units for <span style="white-space: nowrap">epoch <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">e</span></span></span></span>,</span> denoted as the <span style="white-space: nowrap">set <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span>,</span> were cast <span style="white-space: nowrap">for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span>.</span> Since there are <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> voting units for <span style="white-space: nowrap">epoch <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">e</span></span></span></span>,</span> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∩</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> must have size at least <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">/3</span></span></span></span>.</span> In an execution of Crosslink 2 where <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has the <strong>one‑third bound on non‑honest voting</strong> property, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∩</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> must therefore include at least one voting unit that was cast honestly. Since a voting unit for epoch <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">e</span></span></span></span> that is cast honestly is not double-cast, it must be <span style="white-space: nowrap">that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span>.</span></p>
+</div>
+</div>
+<div id="admonition-info-2" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Info</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/security-analysis.html#admonition-info-2"></a></p>
+</div>
+<div>
+<p>In the case of a network partition, votes may not be seen on both/all sides of the partition. Therefore, it is not necessarily the case that honest nodes can directly see double‑voting. The above argument does not depend on being able to do so.</p>
+</div>
+</div>
+<p>Therefore, in an execution of Crosslink 2 for which <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has the <strong>one‑third bound on non‑honest voting</strong> property, for each <span style="white-space: nowrap">epoch <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">e</span></span></span></span></span> there will be at most one bft‑proposal‑valid <span style="white-space: nowrap">proposal <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>,</span> and at least one third of honestly cast voting units must have been cast for it. Let <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.07382em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0738em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> be the (necessarily nonempty) set of nodes that cast these honest votes; then, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2558em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9727em;"><span style="top:-2.4231em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.2421em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">e</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2819em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2769em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span> for <span style="white-space: nowrap">all <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6986em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.07382em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0738em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span> at the <span style="white-space: nowrap">times <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9012em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">e</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span></span> of their votes in epoch <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">e</span></span></span></span>. <span style="white-space: nowrap">(For simplicity,</span> we assume that for each honest <span style="white-space: nowrap">node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span></span> there is only one <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9012em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">e</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span></span> at which it obtains a successful check for the voting condition in <span style="white-space: nowrap">epoch <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">e</span></span></span></span>,</span> which it uses for any votes that it casts in that epoch.)</p>
+<p>Let <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> be any bft‑block for <span style="white-space: nowrap">epoch <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">e</span></span></span></span></span> such that <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mclose">)</span></span></span></span>,</span> where <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span> is some bft‑block‑valid block. <span style="white-space: nowrap">Since <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span>,</span> <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> is bft‑block‑valid.</span> So by the argument above, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> commits to the only bft‑proposal‑valid <span style="white-space: nowrap">proposal <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span> for <span style="white-space: nowrap">epoch <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">e</span></span></span></span>,</span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> was voted for in that epoch by a nonempty subset of honest <span style="white-space: nowrap">nodes <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.07382em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0738em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</span></p>
+<p>Let <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> be any bc‑valid block. We have by definition: <span class="katex-display fleqn"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">fin</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mspace" style="margin-right:-0.1667em;"></span><span class="mspace" style="margin-right:-0.1667em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:-0.1667em;"></span><span class="mspace" style="margin-right:-0.1667em;"></span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mspace" style="margin-right:-0.1667em;"></span><span class="mspace" style="margin-right:-0.1667em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:-0.1667em;"></span><span class="mspace" style="margin-right:-0.1667em;"></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">[</span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mord text"><span class="mord"> for </span></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)]</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">[</span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mord text"><span class="mord"> for </span></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">context_bft</span></span><span class="mclose">)]</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span></span></span></span></span> So, taking <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">context_bft</span></span></span></span></span>, each <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span> for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> of epoch <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">e</span></span></span></span> in the result of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathrm">fin</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span></span></span> satisfies <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2558em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9727em;"><span style="top:-2.4231em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.2421em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">e</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2819em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2769em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span> for all <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> in some nonempty honest set of nodes <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.07382em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0738em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</p>
+<p>For an execution of Crosslink 2 in which <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has the <a href="design/crosslink/./construction.html#prefix-consistency"><strong>Prefix Consistency</strong></a> property at confirmation depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>, for every epoch <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">e</span></span></span></span>, for every such <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mopen">(</span><span class="mord mathnormal">i</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">e</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>, for every node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> that is honest at any time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal">u</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9012em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">e</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>, we have <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2558em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9727em;"><span style="top:-2.4231em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.2421em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">e</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2819em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2769em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1318em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span>. Let <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7651em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mop">min</span><span class="mopen">{</span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">e</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6986em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.07382em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0738em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">}</span></span></span></span>. Then, by transitivity of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.786em;vertical-align:-0.15em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>:</p>
+<div id="admonition-theorem-on-snapshots-in-logfin" class="admonition admonish-success">
+<div class="admonition-title">
+<p>Theorem: On snapshots in LOG<sub>fin</sub></p>
+<p><a class="admonition-anchor-link" href="design/crosslink/security-analysis.html#admonition-theorem-on-snapshots-in-logfin"></a></p>
+</div>
+<div>
+<p>In an execution of Crosslink 2 where <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has the <strong>one‑third bound on non‑honest voting</strong> property and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has the <a href="design/crosslink/./construction.html#prefix-consistency"><strong>Prefix Consistency</strong></a> property at confirmation <span style="white-space: nowrap">depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>,</span> every bc‑chain <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span> in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1279em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathrm">fin</span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> (and therefore every snapshot that contributes to <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span>)</span> is, <span style="white-space: nowrap">at any time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal">u</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7651em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span>,</span> in the bc‑best‑chain of <span style="white-space: nowrap">every node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span></span> that is honest <span style="white-space: nowrap">at time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span></span> (where <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> commits to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> at epoch <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">e</span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7651em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> is the time of the first honest vote <span style="white-space: nowrap">for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>).</span></p>
+</div>
+</div>
+<p>A similar (weaker) statement holds if we replace <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal">u</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7651em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> with <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal">u</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>, since the time of the first honest vote for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span> necessarily precedes the time at which the signed <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord mathsf" style="margin-right:0.06944em;">proof</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2342em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">P</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> is submitted as a bft‑block, which necessarily <span style="white-space: nowrap">precedes <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>.</span> (Whether or not the notarization proof depends on the <em>first</em> honest vote for <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span>’s proposal <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>,</span> it must depend on some honest vote for that proposal that was not made earlier <span style="white-space: nowrap">than <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7651em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.)</span></p>
+<p>Furthermore, we have <span class="katex-display fleqn"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.461em;vertical-align:-0.9805em;"></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4805em;"><span style="top:-3.6405em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord textsf">ba-ctx</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">μ</span><span class="mclose">)</span></span></span><span style="top:-2.4026em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">ba</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">μ</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9805em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4805em;"><span style="top:-3.6405em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mspace" style="margin-right:-0.1667em;"></span><span class="mspace" style="margin-right:-0.1667em;"></span><span class="mspace" style="margin-right:-0.1667em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:-0.1667em;"></span><span class="mspace" style="margin-right:-0.1667em;"></span><span class="mspace" style="margin-right:-0.1667em;"></span></span></span><span style="top:-2.4026em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mspace" style="margin-right:-0.1667em;"></span><span class="mspace" style="margin-right:-0.1667em;"></span><span class="mspace" style="margin-right:-0.1667em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:-0.1667em;"></span><span class="mspace" style="margin-right:-0.1667em;"></span><span class="mspace" style="margin-right:-0.1667em;"></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9805em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4805em;"><span style="top:-3.6405em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord textsf">san-ctx</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathrm">fin</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">∣∣</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em;"><span style="top:-2.3987em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.1809em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3013em;"><span></span></span></span></span></span></span><span class="mclose">])</span></span></span><span style="top:-2.4026em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord textsf">context-txns</span></span><span class="mopen">(</span><span class="mord text"><span class="mord textsf">ba-ctx</span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">μ</span><span class="mclose">))</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9805em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span></span></span></span></span></p>
+<p>So in an execution of Crosslink 2 where <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has the <a href="design/crosslink/./construction.html#prefix-consistency"><strong>Prefix Consistency</strong></a> property at confirmation <span style="white-space: nowrap">depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span></span></span></span>,</span> <span style="white-space: nowrap">if node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span></span> is honest <span style="white-space: nowrap">at time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span></span> then <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0836em;vertical-align:-0.3013em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em;"><span style="top:-2.3987em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.1809em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3013em;"><span></span></span></span></span></span></span></span></span></span> is also, <span style="white-space: nowrap">at any time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal">u</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>,</span> in the bc‑best‑chain of <span style="white-space: nowrap">every node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span></span> that is honest <span style="white-space: nowrap">at time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span>.</span></p>
+<p>If an execution of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has the <a href="design/crosslink/./construction.html#prefix-consistency"><strong>Prefix Consistency</strong></a> property at confirmation <span style="white-space: nowrap">depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8304em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>,</span> then it necessarily also has it at confirmation <span style="white-space: nowrap">depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>.</span> Therefore we have:</p>
+<div id="admonition-theorem-on-snapshots-in-logba" class="admonition admonish-success">
+<div class="admonition-title">
+<p>Theorem: On snapshots in LOG<sub>ba</sub></p>
+<p><a class="admonition-anchor-link" href="design/crosslink/security-analysis.html#admonition-theorem-on-snapshots-in-logba"></a></p>
+</div>
+<div>
+<p>In an execution of Crosslink 2 where <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has the <strong>one‑third bound on non‑honest voting</strong> property and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has the <a href="design/crosslink/./construction.html#prefix-consistency"><strong>Prefix Consistency</strong></a> property at confirmation <span style="white-space: nowrap">depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8304em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>,</span> every bc‑chain snapshot in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1792em;vertical-align:-0.3013em;"></span><span class="mord"><span class="mord mathrm">fin</span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">∣∣</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mopen">[</span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em;"><span style="top:-2.3987em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.1809em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3013em;"><span></span></span></span></span></span></span><span class="mclose">]</span></span></span></span> (and therefore every snapshot that contributes to <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">ba</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">μ</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span>)</span> is, <span style="white-space: nowrap">at any time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal">u</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>,</span> in the bc‑best‑chain of <span style="white-space: nowrap">every node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span></span> that is honest <span style="white-space: nowrap">at time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span>.</span></p>
+</div>
+</div>
+<p>Sketch: we also need the sequence of snapshots output from fin to only be extended in the view of any node. In that case we can infer that the node does not observe a rollback in LOG_ba.</p>
+<p>Recall that in the proof of safety for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>, we showed that in an execution of Crosslink 2 where <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> (or <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>) has <a href="design/crosslink/./construction.html#final-agreement"><strong>Final Agreement</strong></a>, <span style="white-space: nowrap">for all times <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span></span> and <span style="white-space: nowrap">all nodes <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span></span> (potentially the same) such that <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span></span> and <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> is honest</span> at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span>,</span> <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1279em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathrm">fin</span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:-0.5em;"></span></span><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">⪰</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1331em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord mathrm">fin</span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7487em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span></span></span></span>.</span></p>
+<p>What we want to show is that, under some conditions on executions, ...</p>
+<h2 id="disadvantages-of-crosslink"><a class="header" href="#disadvantages-of-crosslink">Disadvantages of Crosslink</a></h2>
+<h3 id="more-invasive-changes"><a class="header" href="#more-invasive-changes">More invasive changes</a></h3>
+<p>Unlike Snap‑and‑Chat, Crosslink 2 requires structural and consensus rule changes to both <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>. On the other hand, several of those changes are arguably necessary to fix a show‑stopping bug in Snap‑and‑Chat (not being able to spend some finalized outputs).</p>
+<h3 id="finalization-latency"><a class="header" href="#finalization-latency">Finalization latency</a></h3>
+<p>For a given choice of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>, the finalization latency is higher. The snapshot of the BFT chain used to obtain <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">μ</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span> is obtained from the block at depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span></span></span></span> on node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>’s best <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> chain, which will on average lead to a finalized view that is about <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> blocks back (in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>), rather than \sigma_{\sac}} blocks in Snap‑and‑Chat. This is essentially the cost of ensuring that safety is given by the stronger of the safety of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> (at <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span></span></span></span> confirmations) and the safety of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</p>
+<p>On the other hand, the relative increase in expected finalization latency is only \frac{\mu + 1 + \sigma}{\sigma_{\sac}}}, i.e. at most slightly more than a factor of 2 for the case \mu = \sigma = \sigma_{\sac}}.</p>
+<h3 id="more-involved-liveness-argument"><a class="header" href="#more-involved-liveness-argument">More involved liveness argument</a></h3>
+<p>See the Liveness section above.</p>
+<h2 id="every-rule-in-crosslink-2-is-individually-necessary"><a class="header" href="#every-rule-in-crosslink-2-is-individually-necessary">Every rule in Crosslink 2 is individually necessary</a></h2>
+<div id="admonition-warning" class="admonition admonish-warning">
+<div class="admonition-title">
+<p>Warning</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/security-analysis.html#admonition-warning"></a></p>
+</div>
+<div>
+<p>In order to show that Crosslink 2 is at a <em>local</em> optimum in the security/complexity trade‑off space, for each rule we show attacks on safety and/or liveness that could be performed if that rule were omitted or simplified.</p>
+</div>
+</div>
+<p>Edit: some rules, e.g. the <a href="design/crosslink/./construction.html#linearity-rule"><strong>Linearity rule</strong></a>, only contribute heuristically to security in the analysis so far.</p>
+<div style="break-before: page; page-break-before: always;"></div><link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.12.0/dist/katex.min.css" integrity="sha384-AfEj0r4/OFrOo5t7NnNe46zW/tFgW6x/bCJG8FqQCEo3+Aro6EYUG4+cU+KJWu/X" crossorigin="anonymous">
+<h1 id="questions-about-crosslink"><a class="header" href="#questions-about-crosslink">Questions about Crosslink</a></h1>
+<h3 id="why-dont-we-have-a-bcblockvalidity-rule-snapshotlfh-bc-h-"><a class="header" href="#why-dont-we-have-a-bcblockvalidity-rule-snapshotlfh-bc-h-">Why don’t we have a bc‑block‑validity rule snapshot(LF(H)) ⪯<sub>bc</sub> H ?</a></h3>
+<p>Rationale: Can we outright prevent rollbacks &gt; <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> from ever appearing in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">Ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span> ?</p>
+<div id="admonition-info" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Info</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/questions.html#admonition-info"></a></p>
+</div>
+<div>
+<p>This document analyzes the effect of this rule <em>on its own</em>. For the effect in combination with an additional <a href="design/crosslink/questions.html#linearity-rule"><strong>Linearity rule</strong></a>, see the <a href="design/crosslink/potential-changes.html#linearity-and-last-final-snapshot-rules">Linearity and Last Final Snapshot rules</a> section of <a href="design/crosslink/potential-changes.html">Potential changes for Crosslink 2</a>.</p>
+</div>
+</div>
+<p>Daira-Emma: In a variant of Crosslink with this rule, an adversary’s strategy would be to keep the <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">context_bft</span></span></span></span></span> fields in its blocks as <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span> such that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mclose">)</span></span></span></span> when the attack starts, and then fork from the bc‑best‑chain that extends <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span>. If its private chain falls behind the public bc‑best‑chain, it resets, just like in a conventional private mining attack.</p>
+<p>Note that the proposed rule does <em>not</em> prevent the adversary’s private chain from just staying at the same <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">context_bft</span></span></span></span></span> block. The reason is that Crosslink does not change the fork-choice rule of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>. That is, even if the adversary’s chain has a <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">context_bft</span></span></span></span></span> that is far behind the current bft‑block, it is still allowed to become the bc‑best‑chain.</p>
+<p>(Eventually the adversary’s chain using this strategy will hit the finality gap bound of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span> blocks. But that must be significantly greater than <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>, to avoid availability problems. So it does not prevent the adversary from performing a rollback longer than <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> blocks before they hit the bound. Also, going into Stalled Mode for new blocks does not prevent the attacker’s chain from having included harmful transactions before that point.)</p>
+<p>It is possible to change the fork-choice rule, for example so that the bc‑best‑chain for a node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> is required to extend <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span> where <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> is the last final block for any bft-chain in node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>’s view.</p>
+<p>This would break the current safety and liveness arguments for Crosslink. But for the sake of argument, suppose we did it anyway.</p>
+<p>The adversary’s strategy would change slightly: it resets if <em>either</em> its private chain falls behind the public bc‑best‑chain, or its private chain is invalidated because it forks before <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span> for some new last final block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> of a bft-chain. During the attack, it also attempts to impede progress of the BFT protocol as far as possible.</p>
+<p>In that case, the proposed rule still does not preclude a rollback of more than <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> blocks, for several reasons:</p>
+<ul>
+<li>
+<p>In general we can’t say anything about how many bc‑blocks are mined in any given interval, so it could be the case that more than <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> blocks are mined on both the honest chain and the adversary’s chain before it would be realistically possible to go through even a single round of the BFT protocol.</p>
+</li>
+<li>
+<p>Nor can we say anything about how quickly those blocks are finalized, unless we enforce it, which we don’t. (In Crosslink we do enforce a finalization gap bound <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span>, but as explained above <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span> must be significantly greater than <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>, so that doesn’t really help.)</p>
+<ul>
+<li>In particular, the adversary could be suppressing publication of final bft‑blocks, or attacking the <em>liveness</em> of the BFT protocol in other ways. An attack against BFT liveness is potentially easier than an attack against BFT safety, and it would be difficult to characterize exactly how much this rule gains you in terms of security given that (at best) it’s dependent on that.</li>
+</ul>
+</li>
+<li>
+<p><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.1em;"></span><span class="mclose">)</span></span></span></span> will typically be at least <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> blocks back from <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span>.
+The argument for that goes:</p>
+<blockquote>
+<p>None of the block hashes in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">headers_bc</span></span></span></span></span> can point to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> because that would be a hash cycle. In a typical case where no block withholding and no <em>other</em> rollback (not caused by the adversary) occurs on the honestly mined chain, the proposer of the last final block before a context bft‑block that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> can point to will have, at the latest, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0972em;vertical-align:-0.2831em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span> as <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">tip</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span>. Under these conditions, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.1em;"></span><span class="mclose">)</span></span></span></span> will point, at the latest, to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> blocks before <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0972em;vertical-align:-0.2831em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span>, i.e. <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> blocks before <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span>.</p>
+</blockquote>
+<p>This means that by the time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.1em;"></span><span class="mclose">)</span></span></span></span> could catch up to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span>, on average <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> block times will have occurred. So, roughly speaking, the rule that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.1em;"></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> does not usefully constrain the adversary until <span style="white-space: nowrap">after <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> block times.</span> <span style="white-space: nowrap">Wlog let’s assume <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span> <span style="white-space: nowrap">uses PoW: if <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span></span> is chosen reasonably, then being able to do <span style="white-space: nowrap">a <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>-block rollback</span> at all probably requires having somewhere close to 50% of network hash rate. And so in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> block times the adversary has a significant chance of being able to do the required rollback before the suggested rule “kicks in”. (It also has however much additional latency is added by the BFT protocol, which is simultaneously under attack to maximize this latency.)</p>
+</li>
+</ul>
+<p>All that said, does the suggested rule help? First we have to ask whether it introduces any weaknesses.</p>
+<ul>
+<li>One potential issue is that the rule cuts both ways: if an adversarial rollback of more than <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> blocks <em>does</em> occur, then the adversary can make a proposal that will “lock in” its success. But it can be argued that this is intended behaviour anyway: the adversary has a confirmed chain, and is entitled to propose to finalize it.</li>
+<li>What happens if <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> is broken?
+<ul>
+<li><strong>Edit: the answer below refers to Crosslink before the Increasing Score rule was added.</strong></li>
+<li>This could definitely be a problem. If either an adversary has a two‑thirds supermajority of validators, <em>or</em> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> is completely broken (e.g. by a bug in the implementation of the validation signature scheme), then they can add a final block to the bft-chain with a <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">headers_bc</span></span></span></span></span> field that does not satisfy the honest voting condition. They do not need this to be bft-valid (although greater vulnerability to bugs in the implementation of bft-validity are also an issue). Then they can use the proposed rule to prevent an arbitrary bc-valid-chain from being extended (e.g. by mining <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> blocks from <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> at the genesis difficulty, then getting that chain into a final bft‑block). <strong>[Edit: they would no longer be able to do this using a chain mined at a much lower difficulty, because of the Increasing Score rule.]</strong>
+<ul>
+<li>In Crosslink, this is very carefully avoided. The only similar rule that depends on <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.1em;"></span><span class="mclose">)</span></span></span></span> and that could potentially affect liveness of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> is the <a href="design/crosslink/./construction.html#%CE%A0bc%E2%80%91block-validity"><strong>Finality depth</strong></a> rule. But that rule <em>always</em> allows the alternative of producing a Stalled Mode block on the current bc‑best‑chain — and honest block producers will do so. Therefore, the effect of trying to exploit even a catastrophic break in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> in order to cause a rollback of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>, as long as <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has not also been broken, is to go into Stalled Mode.</li>
+<li>This does not mean that a break of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> is not a problem for Crosslink. In particular, an adversary that can violate safety of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> can violate safety of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">ba</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> (and of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> if there is also <span style="white-space: nowrap">a <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>‑block rollback</span> in some node’s bc‑best‑chain).</li>
+<li>The difference is that a safety violation of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> can be directly observed by nodes <em>without any chance of false positives</em>, which is not necessarily the case for all possible attacks against <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>. (The attack described above does <em>not</em> violate safety of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>; it just adds a final bft‑block with a suspiciously long <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> rollback. It could alternatively have added a block with a less suspiciously long rollback, say exactly <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> blocks. That is, in pursuit of preventing an attack against <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>, we have enabled attacks against <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> to achieve the same effect — precisely what Crosslink is designed to prevent.)</li>
+<li>This raises an interesting idea: if any node sees a rollback in the chain of final bft‑blocks, it could provide objective evidence of that rollback in the form of a “bft-final-vee”: two final bft‑blocks with the same parent. Similarly, if any node sees more than one third of stake vote for conflicting blocks in a given epoch, then the assumption bounding the adversary’s stake must have been violated. This evidence can be posted in a transaction to the bc-chain. In that case, any node that is synced to the bc-chain can see that the bft-chain suffered a violation of safety or of a safety assumption, without needing to have seen that violation itself. This can be generalized to allow other proofs of flaws in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>. Optionally, a bc-chain that has posted such a proof could be latched into Stalled Mode until manual intervention can occur. (Obviously we need to make sure that this cannot be abused for denial-of-service.)</li>
+<li>This is now described in <a href="design/crosslink/./potential-changes.html#recommended-recording-more-info-about-the-bftchain-in-bcblocks">Potential changes to Crosslink</a>.</li>
+</ul>
+</li>
+</ul>
+</li>
+</ul>
+<p><strong>Okay, but is it a good idea to make that change to the fork-choice rule anyway?</strong></p>
+<p>Probably not. I don’t know how to repair the safety and liveness arguments.</p>
+<p>The change was that the bc‑best‑chain for a node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> would be required to extend <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span> where <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> is the last final bft‑block in node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>’s view.</p>
+<p>From the point of view of any modular analysis that treats <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathsf mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> as potentially subverted, we cannot say anything useful about <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span>. It seems as though any repair would have to assume much more about the BFT protocol than is desirable.</p>
+<p>In general, changes to fork‑choice rules are tricky; it was a fork-choice rule problem that allowed the liveness attack against Casper FFG described in [<a href="https://eprint.iacr.org/2020/1091.pdf">NTT2020</a>, Appendix E].</p>
+<p><strong>What if validators who see that a long rollback occurred, refuse to vote for it?</strong></p>
+<p>Yep that is allowed. The rule is “An honest validator will only vote for a proposal <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span> if ...” (not if‑and‑only‑if). If an honest validator sees a “good” reason not to vote for a proposal, including reasons based on out‑of‑band information, they should not. The <a href="design/crosslink/./the-arguments-for-bounded-availability-and-finality-overrides.html#complementarity">Complementarity argument</a> made in <em>The Argument for Bounded Availability and Finality Overrides</em> actually depends on this. Obviously, it may affect BFT liveness (and that’s okay).</p>
+<p>The only reason why we don’t make this part of the voting condition is that it’s a stateful rule. A new validator could come along and wouldn’t have the state needed to enforce it. Perhaps that could be fixed.</p>
+<div style="break-before: page; page-break-before: always;"></div><link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.12.0/dist/katex.min.css" integrity="sha384-AfEj0r4/OFrOo5t7NnNe46zW/tFgW6x/bCJG8FqQCEo3+Aro6EYUG4+cU+KJWu/X" crossorigin="anonymous">
+<h1 id="potential-changes-to-crosslink"><a class="header" href="#potential-changes-to-crosslink">Potential Changes to Crosslink</a></h1>
+<p>This page documents suggestions that have not had the same attention to security analysis as the <a href="design/crosslink/./construction.html">Crosslink 2 construction</a>. Some of them are broken. Some of them also increase the complexity of the protocol (while some simplify it or have a mixed effect on complexity), and so we need to consider the security/complexity trade‑off of each suggestion before we could include it.</p>
+<div id="admonition-this-page-is-out-of-date" class="admonition admonish-warning">
+<div class="admonition-title">
+<p>This page is out-of-date</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/potential-changes.html#admonition-this-page-is-out-of-date"></a></p>
+</div>
+<div>
+<p>This page has not yet been updated for the changes from Crosslink 1 to Crosslink 2.</p>
+</div>
+</div>
+<h2 id="attempts-to-improve-safety-or-to-simplify-the-protocol"><a class="header" href="#attempts-to-improve-safety-or-to-simplify-the-protocol">Attempts to improve safety or to simplify the protocol</a></h2>
+<h3 id="recommended-recording-more-info-about-the-bftchain-in-bcblocks"><a class="header" href="#recommended-recording-more-info-about-the-bftchain-in-bcblocks">[Recommended] Recording more info about the bft‑chain in bc‑blocks</a></h3>
+<p>We can allow honest bc‑block‑producers to record information about every proposed and notarized bft‑block, rather than just the one in the <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">context_bft</span></span></span></span></span> field.</p>
+<p>Duplicate information that has already been given in an ancestor bc‑block would be omitted.</p>
+<p>This would automatically expose the following shenanigans to public view (as long as enough bc‑block‑proposers are honest, which is already assumed):</p>
+<ul>
+<li>any attempt to double‑propose in the same epoch;</li>
+<li>any successful attempt to double‑notarize.</li>
+</ul>
+<p>We could also expose attempts to double‑vote.</p>
+<p>Note that double‑proposal and double‑voting could be a sign that a proposer or validator’s private key is compromised, rather than that it belongs to the adversary per se. However, the security analysis must treat such a proposer/validator as non‑honest in any case.</p>
+<h3 id="changing-the-increasing-score-rule-to-require-the-score-of-the-tip-rather-than-the-score-of-the-snapshot-to-increase"><a class="header" href="#changing-the-increasing-score-rule-to-require-the-score-of-the-tip-rather-than-the-score-of-the-snapshot-to-increase">Changing the Increasing Score rule to require the score of the tip (rather than the score of the snapshot) to increase</a></h3>
+<p>The current <a href="design/crosslink/./construction.html#%CE%A0bft-proposal-and-block-validity"><strong>Increasing Score rule</strong></a> concerns the score of the snapshot:</p>
+<p><strong>Increasing Snapshot Score rule:</strong> Either <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0972em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord mathsf">score</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">))</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">score</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">))</span></span></span></span> or <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0972em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span>.</p>
+<p>We could instead require the score of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">tip</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">headers_bc</span></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">]</span></span></span></span> to increase:</p>
+<p><strong>Increasing Tip Score rule:</strong> Either <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0972em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord mathsf">score</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">tip</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">))</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">score</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">tip</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">))</span></span></span></span> or <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0972em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord mathsf">tip</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">tip</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span>.</p>
+<p>Pros:</p>
+<ul>
+<li>This more directly reflects the fork‑choice rule in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</li>
+<li>In Crosslink 1, an <a href="design/crosslink/./construction.html#%CE%A0bft-honest-proposal">honest bft‑proposer</a> uses its bc‑best‑chain tip with the highest score <em>provided that</em> it is consistent with the <strong>Increasing Snapshot Score rule</strong>. This change removes the caveat, simplifying honest bft‑proposer behaviour.</li>
+<li>As a result of removing that caveat, we always know about an honest bft‑proposer’s bc‑best‑chain.</li>
+</ul>
+<p>Con:</p>
+<ul>
+<li>The score of the snapshot would not necessarily increase. As a result, it is technically possible that the new snapshot can be an ancestor of a previous snapshot. Whether this can actually happen depends on the value of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> and the difficulty adjustment rule. This causes no particular harm other than adding a corner case in ledger sanitization, but is inelegant.
+<ul>
+<li>This con is removed if we either use the <strong>Combined Increasing Score rule</strong> variant described below, <em>or</em> we also apply the “<a href="design/crosslink/potential-changes.html#Making-bc-rollbacks-more-difficult">Making bc‑rollbacks more difficult</a>” change in the next section.</li>
+</ul>
+</li>
+</ul>
+<p>Apart from the above con, the original motivations for the <strong>Increasing Snapshot Score rule</strong> also apply to the <strong>Increasing Tip Score rule</strong>. In particular,</p>
+<ul>
+<li>it still prevents potential attacks that rely on proposing a bc‑valid‑chain that forks from a much earlier block;</li>
+<li>it still limits the extent of disruption an adversary can feasibly cause to \LOG_{\bda}};</li>
+<li>it still always allows a proposal to be made, which may be needed to preserve liveness of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> relative to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>;</li>
+<li>it still prevents potential validation cost DoS attacks due to switching between snapshots with the same score.</li>
+</ul>
+<p>If we switch to using the Increasing Tip Score rule, then it would be more consistent for block producers to also change the tie‑breaking rule for choosing <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">context_bft</span></span></span></span></span> to use the tip score, <span style="white-space: nowrap">i.e. <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">score</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">tip</span></span><span class="mopen">(</span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mclose">)))</span></span></span></span>.</span></p>
+<p>A variation on this suggestion effectively keeps both the <strong>Increasing Snapshot Score rule</strong> and the <strong>Increasing Tip Score rule</strong>:</p>
+<p><strong>Combined Increasing Score rule:</strong> Either (<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0972em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord mathsf">score</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">))</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">score</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">))</span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0972em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord mathsf">score</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">tip</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">))</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">score</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">tip</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">))</span></span></span></span>), or <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0972em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord mathsf">tip</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">tip</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span>.</p>
+<p>Note that if <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0972em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord mathsf">tip</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">tip</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span>, both scores are necessarily equal.</p>
+<p>This variation does not simplify honest bft‑proposer behaviour.</p>
+<h3 id="making-exploitation-of-bcrollbacks-more-difficult"><a class="header" href="#making-exploitation-of-bcrollbacks-more-difficult">Making exploitation of bc‑rollbacks more difficult</a></h3>
+<p>Basic idea: Detect the case where the bc‑snapshot is rolling back, and impose a longer confirmation depth to switch to the new bc‑chain. Also temporarily stall finalization of the existing bc‑chain until the conflict has been resolved.</p>
+<p>Let <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">baseline_snapshot</span></span></span></span></span> be the Crosslink 1 definition of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span></span></span></span>, i.e. <span class="katex-display fleqn"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">baseline_snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">headers_bc</span></span><span class="mopen">[</span><span class="mord">0</span><span class="mclose">]</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mpunct">,</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord">if </span></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">headers_bc</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord amsrm">∅</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord">otherwise.</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>
+<p>When <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1641em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord"><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span><span class="mrel">⪯</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.242em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathsf">baseline_snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span>, we want to go into a mode where we require a longer confirmation <span style="white-space: nowrap">depth <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Σ</span></span></span></span>,</span> <span style="white-space: nowrap">say <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>.</span> Because we don’t know in this situation whether the old bc‑chain or the new bc‑chain will win, we stop finalizing both until a winner is clear.</p>
+<p>The simplest option is to record the state saying that we are in this mode explicitly, and add a consensus rule requiring it to be correct. That is, add an <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">bc_is_forked</span></span></span></span></span> field to bft‑proposals and bft‑blocks, and add a bft‑proposal and bft‑block validity rule as follows:</p>
+<ul>
+<li><strong>Is Forked rule:</strong>
+<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">bc_is_forked</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1em;vertical-align:-0.35em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">enter_forked</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span> or <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1641em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">bc_is_forked</span></span><span class="mclose">)</span></span></span></span> and not <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">exit_forked</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span></li>
+</ul>
+<p>where:</p>
+<ul>
+<li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">enter_forked</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1641em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord"><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span><span class="mrel">⪯</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.242em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathsf">baseline_snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span></li>
+<li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">exit_forked</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1244em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord mathsf">score</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">tip</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">Σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0972em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord mathsf">score</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">))</span></span></span></span>.</li>
+</ul>
+<p>It is intentional that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">exit_forked</span></span></span></span></span> takes precedence over <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">enter_forked</span></span></span></span></span>.</p>
+<p>Then redefine <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span></span></span></span> as follows: <span class="katex-display fleqn"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mpunct">,</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">baseline_snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mpunct">,</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord">if </span></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">bc_is_forked</span></span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord">otherwise.</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>
+<p>Since <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">bc_is_forked</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathsf">false</span></span></span></span></span>, the recursion will terminate.</p>
+<p>Note that there is an interaction between the <strong>Increasing Snapshot Score rule</strong> and this change: the <strong>Increasing Snapshot Score rule</strong> should arguably use <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">baseline_snapshot</span></span></span></span></span> instead of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span></span></span></span>. The <strong>Increasing Tip Score rule</strong>, on the other hand, works fine as‑is, and so it makes sense to use both of these changes together. The combination of both changes also fixes the con discussed above for the <strong>Increasing Tip Score rule</strong>; it ensures that the score of the snapshot must increase.</p>
+<p>Pros:</p>
+<ul>
+<li>If the winning chain is the chain that was first snapshotted, then there ends up being no disruption whatsoever <span style="white-space: nowrap">to \LOG_{\bda}}.</span></li>
+<li>It becomes extremely difficult for an adversary with less than 50% hash power to get the finalized snapshot to switch between two competing chains more than once.</li>
+</ul>
+<p>Cons:</p>
+<ul>
+<li>It is potentially easier to cause a temporary finalization stall. An adversary could try to provoke this situation on the honest chain, either as a DoS attack, or so that its own chain that is not so encumbered can finalize more quickly than the honest chain.
+<ul>
+<li>This does not seem like a practical attack, because such a stall can only happen when the adversary can cause a <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>‑block rollback or has <span style="white-space: nowrap">subverted <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</span></li>
+</ul>
+</li>
+<li>The definition of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span></span></span></span> becomes more complicated, and there is a risk of this complexity introducing problems.</li>
+<li>It can be argued that, unless <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> is</span> chosen too small, an adversary that can cause a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>‑block rollback</span> likely has 50% of hash power, and therefore can cause a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Σ</span></span></span></span>‑block rollback.</span> That is, increasing confirmation depth does not help beyond a certain point, and therefore (it could be argued) this change will also not help.
+<ul>
+<li>This argument may hold for private mining attacks, but does not necessarily hold for partitioning attacks, as discussed in the next section.</li>
+</ul>
+</li>
+</ul>
+<h3 id="using-tip-information-to-detect-rollbacks-and-partitions"><a class="header" href="#using-tip-information-to-detect-rollbacks-and-partitions">Using tip information to detect rollbacks and partitions</a></h3>
+<p>In the case of a private mining attack, the adversary will typically conceal the existence of the overtaking chain until it can be used to cause a rollback <span style="white-space: nowrap">in \LOG_{\bda}}.</span> So the approach used in the previous section seems to be all we can do against such an attack.</p>
+<p>In the case of a partitioning attack, on the other hand, the adversary relies on honest nodes to do mining work on each side of the partition. This relies on the successful miners on each side knowing about their chain, but not the chain on the other side. Subtly, it does not rely on a perfect network partition. An adversary could, for example, attempt to create partitions <em>around</em> the most successful mining pools. Occasional leaks of information across a partition also do not necessarily foil the attack unless that information gets to a successful miner. Therefore, measures that constrain the adversary’s ability to make use of an <em>incomplete</em> partition can be useful.</p>
+<p>This also has the benefit of making the protocol more robust against <em>non‑malicious</em> incomplete partitions.</p>
+<p>Given that in such an attack the competing chains may be visible to some proposers, there is the possibility of detecting a potential rollback even before it gets snapshotted, by using the fact that previous bft‑blocks created by honest bft‑proposers have been recording the bc‑best‑chain tip <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> blocks ahead.</span> Also, depending on what proportion of validators an adversary has, they may rely on honest validators on each side to ensure that a snapshot of each chain appears in a bft‑valid block; in that case, including information about competing chains in validators' votes (see the next subsection) may be useful.</p>
+<p>It is still possible that if an adversary has several consecutive proposal slots, it can get its chain snapshotted. However, if there is an intervening slot with an honest proposer, we can potentially compare its tip with the adversary’s tip and anticipate the need to go into <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">bc_is_forked</span></span></span></span></span> mode.</span></p>
+<p>In order to get this to work, we need to propose a definition to identify bc‑chains that are competing with the current best chain, such that there is some risk of a “long” rollback to a competing chain. <span style="white-space: nowrap">Let <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03785em;">δ</span></span></span></span> be</span> a measure of how close (in terms of bc‑blocks) a competing chain’s score needs to be to that of the bc‑best‑chain, and <span style="white-space: nowrap">let <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03785em;">δ</span></span></span></span> be</span> a lower bound on the rollback depth we would consider significant if the competing chain were to immediately catch up. <span style="white-space: nowrap">(The condition <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03785em;">δ</span></span></span></span></span> is necessary to avoid false positives that might only be a single‑block fork.)</p>
+<p>A node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> identifies <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03785em;">δ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">μ</span><span class="mclose">)</span></span></span></span>‑competing chains</span> as follows based on its current view at <span style="white-space: nowrap">time <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>:</span></p>
+<ul>
+<li>A bc‑block is a tip if it has no known descendants.</li>
+<li>Let <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1249em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span> be <span style="white-space: nowrap">node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>’s bc‑best‑chain.</span></li>
+<li>Identify all of the <span style="white-space: nowrap">tips <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span></span></span></span></span> such that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">score</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.161em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord mathsf">score</span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03785em;">δ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> and <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1279em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">last-common-ancestor</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1792em;vertical-align:-0.3013em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">ch</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em;"><span style="top:-2.3987em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.1809em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3013em;"><span></span></span></span></span></span></span></span></span></span>.</span></li>
+</ul>
+<p><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="enclosing" style="background-color:yellow;color:black;"><span class="mord text"><span class="mord textsf">TODO:</span></span></span></span></span></span> Details, including how to modify the <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">enter_forked</span></span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">exit_forked</span></span></span></span></span> conditions.</p>
+<p><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="enclosing" style="background-color:yellow;color:black;"><span class="mord text"><span class="mord textsf">TODO:</span></span></span></span></span></span> For now we will assume that all of the competing chain information in a bft‑block has to be checked as bc‑block‑valid in order for that block to be bft‑block‑valid. This might introduce validation DoS attacks and needs to be considered more carefully.</p>
+<h3 id="allowing-validators-to-signal-the-existence-of-a-competing-chain-in-their-votes"><a class="header" href="#allowing-validators-to-signal-the-existence-of-a-competing-chain-in-their-votes">Allowing validators to signal the existence of a competing chain in their votes</a></h3>
+<p>This complements the above idea by letting a validator that has seen a competing chain signal it in its signed vote. Then, as long as the adversary is reliant on some votes from honest validators that are signalling the existence of competing chains, we would go into <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">bc_is_forked</span></span></span></span></span> mode</span> without relying on honest proposers to have an intervening slot.</p>
+<p>The <a href="design/crosslink/./construction.html#%CE%A0bft-proposal-and-block-validity">notarization proof</a> that appears in a bft‑block would need to be modified to preserve these signals. More precisely, it is necessary for a <span style="white-space: nowrap">bft‑block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span></span> to preserve at least:</p>
+<ul>
+<li>the best bc‑chain that credibly competes <span style="white-space: nowrap">with <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">tip</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span>,</span> if any.</li>
+<li>the best bc‑chain that does not credibly compete <span style="white-space: nowrap">with <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">tip</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span></span> (this necessarily exists because <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">tip</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span> does not credibly compete with itself).</li>
+</ul>
+<p>This is also motivated by the suggested change in the next section.</p>
+<p>Enforcing this is relatively straightforward if the evidence is a SNARK. It can also be enforced with aggregate signatures even for schemes that only allow aggregation of signatures over a common message: we just collect the distinct messages (corresponding to either “no competing chain” or each distinct competing chain) and aggregate them separately.</p>
+<h3 id="strengthening-the-increasing-tip-score-rule"><a class="header" href="#strengthening-the-increasing-tip-score-rule">Strengthening the Increasing Tip Score rule</a></h3>
+<p>Assume that votes include competing chain information as discussed above. We can assume that an honest proposer has read all of this information from its parent bft‑block. Therefore, we can require the tip score of its proposal to have at least the score of the best tip implied by that information:</p>
+<p>Let <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">best-tip</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span> be the tip mentioned in <span style="white-space: nowrap">bft‑block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span></span> with the highest score. <span style="white-space: nowrap">A bft‑block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span></span> “mentions” the two best tips defined in the previous section.</p>
+<p><strong>Strong Increasing Tip Score rule:</strong> Either <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0972em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord mathsf">score</span></span><span class="mopen">(</span><span class="mord text"><span class="mord textsf">best-tip</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">))</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">score</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">tip</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">))</span></span></span></span> or <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0972em;vertical-align:-0.2831em;"></span><span class="mord text"><span class="mord textsf">best-tip</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">tip</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span>.</span></p>
+<p>Note that this rule is really quite constraining for a potential adversary, especially in partitioning attacks. It means that if the adversary does not want to acknowledge the existence of a given chain, it cannot use any votes or build on any previous bft‑block that signals the existence of that chain. Essentially, a partitioning adversary with control over only the minimum one‑third of the stake would have to have ensure a perfectly complete partition; it could not get away with any information leakage between honest validators.</p>
+<h2 id="attempts-to-improve-finalization-latency"><a class="header" href="#attempts-to-improve-finalization-latency">Attempts to improve finalization latency</a></h2>
+<h3 id="broken-adjusting-the-last-snapshot-definition"><a class="header" href="#broken-adjusting-the-last-snapshot-definition">[Broken] Adjusting the last snapshot definition</a></h3>
+<p>The Crosslink 1 design imposes a finalization latency of at least <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> block times.</span> Intuitively, this is because in <span class="katex-display fleqn"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathrm">fin</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mord text"><span class="mord"> for </span></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1em;vertical-align:-0.35em;"></span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">context_bft</span></span><span class="mclose">)]</span><span class="mpunct">,</span></span></span></span></span> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">context_bft</span></span><span class="mclose">))</span></span></span></span> is at least <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> blocks</span> back <span style="white-space: nowrap">from <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0331em;vertical-align:-0.2831em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span></span> (as argued in <a href="design/crosslink/./questions.html#Why-don%E2%80%99t-we-have-a-bc-block-validity-rule-snapshotfinal-bftH-%E2%AA%AFbc-H-">Questions about Crosslink 1</a>), and therefore <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> blocks</span> back <span style="white-space: nowrap">from <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span>.</span> So the total finalization latency is <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> block times + BFT overhead + <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span> block times + snapshot overhead.</span></p>
+<p>However, the snapshot headers contain information about the proposer’s bc‑best‑chain.</p>
+<p>Define <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1em;vertical-align:-0.35em;"></span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">context_bft</span></span><span class="mclose">)</span></span></span></span>. Although it is not guaranteed, normally <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0331em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.1em;"></span><span class="mclose">)</span></span></span></span> will be an ancestor <span style="white-space: nowrap">of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0331em;vertical-align:-0.2831em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span>.</span> What if we were to optimistically allow the last snapshot to be taken as <span class="katex-display fleqn"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord textsf">last-common-ancestor</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathsf">tip</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">))</span><span class="mpunct">,</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.1em;"></span><span class="mclose">)</span><span class="mpunct">,</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord">if it extends </span></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.1em;"></span><span class="mclose">)</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord">otherwise</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span>? After all, we know that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0331em;vertical-align:-0.2831em;"></span><span class="mord text"><span class="mord textsf">last-common-ancestor</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathsf">tip</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">))</span></span></span></span> is confirmed.</p>
+<p>Oh, this won’t work. The problem is that we want safety of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> not to depend on safety of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>. So we cannot assume (for this purpose) that nodes see the <span style="white-space: nowrap">same <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0331em;vertical-align:-0.2831em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span>.</span></p>
+<h3 id="replacing-log_bda-with-log_opt"><a class="header" href="#replacing-log_bda-with-log_opt">Replacing \LOG_{\bda}} with \LOG_{\opt}}</a></h3>
+<p>What if we instead take this to be the definition of \LOG_{\opt}}, replacing \LOG_{\bda}} (&quot;opt&quot; meaning optimistic)?</p>
+<p>As stated, a malicious proposer can try to maximize the latency of \LOG_{\opt}} (subject to the Increasing Score rule). For example, if there exists a fork of <span style="white-space: nowrap">length <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span></span></span></span>,</span> the malicious proposer can force the latency of \LOG_{\opt}} to be <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span> block times + BFT overhead.</span> However, this can be improved by applying the idea to each bft‑block in turn after the one pointed to by the best confirmed bc‑block. Then a malicious proposer cannot do anything that it could not do anyway (keeping the finalization point at its current position).</p>
+<p>Pros:</p>
+<ul>
+<li>This is always more conservative in terms of safety than the current design.</li>
+<li>The latency of \LOG_{\opt}} will typically be <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> bc‑block times,</span> rather than <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> block times.</span></li>
+</ul>
+<p>Cons:</p>
+<ul>
+<li>\LOG_{\opt}} is <em>not</em> dynamically available in any sense. It just has lower latency and different security characteristics.</li>
+<li>Even under optimistic conditions, \LOG_{\opt}} will lag slightly behind where it would be for the Crosslink 1 design, because <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0331em;vertical-align:-0.2831em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span> will necessarily be ahead of <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0331em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord mathsf">tip</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">))</span></span></span></span>.</span></li>
+</ul>
+<h3 id="broken-using-snapshots-from-the-lastseen-bftchain-when-it-is-consistent-with-the-bcbestchain"><a class="header" href="#broken-using-snapshots-from-the-lastseen-bftchain-when-it-is-consistent-with-the-bcbestchain">[Broken] Using snapshots from the last‑seen bft‑chain when it is consistent with the bc‑best‑chain</a></h3>
+<p>The following idea is broken for safety when <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has been subverted:</p>
+<div id="admonition-info" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Info</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/potential-changes.html#admonition-info"></a></p>
+</div>
+<div>
+<p>We have two potential sources of information about blocks that could plausibly be considered finalized:</p>
+<ol>
+<li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0331em;vertical-align:-0.2831em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span></li>
+<li>the snapshots on the chain of the last <em>seen</em> <span style="white-space: nowrap">final bft‑block, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.06944em;">lsf</span></span></span></span></span>.</span></li>
+</ol>
+<p>We cannot rely only on 1. because we want assured finalization even under partition.
+We cannot rely only on 2. because if <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has been subverted, then the chain of final bft‑blocks could fork.</p>
+<p>But intuitively, if we combine these sources of information, using them over the Crosslink 1 finalization only when they are consistent, the resulting protocol should still be as safe as the safer of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> <span style="white-space: nowrap">and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</span> In particular, 2. will not roll back <em>unless</em> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has been subverted.</p>
+<p>If this idea were to pan out, it could improve the latency of finalization by <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> block times.</span></p>
+<p>This approach is essentially a hybrid of Snap‑and‑Chat and Crosslink 1:</p>
+<ul>
+<li>the Snap‑and‑Chat construction gives a finalized ledger under the assumption that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has not been subverted;</li>
+<li>the main crosslink idea is used to make sure that outputs from all finalized transactions are <em>eventually</em> spendable;</li>
+<li>safety is still only dependent on the stronger of the safety of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> <span style="white-space: nowrap">and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>,</span> because we use the additional information from snapshots in final bft‑blocks only up to the point at which they agree with the best confirmed bc‑block.</li>
+</ul>
+</div>
+</div>
+<p>To explain the safety problem with this idea: suppose that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has been subverted. In that case it is possible for a snapshot to be finalized without having being confirmed as in any honest node’s bc‑best‑chain; that is, it is possible for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> to include transactions <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span></span></span></span> from a snapshot <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span></span></span></span> in bft‑block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> such that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span></span></span></span> is not on the consensus bc‑best‑chain. And, because <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has been subverted, it is also possible that a conflicting final bft‑block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> <span style="white-space: nowrap">omits <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span></span></span></span>.</span> And so a node that has seen <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> will think that it is consistent with the best bc‑chain (so that its <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> does not include <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span></span></span></span> but does include later transactions on the consensus bc‑best‑chain), but a node that has <span style="white-space: nowrap">seen <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span></span> will compute a <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> that does <span style="white-space: nowrap">include <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span></span></span></span>.</span></p>
+<details id="admonition-more-detailed-specification-of-the-above-broken-idea" class="admonition admonish-info">
+<summary class="admonition-title">
+<p>More detailed specification of the above broken idea.</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/potential-changes.html#admonition-more-detailed-specification-of-the-above-broken-idea"></a></p>
+</summary>
+<div>
+<p>Define <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1em;vertical-align:-0.35em;"></span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">context_bft</span></span><span class="mclose">)</span></span></span></span> as before.</p>
+<p>For simplicity assume that <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.06944em;">lsf</span></span></span></span></span> extends <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0331em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span> by only one bft‑block. (This assumption could have been removed if the idea had panned out.)</p>
+<p>Then this proposal was to consider this bc‑block as contributing the last finalized snapshot:
+<span class="katex-display fleqn"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord textsf">last-common-ancestor</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf" style="margin-right:0.06944em;">lsf</span></span><span class="mclose">))</span><span class="mpunct">,</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.1em;"></span><span class="mclose">)</span><span class="mpunct">,</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord">if it extends </span></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.1em;"></span><span class="mclose">)</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord">otherwise</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>
+<p>There is no need for a tie‑breaking rule for 2.: if we ever see two context bft‑blocks for which the last‑final blocks are conflicting, we know that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has been subverted, so we should stall or crash.</p>
+<p>Caveat: for a given node, <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0331em;vertical-align:-0.2831em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span> can</span> in theory roll back <span style="white-space: nowrap">past <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf" style="margin-right:0.06944em;">lsf</span></span><span class="mclose">)</span></span></span></span>,</span> therefore <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span></span></span> can also roll back. It is okay if we keep state here and refuse to roll back. We should set a “crisis flag”, and unset it if at any point <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0331em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> extends <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">lsf_at_crisis</span></span></span></span></span>.</span> <span style="white-space: nowrap">(If <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span> is safe and live, it will.)</p>
+<p>A similar rule that would give the same result in almost all circumstances is: <span class="katex-display fleqn"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:4.32em;vertical-align:-1.91em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.35em;"><span style="top:-2.2em;"><span class="pstrut" style="height:3.15em;"></span><span class="delimsizinginner delim-size4"><span>⎩</span></span></span><span style="top:-2.192em;"><span class="pstrut" style="height:3.15em;"></span><span style="height:0.316em;width:0.8889em;"><svg xmlns="http://www.w3.org/2000/svg" width='0.8889em' height='0.316em' style='width:0.8889em' viewBox='0 0 888.89 316' preserveAspectRatio='xMinYMin'><path d='M384 0 H504 V316 H384z M384 0 H504 V316 H384z'/></svg></span></span><span style="top:-3.15em;"><span class="pstrut" style="height:3.15em;"></span><span class="delimsizinginner delim-size4"><span>⎨</span></span></span><span style="top:-4.292em;"><span class="pstrut" style="height:3.15em;"></span><span style="height:0.316em;width:0.8889em;"><svg xmlns="http://www.w3.org/2000/svg" width='0.8889em' height='0.316em' style='width:0.8889em' viewBox='0 0 888.89 316' preserveAspectRatio='xMinYMin'><path d='M384 0 H504 V316 H384z M384 0 H504 V316 H384z'/></svg></span></span><span style="top:-4.6em;"><span class="pstrut" style="height:3.15em;"></span><span class="delimsizinginner delim-size4"><span>⎧</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.85em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.41em;"><span style="top:-4.41em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mpunct">,</span></span></span><span style="top:-2.97em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf" style="margin-right:0.06944em;">lsf</span></span><span class="mclose">)</span><span class="mpunct">,</span></span></span><span style="top:-1.53em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.1em;"></span><span class="mclose">)</span><span class="mpunct">,</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.91em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.41em;"><span style="top:-4.41em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord">if </span></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.1em;"></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf" style="margin-right:0.06944em;">lsf</span></span><span class="mclose">)</span></span></span><span style="top:-2.97em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord">if </span></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.1em;"></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf" style="margin-right:0.06944em;">lsf</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span><span style="top:-1.53em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord">otherwise</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.91em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>
+</div>
+</details>
+<h2 id="what-about-making-the-bcblockproducer-the-bftproposer"><a class="header" href="#what-about-making-the-bcblockproducer-the-bftproposer">What about making the bc‑block‑producer the bft‑proposer?</a></h2>
+<p>The answer given for this question at <a href="design/crosslink/./construction.html#%CE%A0bft-proposal-and-block-validity">The Crosslink 2 Construction</a> is:</p>
+<blockquote>
+<p>If this were enforced, it could be an alternative way of ensuring that every bft‑proposal snapshots a new bc‑block with a higher score than previous snapshots, potentially making the <strong>Increasing Score rule</strong> redundant. However, it would require merging bc‑block‑producers and bft‑proposers, which could have concerning knock‑on effects (such as concentrating security into fewer participants).</p>
+</blockquote>
+<p>This may have been too hasty. It is not clear that merging bc‑block‑producers and bft‑proposers actually does “concentrate security into fewer participants” in a way that can have any harmful effect.</p>
+<p>Remember, the job of a bft‑proposer in Crosslink is primarily to snapshot the bc‑best‑chain (even more so if the <a href="design/crosslink/potential-changes.html#changing-the-increasing-score-rule-to-require-the-score-of-the-tip-rather-than-the-score-of-the-snapshot-to-increase">Increasing Tip Score rule</a> is adopted). An honest miner <em>by definition</em> is claiming to build on the best chain, and miners have a strong economic incentive to do so. Therefore, it is entirely reasonable for every newly produced block to be treated as a bft‑proposal. This arguably decentralizes the task of proposing bft‑blocks more effectively than using a leader election protocol would — especially given that in a hybrid protocol we <em>necessarily</em> still rely on there being sufficient honest miners.</p>
+<p><a href="https://arxiv.org/pdf/2010.08154.pdf">[DKT2021]</a>, for example, argues for the importance of “the complete unpredictability of who will get to propose a block next, even by the winner itself.” The main basis of this argument is that it makes subversion of the proposer significantly more difficult. A PoW protocol has that property, and most PoS protocols do not. (It is not that PoS protocols are unable to provide this property; indeed, <a href="https://arxiv.org/pdf/2010.08154.pdf">[DKT2021]</a> constructs a PoS protocol, “PoSAT”, that provides it.)</p>
+<p>So let’s explore this in more detail. A newly produced bc‑block would implicitly be a bft‑proposal with itself as the tip. The <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">bc_headers</span></span></span></span></span> field is therefore not needed. The <a href="design/crosslink/./construction.html#%CE%A0bft-proposal-and-block-validity"><strong>Tail Confirmation rule</strong></a> goes away since its intent is automatically satisfied. This is already a significant simplification.</p>
+<p>The inner proposer signature is also not needed (since the bc‑header is self-authenticating), but the block producer would have to include a <span style="white-space: nowrap">public key <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">pubkey</span></span></span></span></span></span> that can be used to verify its outer signature. It would sign the notarized bft‑block with the corresponding private key. This change is a wash in terms of protocol complexity.</p>
+<p>Considered as a bft‑proposal, a bc‑block needs to refer to a parent bft‑block, which requires a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">parent_bft</span></span></span></span></span> field</span> in the bc‑header. With some caveats depending on the design <span style="white-space: nowrap">of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>,</span> it might be possible to merge this with the <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">context_bft</span></span></span></span></span> field,</span> but for now we will assume that it is not merged.</p>
+<details id="admonition-what-are-the-caveats" class="admonition admonish-info">
+<summary class="admonition-title">
+<p>What are the caveats?</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/potential-changes.html#admonition-what-are-the-caveats"></a></p>
+</summary>
+<div>
+<p>If we are in an execution where <strong>Final Agreement</strong> holds <span style="white-space: nowrap">for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>,</span> then it is possible to show that merging the two fields has no negative effect, <em>provided</em> that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> has no additional rules that could disallow it in some cases.</p>
+<p>This is because, by <strong>Final Agreement</strong>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1019em;vertical-align:-0.35em;"></span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">context_bft</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:-0.5em;"></span></span><span class="base"><span class="strut" style="height:0.786em;vertical-align:-0.15em;"></span><span class="mrel"><span class="mrel">⪰</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span></span></span></span> for any potential <span style="white-space: nowrap">bft‑block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span></span> that the bc‑block‑producer of a new block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> could choose <span style="white-space: nowrap">as <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">context_bft</span></span></span></span></span>.</span> Suppose that the bc‑block‑producer freely chooses <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">parent_bft</span></span></span></span></span> according to the desired honest behaviour for a bft‑proposer <span style="white-space: nowrap">in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>,</span> and then chooses <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span> to be the same block (which is always reasonable as long as it is allowed).</p>
+<p>In the case <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1019em;vertical-align:-0.35em;"></span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">context_bft</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mclose">)</span></span></span></span>,</span> we are done, because this choice of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">context_bft</span></span></span></span></span> is allowed by the <strong>Extension rule</strong>.</p>
+<p>In the case <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1019em;vertical-align:-0.35em;"></span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">context_bft</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord">⪲</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mclose">)</span></span></span></span>,</span> we can argue that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1019em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">context_bft</span></span></span></span></span> would be a better choice than <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span> for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">parent_bft</span></span></span></span></span> as well as for <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">context_bft</span></span></span></span></span>,</span> because it has a later final ancestor. This is where the argument might fall down if <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> <span style="white-space: nowrap">(and therefore <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>)</span> has any additional rules that could disallow this choice. For now let’s suppose that situtation does not arise, but it is one of the caveats.</p>
+<p>Another potential problem is that in an execution where <strong>Final Agreement</strong> does <em>not</em> hold for <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>,</span> we can no longer infer that either <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1019em;vertical-align:-0.35em;"></span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">context_bft</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mclose">)</span></span></span></span> or <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1019em;vertical-align:-0.35em;"></span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">context_bft</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord">⪲</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mclose">)</span></span></span></span>. In particular it could be the case that the producer of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> was adversarial, and chose <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1019em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">context_bft</span></span></span></span></span> in such a way as to favour its own bft‑block that is final in that context.</p>
+<p>However, in that situation it must be possible for the bc‑block‑producer to see (and prove) that the bft‑chain has a final fork. That is, it can produce a <span style="white-space: nowrap">witness <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span></span> to the violation of <strong>Final Agreement</strong>, showing that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1019em;vertical-align:-0.35em;"></span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">context_bft</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⪯</span><span class="mspace" style="margin-right:-0.5em;"></span></span><span class="base"><span class="strut" style="height:0.786em;vertical-align:-0.15em;"></span><span class="mrel"><span class="mrel">⪰</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">∗</span><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">bft-last-final</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span></span></span></span> does not hold, as discussed in the section <a href="design/crosslink/potential-changes.html#recommended-recording-more-info-about-the-bftchain-in-bcblocks">Recording more info about the bft‑chain in bc‑blocks</a> above.</p>
+<p>The second caveat is that in that situation, we still need to set <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">parent_bft</span></span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">context_bft</span></span></span></span></span> in order to be able to recover, and they typically should not be the same in order to do so.</p>
+</div>
+</details>
+<p>The <strong>Increasing Tip Score rule</strong> is still needed, but it can be simplified. A newly produced <span style="white-space: nowrap">bc‑block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span></span> is also a bft‑proposal <span style="white-space: nowrap">such that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span>.</span> This would yield the following bft‑proposal / bc‑block validity rule:</p>
+<blockquote>
+<p>[Candidate rule for discussion] Either <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">score</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">tip</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">parent_bft</span></span><span class="mclose">))</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">score</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span></span></span> <span style="white-space: nowrap">or <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">tip</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">parent_bft</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span>.</span></p>
+</blockquote>
+<p>except that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">tip</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">parent_bft</span></span><span class="mclose">)</span></span></span></span> <em>cannot</em> <span style="white-space: nowrap">be <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span>,</span> because <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> is newly produced. It turns out we can just drop that part:</p>
+<blockquote>
+<p><strong>Increasing Tip Score (producer = proposer) rule:</strong> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">score</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">tip</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">parent_bft</span></span><span class="mclose">))</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">score</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span></span></span>.</p>
+</blockquote>
+<p>This works because, if <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> does not have a higher score than the <span style="white-space: nowrap">bc‑block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">tip</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">parent_bft</span></span><span class="mclose">)</span></span></span></span>,</span> the bc‑block‑producer should instead have built on top of that bc‑block — which was necessarily known to the producer in order for it to set <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">parent_bft</span></span></span></span></span> in the header of the new block.</p>
+<p>The voting would be the same, performed by the same parties. Therefore, there is no concentration of voting into fewer parties. There is no change in the producer/proposer’s incentive to make the bft‑notarization‑proof or its soundness properties. Everything else is roughly the same, including the use of the <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">context_bft</span></span></span></span></span> field</span> of a bc‑block and the validity rules related to it. As far as I can see, all of the security analysis goes through essentially unchanged.</p>
+<p>There may be some complication due to the fact that BFT protocols are typically designed to use epochs with a fixed period, whereas bc‑blocks are found at less predictable intervals. However, as long as BFT messages are labelled with the bc‑block they apply to, it seems like most BFT protocols would be tolerant to this change. In fact the adaptations of Snap‑and‑Chat to Hotstuff and PBFT in <a href="https://eprint.iacr.org/2020/1091.pdf">[NTT2020]</a> already assume that BFT messages can be queued and processed at a later time, and rely on those BFT protocols' tolerance to this.</p>
+<div id="admonition-info-1" class="admonition admonish-info">
+<div class="admonition-title">
+<p>Info</p>
+<p><a class="admonition-anchor-link" href="design/crosslink/potential-changes.html#admonition-info-1"></a></p>
+</div>
+<div>
+<p>In most PoS protocols, the requirement to have a minimum amount of stake in order to make a proposal acts as a gatekeeping filter on <em>proposals</em>, and potentially allows parties that make invalid <em>proposals</em> to be slashed.</p>
+<p>Strictly speaking, whether there is a stake requirement to make a proposal is independent of whether bc‑block‑producers (e.g. miners) are merged with bft‑proposers. It could be, for example, that a miner is still able to produce bc‑blocks, but is not able to make them into a proposal unless they satisfy a stake requirement. (This would have significant effects on the economics of mining that would need to be analyzed, and that might have governance consequences.)</p>
+<p>In a system that uses PoS, validators by definition need to have stake in order to control the ability to vote. This also allows validators to be slashed.</p>
+<p>On the other hand, there is no <em>technical</em> reason why the ability to make a bft‑proposal has to be gatekept by a stake requirement — given the situation of Zcash in which we already have a mining infrastructure, and that in a Snap‑and‑Chat or Crosslink‑style hybrid protocol we necessarily still rely on miners not to censor transactions. The potential to make proposals that are expensive to validate as a denial of service is made sufficiently difficult by proof‑of‑work. This option has probably been underexplored by previous PoS protocols because they cannot assume an existing mining infrastructure.</p>
+</div>
+</div>
+<p>It could be argued that this approach goes less far toward a pure PoS‑based block‑chain protocol, leaving more to be done in the second stage. However, there is a clear route to that second stage, by replacing PoW with a protocol like PoSAT that has full unpredictabilty and dynamic availability. PoSAT does this using a VDF, and as it happens, Halo 2 is a strong candidate to be used to construct such a VDF.</p>
+<p>If the arguments in <a href="https://arxiv.org/pdf/2010.08154.pdf">[DKT2021]</a> about the need for proposer unpredictability are persuasive, then this approach defers the complexity of requiring a VDF without losing any security, since Zcash’s PoW is already unpredictable.</p>
+<h3 id="do-we-need-an-explicit-bftchain-at-all"><a class="header" href="#do-we-need-an-explicit-bftchain-at-all">Do we need an explicit bft‑chain at all?</a></h3>
+<p>Building on the previous idea, we can try to eliminate the explicit bft‑chain by piggybacking the information it would hold onto a bc‑block (in the header and/or the block contents). In the previous section we merged the concepts of a bft‑proposal and a bc‑block; the <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">parent</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.242em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mclose">)</span></span></span></span> and <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">context_bft</span></span></span></span></span> fields</span> of a bft‑proposal were moved into the <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">parent_bft</span></span></span></span></span> and <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">context_bft</span></span></span></span></span> fields</span> of a bc‑header respectively. <span style="white-space: nowrap">A field <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">pubkey</span></span></span></span></span></span> was also added to hold the producer’s public key, so that the producer can sign the bft‑block constructed from it using the corresponding private key.</p>
+<p>This left the concept of a bft‑block intact. Recall that in Crosslink 1, a <span style="white-space: nowrap">bft‑block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span></span> consists of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord mathsf" style="margin-right:0.06944em;">proof</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2342em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">P</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> signed by the proposer. So in “Crosslink with proposer = producer”, a bft‑block consists of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord mathsf" style="margin-right:0.06944em;">proof</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2342em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.08125em;">H</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> signed by the producer.</p>
+<p>What if a <span style="white-space: nowrap">bc‑block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span></span> were to “inline” its parent and context bft‑blocks rather than referring to them? <span style="white-space: nowrap">I.e. a bc‑block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span></span> with <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">parent_bft</span></span></span></span></span> <em>referring to</em> <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord mathsf" style="margin-right:0.06944em;">proof</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2342em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6828em;"><span style="top:-2.786em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> signed for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9463em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">pubkey</span></span></span></span></span>,</span> would instead <span style="white-space: nowrap">
+<em>literally include</em></span> (either in the header or the coinbase transaction) <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord mathsf" style="margin-right:0.06944em;">proof</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2342em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6828em;"><span style="top:-2.786em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> signed <span style="white-space: nowrap">for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9463em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">pubkey</span></span></span></span></span> —</span> and similarly <span style="white-space: nowrap">for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">context_bft</span></span></span></span></span>.</span></p>
+<p>It would still be necessary to have the message type that the proposer/producer previously used to submit a notarized bft‑block. (It cannot be merged with a bc‑block announcement: the producer of a new block is not in general the producer of its parent, and their incentives may differ; also we cannot wait until a new block is found before publishing the previous notarization.) It would also still be necessary for Crosslink nodes to keep track of notarizations that have not made it into any bc‑block. Nevertheless, this is a potential simplification.</p>
+<p>Note that unless notarization proofs are particularly short and constant-length, it would not be appropriate to include them in the bc‑block headers, and so they would have to go into the coinbase transaction or another similar variable-length data structure. In that case we would still have an indirection to obtain the bft‑block information; it would just be merged with the indirection to obtain a coinbase transaction (or similar) — which is already needed in order to check validity of the bc‑block.</p>
+<p>As discussed under <a href="design/crosslink/potential-changes.html#recommended-recording-more-info-about-the-bftchain-in-bcblocks">Recording more info about the bft‑chain in bc‑blocks</a> above, we might in any case want to record information about other proposed and notarized bft‑blocks, and the data structure needed for this would necessarily be variable-length. The complexity burden of doing so would be shared between these two changes.</p>
+<p>It would be possible to save some space in headers (while keeping them fixed-length), by inlining only one of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">parent_bft</span></span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathsf">.</span><span class="mord"><span class="mord mathsf">context_bft</span></span></span></span></span> in the header and keeping the other as a hash. As discussed under “What are the caveats?” above, the only reason for the two bft‑blocks referred to by these fields to be different, is that the bc‑block producer has observed a violation of <strong>Final Agreement</strong> <span style="white-space: nowrap">in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</span> In that case, we can include an inlining of the other <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathsf">.</span><span class="mord"><span class="mord"><span class="mord">∗</span></span><span class="mord mathsf">_bft</span></span></span></span></span> block,</span> and any other information needed to prove that a violation of <strong>Final Agreement</strong> has occurred, in a variable-length overflow structure.</p>
+<p>Pros:</p>
+<ul>
+<li>No additional mechanism or messages are needed to obtain bft‑blocks given their hashes.</li>
+<li>It could be a performance/latency improvement to not have to separately fetch bft‑blocks.</li>
+</ul>
+<p>Con:</p>
+<ul>
+<li>Additional complexity of the variable-length overflow mechanism suggested above, if it is used.</li>
+<li>Assumes that notarization proofs are not too large.</li>
+</ul>
+<h2 id="linearity-and-last-final-snapshot-rules"><a class="header" href="#linearity-and-last-final-snapshot-rules">Linearity and Last Final Snapshot rules</a></h2>
+<p>A potential simplification can be obtained by combining the following two ideas:</p>
+<ul>
+<li>Str4d suggested that the snapshot of each bft‑block should descend from the snapshot of its parent bft‑block.</li>
+<li>Nate suggested that each bc‑block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> should descend from <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.1em;"></span><span class="mclose">)</span></span></span></span>.</li>
+</ul>
+<p>Str4d’s suggestion can be written as:</p>
+<p><strong>Linearity rule:</strong> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0972em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span>.</p>
+<p>Notice that this implies the existing <strong>Increasing Score rule</strong> in Crosslink 1, because score necessarily increases within a bc‑valid‑chain. Therefore it would in practice be a replacement for the <strong>Increasing Score rule</strong>. It does not imply the <strong>Increasing Tip Score rule</strong> discussed <a href="design/crosslink/potential-changes.html#changing-the-increasing-score-rule-to-require-the-score-of-the-tip-rather-than-the-score-of-the-snapshot-to-increase">above</a>, and in fact it could make sense to enforce both the <a href="design/crosslink/potential-changes.html#linearity-rule"><strong>Linearity rule</strong></a> and the <strong>Increasing Tip Score rule</strong>.</p>
+<p>The <a href="design/crosslink/potential-changes.html#linearity-rule"><strong>Linearity rule</strong></a> implies that it is no longer possible for a bft‑valid‑chain to snapshot a bc‑chain that rolls back relative to the previous snapshot. This makes it unnecessary to sanitize <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathsf">LO</span><span class="mord"><span class="mord mathsf">G</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathsf mtight">fin</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span>: the sequence of snapshots considered by <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5714em;"></span><span class="mord text"><span class="mord textsf">san-ctx</span></span></span></span></span> is linear, and so the “sanitization” would just return the transactions in the last snapshot.</p>
+<p>To remove the need to sanitize <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathsf">LO</span><span class="mord"><span class="mord mathsf">G</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathsf mtight">bda</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span> as well, we need a further modification to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>. Recall that in Crosslink 1 we define: <span class="katex-display fleqn"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">fin</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord textsf">bda-ctx</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">μ</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mrel">:=</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mrel">:=</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">[</span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mord text"><span class="mord"> for </span></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bft</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)]</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord textsf">san-ctx</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathrm">fin</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">∣∣</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em;"><span style="top:-2.3987em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.1809em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3013em;"><span></span></span></span></span></span></span><span class="mclose">])</span><span class="mord">.</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span></span></span></span></span>
+The <a href="design/crosslink/potential-changes.html#linearity-rule"><strong>Linearity rule</strong></a> ensures that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathrm">fin</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span></span></span> is a linear sequence of snapshots, but for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0836em;vertical-align:-0.3013em;"></span><span class="mord"><span class="mord mathrm">fin</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">∣∣</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em;"><span style="top:-2.3987em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.1809em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3013em;"><span></span></span></span></span></span></span><span class="mclose">]</span></span></span></span> to be linear, we also need <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">last</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathrm">fin</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">))</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0331em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.1em;"></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0836em;vertical-align:-0.3013em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em;"><span style="top:-2.3987em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.1809em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3013em;"><span></span></span></span></span></span></span></span></span></span>. In order for this to hold for any choice of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span></span></span></span> with <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7804em;vertical-align:-0.136em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8304em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>, we require the strongest version of this condition with <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>, i.e. <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0331em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.1em;"></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0331em;vertical-align:-0.2831em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span>.</p>
+<p>Since we can only enforce that this holds for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0331em;vertical-align:-0.2831em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span> by enforcing that it holds for an arbitrary bc‑valid‑block <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span>, the rule becomes:</p>
+<p><strong>Last Final Snapshot rule:</strong> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.1em;"></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">⪯</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span>.</p>
+<p>This is exactly Nate’s suggestion discussed in <a href="design/crosslink/questions.html">Questions about Crosslink</a>. In that document we argued against this rule, but that argument was made in the context of a protocol without the <a href="design/crosslink/./construction.html##linearity-rule"><strong>Linearity rule</strong></a> (and originally, even without the <strong>Increasing Score rule</strong>).</p>
+<p>Combining the <a href="design/crosslink/./construction.html#linearity-rule"><strong>Linearity rule</strong></a> and <a href="design/crosslink/./construction.html#last-final-snapshot-rule"><strong>Last Final Snapshot rule</strong></a>, on the other hand, completely eliminates the need for sanitization. This could be a huge simplification — and potentially safer, since it would avoid breaking assumptions that may be made by existing Zcash node implementations and other consumers of the Zcash block chain.</p>
+<p>To spell out the resulting simplifications to the definitions of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span> and \LOG^t_{\bda,\mu,i}, we would just have: 
+\begin{array}{rcl}
+\LOG_{\fin,i}^t &amp;:=&amp; \snapshotlf{\ch_i^t \trunc_{\bc}^\sigma} \
+\LOG_{\bda,\mu,i}^t &amp;:=&amp; \ch_i^t \trunc_{\bc}^\mu
+\end{array}</p>
+<p>Here it is no longer necessary to define <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.261em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">LOG</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8779em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">fin</span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span> and \LOG^t_{\bda,\mu,i} as sequences of transactions, since the final and bounded-available chains are both just bc‑valid‑chains.</p>
+<p>The definition of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord textsf">finality-depth</span></span></span></span></span> in the <a href="design/crosslink/./construction.html#finality-depth-rule"><strong>Finality depth rule</strong></a> becomes much simpler: <span class="katex-display fleqn"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">finality-depth</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">height</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">height</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">snapshot</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">LF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen"><span class="mopen">⌈</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.1em;"></span><span class="mclose">))</span></span></span></span></span> As before, either <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">finality-depth</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span> or <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textsf">is-stalled-block</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span></span></span>.</span></p>
+<p>Avoiding sanitization also means that the bug we described in Snap‑and‑Chat, that could prevent spending outputs from a snapshotted chain after a <span style="white-space: nowrap"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>-block</span> rollback, cannot occur by construction. That is, the changes in <a href="design/crosslink/./construction.html#%CE%A0bc-contextual-validity"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> contexual validity</a> relative to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> are not needed any more.</p>
+<p>This almost seems too simple, and indeed we should be skeptical, because the security analysis essentially has to be redone. The reason why Snap‑and‑Chat didn’t take this approach is that it requires a more complicated argument to show that it is reasonable to believe in the safety assumptions of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> whenever it is reasonable to believe in the corresponding assumptions for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">origbc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>. This is because the ... We will need to do some work to show that the changes are benign.</p>
+<h3 id="security-analysis"><a class="header" href="#security-analysis">Security Analysis</a></h3>
+<p>The key observation needed for this analysis is that neither the <a href="design/crosslink/potential-changes.html#linearity-rule"><strong>Linearity rule</strong></a> nor the <strong>Last Final Snapshot rule</strong> affect the evolution of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">bc</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> <em>unless</em> we are in a situation where its <strong>Prefix Consistency</strong> or <strong>Prefix Agreement</strong> properties would be violated.</p>
+<p>This implies that any safety property that we can prove given <strong>Prefix Consistency</strong> plus ****.</p>
+<div style="break-before: page; page-break-before: always;"></div><link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.12.0/dist/katex.min.css" integrity="sha384-AfEj0r4/OFrOo5t7NnNe46zW/tFgW6x/bCJG8FqQCEo3+Aro6EYUG4+cU+KJWu/X" crossorigin="anonymous">
+<h1 id="references"><a class="header" href="#references">References</a></h1>
+<h2 id="ebb-and-flow-protocols"><a class="header" href="#ebb-and-flow-protocols">Ebb-and-Flow Protocols</a></h2>
+<p><a href="https://arxiv.org/abs/2009.04987">Ebb-and-Flow Protocols: A Resolution of the Availability-Finality Dilemma</a></p>
+<div style="break-before: page; page-break-before: always;"></div><link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.12.0/dist/katex.min.css" integrity="sha384-AfEj0r4/OFrOo5t7NnNe46zW/tFgW6x/bCJG8FqQCEo3+Aro6EYUG4+cU+KJWu/X" crossorigin="anonymous">
+<h1 id="version-history"><a class="header" href="#version-history">Version History</a></h1>
+<h2 id="020---crosslink-2"><a class="header" href="#020---crosslink-2">0.2.0 - Crosslink 2</a></h2>
+<p>This version updated the <a href="./design/crosslink/construction.html">Crosslink construction</a> and its <a href="./design/crosslink/security-analysis.html">security analysis</a> to version 2.</p>
+<h2 id="010---introducing-crosslink"><a class="header" href="#010---introducing-crosslink">0.1.0 - Introducing Crosslink</a></h2>
+<p><a href="https://github.com/Electric-Coin-Company/tfl-book/milestone/1?closed=1">Issue Tracking</a></p>
+<p>This version introduced the <a href="./terminology.html#crosslink">Crosslink construction</a> by linking to an external write-up. It transitioned away from the implicit design presented at Zcon4 to a rigorously defined <a href="./terminology.html#definition-hybrid-construction">hybrid construction</a>.</p>
+<h2 id="prehistory---zcon4-presentation"><a class="header" href="#prehistory---zcon4-presentation">Prehistory - Zcon4 Presentation</a></h2>
+<p>Prior to the initial release of this book, a design sketch was presented in <a href="https://www.youtube.com/watch?v=qhMzMYeEPMM&amp;list=PL40dyJ0UYTLII7oQRQmNOFf0d2iKT35tL&amp;index=19">Interactive Design of a Zcash Trailing Finality Layer (Hybrid PoW+PoS) by Nate Wilcox: Zcon4</a>.</p>
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+                <ol class="chapter"><li class="chapter-item expanded "><a href="introduction.html"><strong aria-hidden="true">1.</strong> Introduction</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="introduction/a-path-to-pos-zcash.html"><strong aria-hidden="true">1.1.</strong> A Path to Proof-of-Stake Zcash</a></li><li class="chapter-item expanded "><a href="introduction/trailing-finality-layer-in-a-nutshell.html"><strong aria-hidden="true">1.2.</strong> Trailing Finality Layer in a Nutshell</a></li><li class="chapter-item expanded "><a href="introduction/status-and-next-steps.html"><strong aria-hidden="true">1.3.</strong> Status and Next Steps</a></li><li class="chapter-item expanded "><a href="introduction/motivating-finality.html"><strong aria-hidden="true">1.4.</strong> Motivating Finality</a></li><li class="chapter-item expanded "><a href="introduction/get-involved.html"><strong aria-hidden="true">1.5.</strong> Get Involved</a></li></ol></li><li class="chapter-item expanded "><a href="terminology.html"><strong aria-hidden="true">2.</strong> Terminology</a></li><li class="chapter-item expanded "><a href="design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/design-at-a-glance.html"><strong aria-hidden="true">3.1.</strong> Design at a Glance</a></li><li class="chapter-item expanded "><a href="design/goals.html"><strong aria-hidden="true">3.2.</strong> Design Goals</a></li><li class="chapter-item expanded "><a href="design/crosslink.html"><strong aria-hidden="true">3.3.</strong> Crosslink</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/crosslink/the-arguments-for-bounded-availability-and-finality-overrides.html"><strong aria-hidden="true">3.3.1.</strong> The Arguments for Bounded Availability and Finality Overrides</a></li><li class="chapter-item expanded "><a href="design/crosslink/notes-on-snap-and-chat.html"><strong aria-hidden="true">3.3.2.</strong> Notes On Snap‑and‑Chat</a></li><li class="chapter-item expanded "><a href="design/crosslink/construction.html"><strong aria-hidden="true">3.3.3.</strong> The Crosslink 2 Construction</a></li><li class="chapter-item expanded "><a href="design/crosslink/security-analysis.html"><strong aria-hidden="true">3.3.4.</strong> Security Analysis of Crosslink 2</a></li><li class="chapter-item expanded "><a href="design/crosslink/questions.html"><strong aria-hidden="true">3.3.5.</strong> Questions About Crosslink</a></li><li class="chapter-item expanded "><a href="design/crosslink/potential-changes.html"><strong aria-hidden="true">3.3.6.</strong> Potential Changes to Crosslink</a></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">3.4.</strong> Model Code Architecture</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">3.5.</strong> Network Architecture</div></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">4.</strong> Protocol Specification</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">4.1.</strong> Abstract Protocol</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">4.1.1.</strong> Token Dynamics</div></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">4.2.</strong> Concrete Protocol</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">4.2.1.</strong> Token Dynamics</div></li></ol></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.</strong> Security</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.</strong> Abstract Analysis</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.1.</strong> Ebb-and-Flow analysis</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.2.</strong> Subprotocol Compromise Analysis</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.3.</strong> Model Code Architecture Analysis</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.4.</strong> Network Architecture Analysis</div></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.2.</strong> Concrete Analysis</div></li></ol></li><li class="chapter-item expanded "><a href="references.html" class="active"><strong aria-hidden="true">6.</strong> References</a></li><li class="chapter-item expanded "><a href="version-history.html"><strong aria-hidden="true">7.</strong> Version History</a></li></ol>
+            </div>
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+                            <i class="fa fa-search"></i>
+                        </button>
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+
+                    <h1 class="menu-title">Zcash Trailing Finality Layer - v0.1.0</h1>
+
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+                        <a href="print.html" title="Print this book" aria-label="Print this book">
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+                <!-- Apply ARIA attributes after the sidebar and the sidebar toggle button are added to the DOM -->
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+                    document.getElementById('sidebar-toggle').setAttribute('aria-expanded', sidebar === 'visible');
+                    document.getElementById('sidebar').setAttribute('aria-hidden', sidebar !== 'visible');
+                    Array.from(document.querySelectorAll('#sidebar a')).forEach(function(link) {
+                        link.setAttribute('tabIndex', sidebar === 'visible' ? 0 : -1);
+                    });
+                </script>
+
+                <div id="content" class="content">
+                    <main>
+                        <link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.12.0/dist/katex.min.css" integrity="sha384-AfEj0r4/OFrOo5t7NnNe46zW/tFgW6x/bCJG8FqQCEo3+Aro6EYUG4+cU+KJWu/X" crossorigin="anonymous">
+<h1 id="references"><a class="header" href="#references">References</a></h1>
+<h2 id="ebb-and-flow-protocols"><a class="header" href="#ebb-and-flow-protocols">Ebb-and-Flow Protocols</a></h2>
+<p><a href="https://arxiv.org/abs/2009.04987">Ebb-and-Flow Protocols: A Resolution of the Availability-Finality Dilemma</a></p>
+
+                    </main>
+
+                    <nav class="nav-wrapper" aria-label="Page navigation">
+                        <!-- Mobile navigation buttons -->
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+        <script>
+            window.playground_copyable = true;
+        </script>
+
+
+        <script src="elasticlunr.min.js"></script>
+        <script src="mark.min.js"></script>
+        <script src="searcher.js"></script>
+
+        <script src="clipboard.min.js"></script>
+        <script src="highlight.js"></script>
+        <script src="book.js"></script>
+
+        <!-- Custom JS scripts -->
+
+
+    </div>
+    </body>
+</html>
diff --git a/docs/searcher.js b/docs/searcher.js
new file mode 100644
index 0000000..d2b0aee
--- /dev/null
+++ b/docs/searcher.js
@@ -0,0 +1,483 @@
+"use strict";
+window.search = window.search || {};
+(function search(search) {
+    // Search functionality
+    //
+    // You can use !hasFocus() to prevent keyhandling in your key
+    // event handlers while the user is typing their search.
+
+    if (!Mark || !elasticlunr) {
+        return;
+    }
+
+    //IE 11 Compatibility from https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/String/startsWith
+    if (!String.prototype.startsWith) {
+        String.prototype.startsWith = function(search, pos) {
+            return this.substr(!pos || pos < 0 ? 0 : +pos, search.length) === search;
+        };
+    }
+
+    var search_wrap = document.getElementById('search-wrapper'),
+        searchbar = document.getElementById('searchbar'),
+        searchbar_outer = document.getElementById('searchbar-outer'),
+        searchresults = document.getElementById('searchresults'),
+        searchresults_outer = document.getElementById('searchresults-outer'),
+        searchresults_header = document.getElementById('searchresults-header'),
+        searchicon = document.getElementById('search-toggle'),
+        content = document.getElementById('content'),
+
+        searchindex = null,
+        doc_urls = [],
+        results_options = {
+            teaser_word_count: 30,
+            limit_results: 30,
+        },
+        search_options = {
+            bool: "AND",
+            expand: true,
+            fields: {
+                title: {boost: 1},
+                body: {boost: 1},
+                breadcrumbs: {boost: 0}
+            }
+        },
+        mark_exclude = [],
+        marker = new Mark(content),
+        current_searchterm = "",
+        URL_SEARCH_PARAM = 'search',
+        URL_MARK_PARAM = 'highlight',
+        teaser_count = 0,
+
+        SEARCH_HOTKEY_KEYCODE = 83,
+        ESCAPE_KEYCODE = 27,
+        DOWN_KEYCODE = 40,
+        UP_KEYCODE = 38,
+        SELECT_KEYCODE = 13;
+
+    function hasFocus() {
+        return searchbar === document.activeElement;
+    }
+
+    function removeChildren(elem) {
+        while (elem.firstChild) {
+            elem.removeChild(elem.firstChild);
+        }
+    }
+
+    // Helper to parse a url into its building blocks.
+    function parseURL(url) {
+        var a =  document.createElement('a');
+        a.href = url;
+        return {
+            source: url,
+            protocol: a.protocol.replace(':',''),
+            host: a.hostname,
+            port: a.port,
+            params: (function(){
+                var ret = {};
+                var seg = a.search.replace(/^\?/,'').split('&');
+                var len = seg.length, i = 0, s;
+                for (;i<len;i++) {
+                    if (!seg[i]) { continue; }
+                    s = seg[i].split('=');
+                    ret[s[0]] = s[1];
+                }
+                return ret;
+            })(),
+            file: (a.pathname.match(/\/([^/?#]+)$/i) || [,''])[1],
+            hash: a.hash.replace('#',''),
+            path: a.pathname.replace(/^([^/])/,'/$1')
+        };
+    }
+    
+    // Helper to recreate a url string from its building blocks.
+    function renderURL(urlobject) {
+        var url = urlobject.protocol + "://" + urlobject.host;
+        if (urlobject.port != "") {
+            url += ":" + urlobject.port;
+        }
+        url += urlobject.path;
+        var joiner = "?";
+        for(var prop in urlobject.params) {
+            if(urlobject.params.hasOwnProperty(prop)) {
+                url += joiner + prop + "=" + urlobject.params[prop];
+                joiner = "&";
+            }
+        }
+        if (urlobject.hash != "") {
+            url += "#" + urlobject.hash;
+        }
+        return url;
+    }
+    
+    // Helper to escape html special chars for displaying the teasers
+    var escapeHTML = (function() {
+        var MAP = {
+            '&': '&amp;',
+            '<': '&lt;',
+            '>': '&gt;',
+            '"': '&#34;',
+            "'": '&#39;'
+        };
+        var repl = function(c) { return MAP[c]; };
+        return function(s) {
+            return s.replace(/[&<>'"]/g, repl);
+        };
+    })();
+    
+    function formatSearchMetric(count, searchterm) {
+        if (count == 1) {
+            return count + " search result for '" + searchterm + "':";
+        } else if (count == 0) {
+            return "No search results for '" + searchterm + "'.";
+        } else {
+            return count + " search results for '" + searchterm + "':";
+        }
+    }
+    
+    function formatSearchResult(result, searchterms) {
+        var teaser = makeTeaser(escapeHTML(result.doc.body), searchterms);
+        teaser_count++;
+
+        // The ?URL_MARK_PARAM= parameter belongs inbetween the page and the #heading-anchor
+        var url = doc_urls[result.ref].split("#");
+        if (url.length == 1) { // no anchor found
+            url.push("");
+        }
+
+        // encodeURIComponent escapes all chars that could allow an XSS except
+        // for '. Due to that we also manually replace ' with its url-encoded
+        // representation (%27).
+        var searchterms = encodeURIComponent(searchterms.join(" ")).replace(/\'/g, "%27");
+
+        return '<a href="' + path_to_root + url[0] + '?' + URL_MARK_PARAM + '=' + searchterms + '#' + url[1]
+            + '" aria-details="teaser_' + teaser_count + '">' + result.doc.breadcrumbs + '</a>'
+            + '<span class="teaser" id="teaser_' + teaser_count + '" aria-label="Search Result Teaser">' 
+            + teaser + '</span>';
+    }
+    
+    function makeTeaser(body, searchterms) {
+        // The strategy is as follows:
+        // First, assign a value to each word in the document:
+        //  Words that correspond to search terms (stemmer aware): 40
+        //  Normal words: 2
+        //  First word in a sentence: 8
+        // Then use a sliding window with a constant number of words and count the
+        // sum of the values of the words within the window. Then use the window that got the
+        // maximum sum. If there are multiple maximas, then get the last one.
+        // Enclose the terms in <em>.
+        var stemmed_searchterms = searchterms.map(function(w) {
+            return elasticlunr.stemmer(w.toLowerCase());
+        });
+        var searchterm_weight = 40;
+        var weighted = []; // contains elements of ["word", weight, index_in_document]
+        // split in sentences, then words
+        var sentences = body.toLowerCase().split('. ');
+        var index = 0;
+        var value = 0;
+        var searchterm_found = false;
+        for (var sentenceindex in sentences) {
+            var words = sentences[sentenceindex].split(' ');
+            value = 8;
+            for (var wordindex in words) {
+                var word = words[wordindex];
+                if (word.length > 0) {
+                    for (var searchtermindex in stemmed_searchterms) {
+                        if (elasticlunr.stemmer(word).startsWith(stemmed_searchterms[searchtermindex])) {
+                            value = searchterm_weight;
+                            searchterm_found = true;
+                        }
+                    };
+                    weighted.push([word, value, index]);
+                    value = 2;
+                }
+                index += word.length;
+                index += 1; // ' ' or '.' if last word in sentence
+            };
+            index += 1; // because we split at a two-char boundary '. '
+        };
+
+        if (weighted.length == 0) {
+            return body;
+        }
+
+        var window_weight = [];
+        var window_size = Math.min(weighted.length, results_options.teaser_word_count);
+
+        var cur_sum = 0;
+        for (var wordindex = 0; wordindex < window_size; wordindex++) {
+            cur_sum += weighted[wordindex][1];
+        };
+        window_weight.push(cur_sum);
+        for (var wordindex = 0; wordindex < weighted.length - window_size; wordindex++) {
+            cur_sum -= weighted[wordindex][1];
+            cur_sum += weighted[wordindex + window_size][1];
+            window_weight.push(cur_sum);
+        };
+
+        if (searchterm_found) {
+            var max_sum = 0;
+            var max_sum_window_index = 0;
+            // backwards
+            for (var i = window_weight.length - 1; i >= 0; i--) {
+                if (window_weight[i] > max_sum) {
+                    max_sum = window_weight[i];
+                    max_sum_window_index = i;
+                }
+            };
+        } else {
+            max_sum_window_index = 0;
+        }
+
+        // add <em/> around searchterms
+        var teaser_split = [];
+        var index = weighted[max_sum_window_index][2];
+        for (var i = max_sum_window_index; i < max_sum_window_index+window_size; i++) {
+            var word = weighted[i];
+            if (index < word[2]) {
+                // missing text from index to start of `word`
+                teaser_split.push(body.substring(index, word[2]));
+                index = word[2];
+            }
+            if (word[1] == searchterm_weight) {
+                teaser_split.push("<em>")
+            }
+            index = word[2] + word[0].length;
+            teaser_split.push(body.substring(word[2], index));
+            if (word[1] == searchterm_weight) {
+                teaser_split.push("</em>")
+            }
+        };
+
+        return teaser_split.join('');
+    }
+
+    function init(config) {
+        results_options = config.results_options;
+        search_options = config.search_options;
+        searchbar_outer = config.searchbar_outer;
+        doc_urls = config.doc_urls;
+        searchindex = elasticlunr.Index.load(config.index);
+
+        // Set up events
+        searchicon.addEventListener('click', function(e) { searchIconClickHandler(); }, false);
+        searchbar.addEventListener('keyup', function(e) { searchbarKeyUpHandler(); }, false);
+        document.addEventListener('keydown', function(e) { globalKeyHandler(e); }, false);
+        // If the user uses the browser buttons, do the same as if a reload happened
+        window.onpopstate = function(e) { doSearchOrMarkFromUrl(); };
+        // Suppress "submit" events so the page doesn't reload when the user presses Enter
+        document.addEventListener('submit', function(e) { e.preventDefault(); }, false);
+
+        // If reloaded, do the search or mark again, depending on the current url parameters
+        doSearchOrMarkFromUrl();
+    }
+    
+    function unfocusSearchbar() {
+        // hacky, but just focusing a div only works once
+        var tmp = document.createElement('input');
+        tmp.setAttribute('style', 'position: absolute; opacity: 0;');
+        searchicon.appendChild(tmp);
+        tmp.focus();
+        tmp.remove();
+    }
+    
+    // On reload or browser history backwards/forwards events, parse the url and do search or mark
+    function doSearchOrMarkFromUrl() {
+        // Check current URL for search request
+        var url = parseURL(window.location.href);
+        if (url.params.hasOwnProperty(URL_SEARCH_PARAM)
+            && url.params[URL_SEARCH_PARAM] != "") {
+            showSearch(true);
+            searchbar.value = decodeURIComponent(
+                (url.params[URL_SEARCH_PARAM]+'').replace(/\+/g, '%20'));
+            searchbarKeyUpHandler(); // -> doSearch()
+        } else {
+            showSearch(false);
+        }
+
+        if (url.params.hasOwnProperty(URL_MARK_PARAM)) {
+            var words = decodeURIComponent(url.params[URL_MARK_PARAM]).split(' ');
+            marker.mark(words, {
+                exclude: mark_exclude
+            });
+
+            var markers = document.querySelectorAll("mark");
+            function hide() {
+                for (var i = 0; i < markers.length; i++) {
+                    markers[i].classList.add("fade-out");
+                    window.setTimeout(function(e) { marker.unmark(); }, 300);
+                }
+            }
+            for (var i = 0; i < markers.length; i++) {
+                markers[i].addEventListener('click', hide);
+            }
+        }
+    }
+    
+    // Eventhandler for keyevents on `document`
+    function globalKeyHandler(e) {
+        if (e.altKey || e.ctrlKey || e.metaKey || e.shiftKey || e.target.type === 'textarea' || e.target.type === 'text') { return; }
+
+        if (e.keyCode === ESCAPE_KEYCODE) {
+            e.preventDefault();
+            searchbar.classList.remove("active");
+            setSearchUrlParameters("",
+                (searchbar.value.trim() !== "") ? "push" : "replace");
+            if (hasFocus()) {
+                unfocusSearchbar();
+            }
+            showSearch(false);
+            marker.unmark();
+        } else if (!hasFocus() && e.keyCode === SEARCH_HOTKEY_KEYCODE) {
+            e.preventDefault();
+            showSearch(true);
+            window.scrollTo(0, 0);
+            searchbar.select();
+        } else if (hasFocus() && e.keyCode === DOWN_KEYCODE) {
+            e.preventDefault();
+            unfocusSearchbar();
+            searchresults.firstElementChild.classList.add("focus");
+        } else if (!hasFocus() && (e.keyCode === DOWN_KEYCODE
+                                || e.keyCode === UP_KEYCODE
+                                || e.keyCode === SELECT_KEYCODE)) {
+            // not `:focus` because browser does annoying scrolling
+            var focused = searchresults.querySelector("li.focus");
+            if (!focused) return;
+            e.preventDefault();
+            if (e.keyCode === DOWN_KEYCODE) {
+                var next = focused.nextElementSibling;
+                if (next) {
+                    focused.classList.remove("focus");
+                    next.classList.add("focus");
+                }
+            } else if (e.keyCode === UP_KEYCODE) {
+                focused.classList.remove("focus");
+                var prev = focused.previousElementSibling;
+                if (prev) {
+                    prev.classList.add("focus");
+                } else {
+                    searchbar.select();
+                }
+            } else { // SELECT_KEYCODE
+                window.location.assign(focused.querySelector('a'));
+            }
+        }
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book introduces and specifies a Trailing Finality Layer for the Zcash network. This is version 0.1.0 of the book. This design augments the existing Zcash Proof‑of‑Work ( PoW ) network with a new consensus layer which provides trailing finality . This layer enables transactions included via PoW to become final which assures that they cannot be reverted by the protocol. This enables safer and simpler wallets and other infrastructure, and aids trust-minimized cross-chain bridges. This consensus layer uses Proof-of-Stake consensus, and enables ZEC holders to earn protocol rewards for contributing to the security of the Zcash network. By integrating a PoS layer with the current PoW Zcash protocol, this design specifies a hybrid consensus protocol dubbed PoW+TFL . The rest of this introductory chapter is aimed at a general audience interested in the context of this proposal within Zcash development, status and next steps, motivations, a primer on finality, and tips to get involved.","breadcrumbs":"Introduction » A Trailing Finality Layer for Zcash","id":"0","title":"A Trailing Finality Layer for Zcash"},"1":{"body":"The TFL design provides a possible first step in transitioning Zcash to a PoS protocol. Here we describe how a transition to PoS relates to “the Zcash roadmap” and how TFL fits into one approach to a PoS transition.","breadcrumbs":"Introduction » A Path to Proof-of-Stake Zcash » A Path to Proof-of-Stake Zcash","id":"1","title":"A Path to Proof-of-Stake Zcash"},"10":{"body":"Existing ZEC users who are primarily concerned with storing or receiving ZEC, whether private or transparent may benefit from this change in the short or medium term, because it may help lower delay times for some services, such as exchange deposits. As exchanges come to rely on the new finality guarantee, they can often reduce their deposit wait times. Other services with similar confirmation-depth-based wait times can be improved in a similar way to lower these wait times. Other than this improvement, these users should notice no other changes. In the longer term, providing finality will be useful in establishing trust‑minimized bridges to other block‑chain protocols. We anticipate this can enable better connecting ZEC to the Defi ecosystem, and with the introduction of Zcash Shielded Assets, this can enable other assets to connect to the Zcash shielded pool.","breadcrumbs":"Introduction » Trailing Finality Layer in a Nutshell » Current ZEC Users","id":"10","title":"Current ZEC Users"},"100":{"body":"See the Liveness section above.","breadcrumbs":"Design » Crosslink » Security Analysis of Crosslink 2 » More involved liveness argument","id":"100","title":"More involved liveness argument"},"101":{"body":"Warning In order to show that Crosslink 2 is at a local optimum in the security/complexity trade‑off space, for each rule we show attacks on safety and/or liveness that could be performed if that rule were omitted or simplified. Edit: some rules, e.g. the Linearity rule , only contribute heuristically to security in the analysis so far.","breadcrumbs":"Design » Crosslink » Security Analysis of Crosslink 2 » Every rule in Crosslink 2 is individually necessary","id":"101","title":"Every rule in Crosslink 2 is individually necessary"},"102":{"body":"","breadcrumbs":"Design » Crosslink » Questions About Crosslink » Questions about Crosslink","id":"102","title":"Questions about Crosslink"},"103":{"body":"Rationale: Can we outright prevent rollbacks > σ from ever appearing in Chit​ ? Info This document analyzes the effect of this rule on its own . For the effect in combination with an additional Linearity rule , see the Linearity and Last Final Snapshot rules section of Potential changes for Crosslink 2 . Daira-Emma: In a variant of Crosslink with this rule, an adversary’s strategy would be to keep the context_bft fields in its blocks as C such that B=bft-last-final(C) when the attack starts, and then fork from the bc‑best‑chain that extends snapshot(B). If its private chain falls behind the public bc‑best‑chain, it resets, just like in a conventional private mining attack. Note that the proposed rule does not prevent the adversary’s private chain from just staying at the same context_bft block. The reason is that Crosslink does not change the fork-choice rule of Πbc​. That is, even if the adversary’s chain has a context_bft that is far behind the current bft‑block, it is still allowed to become the bc‑best‑chain. (Eventually the adversary’s chain using this strategy will hit the finality gap bound of L blocks. But that must be significantly greater than σ, to avoid availability problems. So it does not prevent the adversary from performing a rollback longer than σ blocks before they hit the bound. Also, going into Stalled Mode for new blocks does not prevent the attacker’s chain from having included harmful transactions before that point.) It is possible to change the fork-choice rule, for example so that the bc‑best‑chain for a node i is required to extend snapshot(B) where B is the last final block for any bft-chain in node i’s view. This would break the current safety and liveness arguments for Crosslink. But for the sake of argument, suppose we did it anyway. The adversary’s strategy would change slightly: it resets if either its private chain falls behind the public bc‑best‑chain, or its private chain is invalidated because it forks before snapshot(B) for some new last final block B of a bft-chain. During the attack, it also attempts to impede progress of the BFT protocol as far as possible. In that case, the proposed rule still does not preclude a rollback of more than σ blocks, for several reasons: In general we can’t say anything about how many bc‑blocks are mined in any given interval, so it could be the case that more than σ blocks are mined on both the honest chain and the adversary’s chain before it would be realistically possible to go through even a single round of the BFT protocol. Nor can we say anything about how quickly those blocks are finalized, unless we enforce it, which we don’t. (In Crosslink we do enforce a finalization gap bound L, but as explained above L must be significantly greater than σ, so that doesn’t really help.) In particular, the adversary could be suppressing publication of final bft‑blocks, or attacking the liveness of the BFT protocol in other ways. An attack against BFT liveness is potentially easier than an attack against BFT safety, and it would be difficult to characterize exactly how much this rule gains you in terms of security given that (at best) it’s dependent on that. snapshot(LF(H)) will typically be at least σ+1 blocks back from H. The argument for that goes: None of the block hashes in LF(H).headers_bc can point to H because that would be a hash cycle. In a typical case where no block withholding and no other rollback (not caused by the adversary) occurs on the honestly mined chain, the proposer of the last final block before a context bft‑block that H can point to will have, at the latest, H⌈bc1​ as tip(B). Under these conditions, snapshot(LF(H)) will point, at the latest, to σ blocks before H⌈bc1​, i.e. σ+1 blocks before H. This means that by the time snapshot(LF(H)) could catch up to H, on average σ+1 block times will have occurred. So, roughly speaking, the rule that snapshot(LF(H))⪯bc​H does not usefully constrain the adversary until after σ+1 block times. Wlog let’s assume Πbc​ uses PoW: if σ is chosen reasonably, then being able to do a σ+1-block rollback at all probably requires having somewhere close to 50% of network hash rate. And so in σ+1 block times the adversary has a significant chance of being able to do the required rollback before the suggested rule “kicks in”. (It also has however much additional latency is added by the BFT protocol, which is simultaneously under attack to maximize this latency.) All that said, does the suggested rule help? First we have to ask whether it introduces any weaknesses. One potential issue is that the rule cuts both ways: if an adversarial rollback of more than σ blocks does occur, then the adversary can make a proposal that will “lock in” its success. But it can be argued that this is intended behaviour anyway: the adversary has a confirmed chain, and is entitled to propose to finalize it. What happens if Πbft​ is broken? Edit: the answer below refers to Crosslink before the Increasing Score rule was added. This could definitely be a problem. If either an adversary has a two‑thirds supermajority of validators, or Πbft​ is completely broken (e.g. by a bug in the implementation of the validation signature scheme), then they can add a final block to the bft-chain with a headers_bc field that does not satisfy the honest voting condition. They do not need this to be bft-valid (although greater vulnerability to bugs in the implementation of bft-validity are also an issue). Then they can use the proposed rule to prevent an arbitrary bc-valid-chain from being extended (e.g. by mining σ+1 blocks from Obc​ at the genesis difficulty, then getting that chain into a final bft‑block). [Edit: they would no longer be able to do this using a chain mined at a much lower difficulty, because of the Increasing Score rule.] In Crosslink, this is very carefully avoided. The only similar rule that depends on snapshot(LF(H)) and that could potentially affect liveness of Πbc​ is the Finality depth rule. But that rule always allows the alternative of producing a Stalled Mode block on the current bc‑best‑chain — and honest block producers will do so. Therefore, the effect of trying to exploit even a catastrophic break in Πbft​ in order to cause a rollback of LOGfin​, as long as Πbc​ has not also been broken, is to go into Stalled Mode. This does not mean that a break of Πbft​ is not a problem for Crosslink. In particular, an adversary that can violate safety of Πbft​ can violate safety of LOGba​ (and of LOGfin​ if there is also a σ‑block rollback in some node’s bc‑best‑chain). The difference is that a safety violation of Πbft​ can be directly observed by nodes without any chance of false positives , which is not necessarily the case for all possible attacks against Πbft​. (The attack described above does not violate safety of Πbft​; it just adds a final bft‑block with a suspiciously long Πbc​ rollback. It could alternatively have added a block with a less suspiciously long rollback, say exactly σ+1 blocks. That is, in pursuit of preventing an attack against Πbc​, we have enabled attacks against Πbft​ to achieve the same effect — precisely what Crosslink is designed to prevent.) This raises an interesting idea: if any node sees a rollback in the chain of final bft‑blocks, it could provide objective evidence of that rollback in the form of a “bft-final-vee”: two final bft‑blocks with the same parent. Similarly, if any node sees more than one third of stake vote for conflicting blocks in a given epoch, then the assumption bounding the adversary’s stake must have been violated. This evidence can be posted in a transaction to the bc-chain. In that case, any node that is synced to the bc-chain can see that the bft-chain suffered a violation of safety or of a safety assumption, without needing to have seen that violation itself. This can be generalized to allow other proofs of flaws in Πbft​. Optionally, a bc-chain that has posted such a proof could be latched into Stalled Mode until manual intervention can occur. (Obviously we need to make sure that this cannot be abused for denial-of-service.) This is now described in Potential changes to Crosslink . Okay, but is it a good idea to make that change to the fork-choice rule anyway? Probably not. I don’t know how to repair the safety and liveness arguments. The change was that the bc‑best‑chain for a node i would be required to extend snapshot(B) where B is the last final bft‑block in node i’s view. From the point of view of any modular analysis that treats Πbft​ as potentially subverted, we cannot say anything useful about snapshot(B). It seems as though any repair would have to assume much more about the BFT protocol than is desirable. In general, changes to fork‑choice rules are tricky; it was a fork-choice rule problem that allowed the liveness attack against Casper FFG described in [ NTT2020 , Appendix E]. What if validators who see that a long rollback occurred, refuse to vote for it? Yep that is allowed. The rule is “An honest validator will only vote for a proposal P if ...” (not if‑and‑only‑if). If an honest validator sees a “good” reason not to vote for a proposal, including reasons based on out‑of‑band information, they should not. The Complementarity argument made in The Argument for Bounded Availability and Finality Overrides actually depends on this. Obviously, it may affect BFT liveness (and that’s okay). The only reason why we don’t make this part of the voting condition is that it’s a stateful rule. A new validator could come along and wouldn’t have the state needed to enforce it. Perhaps that could be fixed.","breadcrumbs":"Design » Crosslink » Questions About Crosslink » Why don’t we have a bc‑block‑validity rule snapshot(LF(H)) ⪯bc H ?","id":"103","title":"Why don’t we have a bc‑block‑validity rule snapshot(LF(H)) ⪯bc H ?"},"104":{"body":"This page documents suggestions that have not had the same attention to security analysis as the Crosslink 2 construction . Some of them are broken. Some of them also increase the complexity of the protocol (while some simplify it or have a mixed effect on complexity), and so we need to consider the security/complexity trade‑off of each suggestion before we could include it. This page is out-of-date This page has not yet been updated for the changes from Crosslink 1 to Crosslink 2.","breadcrumbs":"Design » Crosslink » Potential Changes to Crosslink » Potential Changes to Crosslink","id":"104","title":"Potential Changes to Crosslink"},"105":{"body":"","breadcrumbs":"Design » Crosslink » Potential Changes to Crosslink » Attempts to improve safety or to simplify the protocol","id":"105","title":"Attempts to improve safety or to simplify the protocol"},"106":{"body":"We can allow honest bc‑block‑producers to record information about every proposed and notarized bft‑block, rather than just the one in the context_bft field. Duplicate information that has already been given in an ancestor bc‑block would be omitted. This would automatically expose the following shenanigans to public view (as long as enough bc‑block‑proposers are honest, which is already assumed): any attempt to double‑propose in the same epoch; any successful attempt to double‑notarize. We could also expose attempts to double‑vote. Note that double‑proposal and double‑voting could be a sign that a proposer or validator’s private key is compromised, rather than that it belongs to the adversary per se. However, the security analysis must treat such a proposer/validator as non‑honest in any case.","breadcrumbs":"Design » Crosslink » Potential Changes to Crosslink » [Recommended] Recording more info about the bft‑chain in bc‑blocks","id":"106","title":"[Recommended] Recording more info about the bft‑chain in bc‑blocks"},"107":{"body":"The current Increasing Score rule concerns the score of the snapshot: Increasing Snapshot Score rule: Either score(snapshot(B⌈bft1​))<score(snapshot(B)) or snapshot(B⌈bft1​)=snapshot(B). We could instead require the score of tip(B)=B.headers_bc[σ−1] to increase: Increasing Tip Score rule: Either score(tip(B⌈bft1​))<score(tip(B)) or tip(B⌈bft1​)=tip(B). Pros: This more directly reflects the fork‑choice rule in Πbc​. In Crosslink 1, an honest bft‑proposer uses its bc‑best‑chain tip with the highest score provided that it is consistent with the Increasing Snapshot Score rule . This change removes the caveat, simplifying honest bft‑proposer behaviour. As a result of removing that caveat, we always know about an honest bft‑proposer’s bc‑best‑chain. Con: The score of the snapshot would not necessarily increase. As a result, it is technically possible that the new snapshot can be an ancestor of a previous snapshot. Whether this can actually happen depends on the value of σ and the difficulty adjustment rule. This causes no particular harm other than adding a corner case in ledger sanitization, but is inelegant. This con is removed if we either use the Combined Increasing Score rule variant described below, or we also apply the “ Making bc‑rollbacks more difficult ” change in the next section. Apart from the above con, the original motivations for the Increasing Snapshot Score rule also apply to the Increasing Tip Score rule . In particular, it still prevents potential attacks that rely on proposing a bc‑valid‑chain that forks from a much earlier block; it still limits the extent of disruption an adversary can feasibly cause to \\LOG_{\\bda}}; it still always allows a proposal to be made, which may be needed to preserve liveness of Πbft​ relative to Πorigbft​; it still prevents potential validation cost DoS attacks due to switching between snapshots with the same score. If we switch to using the Increasing Tip Score rule, then it would be more consistent for block producers to also change the tie‑breaking rule for choosing context_bft to use the tip score, i.e. score(tip(bft-last-final(C))). A variation on this suggestion effectively keeps both the Increasing Snapshot Score rule and the Increasing Tip Score rule : Combined Increasing Score rule: Either (score(snapshot(B⌈bft1​))<score(snapshot(B)) and score(tip(B⌈bft1​))<score(tip(B))), or tip(B⌈bft1​)=tip(B). Note that if tip(B⌈bft1​)=tip(B), both scores are necessarily equal. This variation does not simplify honest bft‑proposer behaviour.","breadcrumbs":"Design » Crosslink » Potential Changes to Crosslink » Changing the Increasing Score rule to require the score of the tip (rather than the score of the snapshot) to increase","id":"107","title":"Changing the Increasing Score rule to require the score of the tip (rather than the score of the snapshot) to increase"},"108":{"body":"Basic idea: Detect the case where the bc‑snapshot is rolling back, and impose a longer confirmation depth to switch to the new bc‑chain. Also temporarily stall finalization of the existing bc‑chain until the conflict has been resolved. Let baseline_snapshot be the Crosslink 1 definition of snapshot, i.e. baseline_snapshot(B)={Obc​,B.headers_bc[0]⌈bc1​,​if B.headers_bc=∅otherwise.​ When snapshot(B⌈bft1​)⪯bc​baseline_snapshot(B), we want to go into a mode where we require a longer confirmation depth Σ, say 2σ. Because we don’t know in this situation whether the old bc‑chain or the new bc‑chain will win, we stop finalizing both until a winner is clear. The simplest option is to record the state saying that we are in this mode explicitly, and add a consensus rule requiring it to be correct. That is, add an bc_is_forked field to bft‑proposals and bft‑blocks, and add a bft‑proposal and bft‑block validity rule as follows: Is Forked rule: B.bc_is_forked=(enter_forked(B) or B⌈bft1​.bc_is_forked) and not exit_forked(B) where: enter_forked(B):=snapshot(B⌈bft1​)⪯bc​baseline_snapshot(B) exit_forked(B):=score(tip(B)⌈bcΣ​)>score(snapshot(B⌈bft1​)). It is intentional that exit_forked takes precedence over enter_forked. Then redefine snapshot as follows: snapshot(B)={snapshot(B⌈bft1​),baseline_snapshot(B),​if B.bc_is_forkedotherwise.​ Since Obft​.bc_is_forked=false, the recursion will terminate. Note that there is an interaction between the Increasing Snapshot Score rule and this change: the Increasing Snapshot Score rule should arguably use baseline_snapshot instead of snapshot. The Increasing Tip Score rule , on the other hand, works fine as‑is, and so it makes sense to use both of these changes together. The combination of both changes also fixes the con discussed above for the Increasing Tip Score rule ; it ensures that the score of the snapshot must increase. Pros: If the winning chain is the chain that was first snapshotted, then there ends up being no disruption whatsoever to \\LOG_{\\bda}}. It becomes extremely difficult for an adversary with less than 50% hash power to get the finalized snapshot to switch between two competing chains more than once. Cons: It is potentially easier to cause a temporary finalization stall. An adversary could try to provoke this situation on the honest chain, either as a DoS attack, or so that its own chain that is not so encumbered can finalize more quickly than the honest chain. This does not seem like a practical attack, because such a stall can only happen when the adversary can cause a σ‑block rollback or has subverted Πbft​. The definition of snapshot becomes more complicated, and there is a risk of this complexity introducing problems. It can be argued that, unless σ is chosen too small, an adversary that can cause a σ‑block rollback likely has 50% of hash power, and therefore can cause a Σ‑block rollback. That is, increasing confirmation depth does not help beyond a certain point, and therefore (it could be argued) this change will also not help. This argument may hold for private mining attacks, but does not necessarily hold for partitioning attacks, as discussed in the next section.","breadcrumbs":"Design » Crosslink » Potential Changes to Crosslink » Making exploitation of bc‑rollbacks more difficult","id":"108","title":"Making exploitation of bc‑rollbacks more difficult"},"109":{"body":"In the case of a private mining attack, the adversary will typically conceal the existence of the overtaking chain until it can be used to cause a rollback in \\LOG_{\\bda}}. So the approach used in the previous section seems to be all we can do against such an attack. In the case of a partitioning attack, on the other hand, the adversary relies on honest nodes to do mining work on each side of the partition. This relies on the successful miners on each side knowing about their chain, but not the chain on the other side. Subtly, it does not rely on a perfect network partition. An adversary could, for example, attempt to create partitions around the most successful mining pools. Occasional leaks of information across a partition also do not necessarily foil the attack unless that information gets to a successful miner. Therefore, measures that constrain the adversary’s ability to make use of an incomplete partition can be useful. This also has the benefit of making the protocol more robust against non‑malicious incomplete partitions. Given that in such an attack the competing chains may be visible to some proposers, there is the possibility of detecting a potential rollback even before it gets snapshotted, by using the fact that previous bft‑blocks created by honest bft‑proposers have been recording the bc‑best‑chain tip σ blocks ahead. Also, depending on what proportion of validators an adversary has, they may rely on honest validators on each side to ensure that a snapshot of each chain appears in a bft‑valid block; in that case, including information about competing chains in validators' votes (see the next subsection) may be useful. It is still possible that if an adversary has several consecutive proposal slots, it can get its chain snapshotted. However, if there is an intervening slot with an honest proposer, we can potentially compare its tip with the adversary’s tip and anticipate the need to go into bc_is_forked mode. In order to get this to work, we need to propose a definition to identify bc‑chains that are competing with the current best chain, such that there is some risk of a “long” rollback to a competing chain. Let δ be a measure of how close (in terms of bc‑blocks) a competing chain’s score needs to be to that of the bc‑best‑chain, and let μ>δ be a lower bound on the rollback depth we would consider significant if the competing chain were to immediately catch up. (The condition μ>δ is necessary to avoid false positives that might only be a single‑block fork.) A node i identifies (δ,μ)‑competing chains as follows based on its current view at time t: A bc‑block is a tip if it has no known descendants. Let chit​ be node i’s bc‑best‑chain. Identify all of the tips T such that score(T)≥score(chit​⌈bcδ​) and last-common-ancestor(T,chit​)⪯bc​chit​⌈bcμ​. TODO: Details, including how to modify the enter_forked and exit_forked conditions. TODO: For now we will assume that all of the competing chain information in a bft‑block has to be checked as bc‑block‑valid in order for that block to be bft‑block‑valid. This might introduce validation DoS attacks and needs to be considered more carefully.","breadcrumbs":"Design » Crosslink » Potential Changes to Crosslink » Using tip information to detect rollbacks and partitions","id":"109","title":"Using tip information to detect rollbacks and partitions"},"11":{"body":"Users who are interested in providing finality infrastructure, or users who want to delegate ZEC towards finality, will be able to earn rewards from the protocol for doing so, while also taking on some risk to their funds (to prevent malicious abuse of the protocol). This may be an important new category of ZEC users and use cases.","breadcrumbs":"Introduction » Trailing Finality Layer in a Nutshell » Proof-of-Stake Users","id":"11","title":"Proof-of-Stake Users"},"110":{"body":"This complements the above idea by letting a validator that has seen a competing chain signal it in its signed vote. Then, as long as the adversary is reliant on some votes from honest validators that are signalling the existence of competing chains, we would go into bc_is_forked mode without relying on honest proposers to have an intervening slot. The notarization proof that appears in a bft‑block would need to be modified to preserve these signals. More precisely, it is necessary for a bft‑block B to preserve at least: the best bc‑chain that credibly competes with tip(B), if any. the best bc‑chain that does not credibly compete with tip(B) (this necessarily exists because tip(B) does not credibly compete with itself). This is also motivated by the suggested change in the next section. Enforcing this is relatively straightforward if the evidence is a SNARK. It can also be enforced with aggregate signatures even for schemes that only allow aggregation of signatures over a common message: we just collect the distinct messages (corresponding to either “no competing chain” or each distinct competing chain) and aggregate them separately.","breadcrumbs":"Design » Crosslink » Potential Changes to Crosslink » Allowing validators to signal the existence of a competing chain in their votes","id":"110","title":"Allowing validators to signal the existence of a competing chain in their votes"},"111":{"body":"Assume that votes include competing chain information as discussed above. We can assume that an honest proposer has read all of this information from its parent bft‑block. Therefore, we can require the tip score of its proposal to have at least the score of the best tip implied by that information: Let best-tip(B) be the tip mentioned in bft‑block B with the highest score. A bft‑block B “mentions” the two best tips defined in the previous section. Strong Increasing Tip Score rule: Either score(best-tip(B⌈bft1​))<score(tip(B)) or best-tip(B⌈bft1​)=tip(B). Note that this rule is really quite constraining for a potential adversary, especially in partitioning attacks. It means that if the adversary does not want to acknowledge the existence of a given chain, it cannot use any votes or build on any previous bft‑block that signals the existence of that chain. Essentially, a partitioning adversary with control over only the minimum one‑third of the stake would have to have ensure a perfectly complete partition; it could not get away with any information leakage between honest validators.","breadcrumbs":"Design » Crosslink » Potential Changes to Crosslink » Strengthening the Increasing Tip Score rule","id":"111","title":"Strengthening the Increasing Tip Score rule"},"112":{"body":"","breadcrumbs":"Design » Crosslink » Potential Changes to Crosslink » Attempts to improve finalization latency","id":"112","title":"Attempts to improve finalization latency"},"113":{"body":"The Crosslink 1 design imposes a finalization latency of at least 2σ+1 block times. Intuitively, this is because in fin(H):=[snapshot(B) for B⪯bft​bft-last-final(H⌈bcσ​.context_bft)], snapshot(bft-last-final(H⌈bcσ​.context_bft)) is at least σ+1 blocks back from H⌈bcσ​ (as argued in Questions about Crosslink 1 ), and therefore 2σ+1 blocks back from H. So the total finalization latency is σ block times + BFT overhead + (σ+1) block times + snapshot overhead. However, the snapshot headers contain information about the proposer’s bc‑best‑chain. Define LF(H):=bft-last-final(H.context_bft). Although it is not guaranteed, normally snapshot(LF(H⌈bcσ​)) will be an ancestor of H⌈bcσ​. What if we were to optimistically allow the last snapshot to be taken as S(H):={last-common-ancestor(H⌈bcσ​,tip(LF(H⌈bcσ​)),snapshot(LF(H⌈bcσ​)),​if it extends snapshot(LF(H⌈bcσ​))otherwise​? After all, we know that last-common-ancestor(H⌈bcσ​,tip(LF(H⌈bcσ​)) is confirmed. Oh, this won’t work. The problem is that we want safety of LOGfin​ not to depend on safety of Πbc​. So we cannot assume (for this purpose) that nodes see the same H⌈bcσ​.","breadcrumbs":"Design » Crosslink » Potential Changes to Crosslink » [Broken] Adjusting the last snapshot definition","id":"113","title":"[Broken] Adjusting the last snapshot definition"},"114":{"body":"What if we instead take this to be the definition of \\LOG_{\\opt}}, replacing \\LOG_{\\bda}} (\"opt\" meaning optimistic)? As stated, a malicious proposer can try to maximize the latency of \\LOG_{\\opt}} (subject to the Increasing Score rule). For example, if there exists a fork of length μ, the malicious proposer can force the latency of \\LOG_{\\opt}} to be (σ+μ+1) block times + BFT overhead. However, this can be improved by applying the idea to each bft‑block in turn after the one pointed to by the best confirmed bc‑block. Then a malicious proposer cannot do anything that it could not do anyway (keeping the finalization point at its current position). Pros: This is always more conservative in terms of safety than the current design. The latency of \\LOG_{\\opt}} will typically be σ+1 bc‑block times, rather than 2σ+1 block times. Cons: \\LOG_{\\opt}} is not dynamically available in any sense. It just has lower latency and different security characteristics. Even under optimistic conditions, \\LOG_{\\opt}} will lag slightly behind where it would be for the Crosslink 1 design, because H⌈bcσ​ will necessarily be ahead of tip(LF(H⌈bcσ​)).","breadcrumbs":"Design » Crosslink » Potential Changes to Crosslink » Replacing \\LOG_{\\bda}} with \\LOG_{\\opt}}","id":"114","title":"Replacing \\LOG_{\\bda}} with \\LOG_{\\opt}}"},"115":{"body":"The following idea is broken for safety when Πbft​ has been subverted: Info We have two potential sources of information about blocks that could plausibly be considered finalized: H⌈bcσ​ the snapshots on the chain of the last seen final bft‑block, lsf. We cannot rely only on 1. because we want assured finalization even under partition. We cannot rely only on 2. because if Πbft​ has been subverted, then the chain of final bft‑blocks could fork. But intuitively, if we combine these sources of information, using them over the Crosslink 1 finalization only when they are consistent, the resulting protocol should still be as safe as the safer of Πbft​ and Πbc​. In particular, 2. will not roll back unless Πbft​ has been subverted. If this idea were to pan out, it could improve the latency of finalization by σ block times. This approach is essentially a hybrid of Snap‑and‑Chat and Crosslink 1: the Snap‑and‑Chat construction gives a finalized ledger under the assumption that Πbft​ has not been subverted; the main crosslink idea is used to make sure that outputs from all finalized transactions are eventually spendable; safety is still only dependent on the stronger of the safety of Πbft​ and Πbc​, because we use the additional information from snapshots in final bft‑blocks only up to the point at which they agree with the best confirmed bc‑block. To explain the safety problem with this idea: suppose that Πbft​ has been subverted. In that case it is possible for a snapshot to be finalized without having being confirmed as in any honest node’s bc‑best‑chain; that is, it is possible for LOGfin​ to include transactions T from a snapshot S in bft‑block A such that S is not on the consensus bc‑best‑chain. And, because Πbft​ has been subverted, it is also possible that a conflicting final bft‑block B omits S. And so a node that has seen B will think that it is consistent with the best bc‑chain (so that its LOGfin​ does not include T but does include later transactions on the consensus bc‑best‑chain), but a node that has seen A will compute a LOGfin​ that does include T. More detailed specification of the above broken idea. Define LF(H):=bft-last-final(H.context_bft) as before. For simplicity assume that lsf extends LF(H⌈bcσ​) by only one bft‑block. (This assumption could have been removed if the idea had panned out.) Then this proposal was to consider this bc‑block as contributing the last finalized snapshot: S(H):={last-common-ancestor(H⌈bcσ​,snapshot(lsf)),snapshot(LF(H⌈bcσ​)),​if it extends snapshot(LF(H⌈bcσ​))otherwise​ There is no need for a tie‑breaking rule for 2.: if we ever see two context bft‑blocks for which the last‑final blocks are conflicting, we know that Πbft​ has been subverted, so we should stall or crash. Caveat: for a given node, H⌈bcσ​ can in theory roll back past snapshot(lsf), therefore S(H) can also roll back. It is okay if we keep state here and refuse to roll back. We should set a “crisis flag”, and unset it if at any point LF(H⌈bcσ​) extends lsf_at_crisis. (If Πbft​ is safe and live, it will.) A similar rule that would give the same result in almost all circumstances is: S(H):=⎩⎨⎧​H⌈bcσ​,snapshot(lsf),snapshot(LF(H⌈bcσ​)),​if snapshot(LF(H⌈bcσ​))⪯bc​H⌈bcσ​⪯bc​snapshot(lsf)if snapshot(LF(H⌈bcσ​))⪯bc​snapshot(lsf)⪯bc​H⌈bcσ​otherwise​","breadcrumbs":"Design » Crosslink » Potential Changes to Crosslink » [Broken] Using snapshots from the last‑seen bft‑chain when it is consistent with the bc‑best‑chain","id":"115","title":"[Broken] Using snapshots from the last‑seen bft‑chain when it is consistent with the bc‑best‑chain"},"116":{"body":"The answer given for this question at The Crosslink 2 Construction is: If this were enforced, it could be an alternative way of ensuring that every bft‑proposal snapshots a new bc‑block with a higher score than previous snapshots, potentially making the Increasing Score rule redundant. However, it would require merging bc‑block‑producers and bft‑proposers, which could have concerning knock‑on effects (such as concentrating security into fewer participants). This may have been too hasty. It is not clear that merging bc‑block‑producers and bft‑proposers actually does “concentrate security into fewer participants” in a way that can have any harmful effect. Remember, the job of a bft‑proposer in Crosslink is primarily to snapshot the bc‑best‑chain (even more so if the Increasing Tip Score rule is adopted). An honest miner by definition is claiming to build on the best chain, and miners have a strong economic incentive to do so. Therefore, it is entirely reasonable for every newly produced block to be treated as a bft‑proposal. This arguably decentralizes the task of proposing bft‑blocks more effectively than using a leader election protocol would — especially given that in a hybrid protocol we necessarily still rely on there being sufficient honest miners. [DKT2021] , for example, argues for the importance of “the complete unpredictability of who will get to propose a block next, even by the winner itself.” The main basis of this argument is that it makes subversion of the proposer significantly more difficult. A PoW protocol has that property, and most PoS protocols do not. (It is not that PoS protocols are unable to provide this property; indeed, [DKT2021] constructs a PoS protocol, “PoSAT”, that provides it.) So let’s explore this in more detail. A newly produced bc‑block would implicitly be a bft‑proposal with itself as the tip. The bc_headers field is therefore not needed. The Tail Confirmation rule goes away since its intent is automatically satisfied. This is already a significant simplification. The inner proposer signature is also not needed (since the bc‑header is self-authenticating), but the block producer would have to include a public key H.pubkey that can be used to verify its outer signature. It would sign the notarized bft‑block with the corresponding private key. This change is a wash in terms of protocol complexity. Considered as a bft‑proposal, a bc‑block needs to refer to a parent bft‑block, which requires a parent_bft field in the bc‑header. With some caveats depending on the design of Πorigbft​, it might be possible to merge this with the context_bft field, but for now we will assume that it is not merged. What are the caveats? If we are in an execution where Final Agreement holds for Πbft​, then it is possible to show that merging the two fields has no negative effect, provided that Πorigbft​ has no additional rules that could disallow it in some cases. This is because, by Final Agreement , bft-last-final(H′.context_bft)⪯⪰∗bft​bft-last-final(C) for any potential bft‑block C that the bc‑block‑producer of a new block H could choose as H.context_bft. Suppose that the bc‑block‑producer freely chooses H.parent_bft according to the desired honest behaviour for a bft‑proposer in Πbft​, and then chooses C to be the same block (which is always reasonable as long as it is allowed). In the case bft-last-final(H′.context_bft)⪯∗bft​bft-last-final(C), we are done, because this choice of H.context_bft is allowed by the Extension rule . In the case bft-last-final(H′.context_bft)⪲∗bft​bft-last-final(C), we can argue that H′.context_bft would be a better choice than C for H.parent_bft as well as for H.context_bft, because it has a later final ancestor. This is where the argument might fall down if Πorigbft​ (and therefore Πbft​) has any additional rules that could disallow this choice. For now let’s suppose that situtation does not arise, but it is one of the caveats. Another potential problem is that in an execution where Final Agreement does not hold for Πbft​, we can no longer infer that either bft-last-final(H′.context_bft)⪯∗bft​bft-last-final(C) or bft-last-final(H′.context_bft)⪲∗bft​bft-last-final(C). In particular it could be the case that the producer of H′ was adversarial, and chose H′.context_bft in such a way as to favour its own bft‑block that is final in that context. However, in that situation it must be possible for the bc‑block‑producer to see (and prove) that the bft‑chain has a final fork. That is, it can produce a witness C to the violation of Final Agreement , showing that bft-last-final(H′.context_bft)⪯⪰∗bft​bft-last-final(C) does not hold, as discussed in the section Recording more info about the bft‑chain in bc‑blocks above. The second caveat is that in that situation, we still need to set H.parent_bft and H.context_bft in order to be able to recover, and they typically should not be the same in order to do so. The Increasing Tip Score rule is still needed, but it can be simplified. A newly produced bc‑block H is also a bft‑proposal such that snapshot(H)=H. This would yield the following bft‑proposal / bc‑block validity rule: [Candidate rule for discussion] Either score(tip(H.parent_bft))<score(H) or tip(H.parent_bft)=H. except that tip(H.parent_bft) cannot be H, because H is newly produced. It turns out we can just drop that part: Increasing Tip Score (producer = proposer) rule: score(tip(H.parent_bft))<score(H). This works because, if H does not have a higher score than the bc‑block tip(H.parent_bft), the bc‑block‑producer should instead have built on top of that bc‑block — which was necessarily known to the producer in order for it to set H.parent_bft in the header of the new block. The voting would be the same, performed by the same parties. Therefore, there is no concentration of voting into fewer parties. There is no change in the producer/proposer’s incentive to make the bft‑notarization‑proof or its soundness properties. Everything else is roughly the same, including the use of the context_bft field of a bc‑block and the validity rules related to it. As far as I can see, all of the security analysis goes through essentially unchanged. There may be some complication due to the fact that BFT protocols are typically designed to use epochs with a fixed period, whereas bc‑blocks are found at less predictable intervals. However, as long as BFT messages are labelled with the bc‑block they apply to, it seems like most BFT protocols would be tolerant to this change. In fact the adaptations of Snap‑and‑Chat to Hotstuff and PBFT in [NTT2020] already assume that BFT messages can be queued and processed at a later time, and rely on those BFT protocols' tolerance to this. Info In most PoS protocols, the requirement to have a minimum amount of stake in order to make a proposal acts as a gatekeeping filter on proposals , and potentially allows parties that make invalid proposals to be slashed. Strictly speaking, whether there is a stake requirement to make a proposal is independent of whether bc‑block‑producers (e.g. miners) are merged with bft‑proposers. It could be, for example, that a miner is still able to produce bc‑blocks, but is not able to make them into a proposal unless they satisfy a stake requirement. (This would have significant effects on the economics of mining that would need to be analyzed, and that might have governance consequences.) In a system that uses PoS, validators by definition need to have stake in order to control the ability to vote. This also allows validators to be slashed. On the other hand, there is no technical reason why the ability to make a bft‑proposal has to be gatekept by a stake requirement — given the situation of Zcash in which we already have a mining infrastructure, and that in a Snap‑and‑Chat or Crosslink‑style hybrid protocol we necessarily still rely on miners not to censor transactions. The potential to make proposals that are expensive to validate as a denial of service is made sufficiently difficult by proof‑of‑work. This option has probably been underexplored by previous PoS protocols because they cannot assume an existing mining infrastructure. It could be argued that this approach goes less far toward a pure PoS‑based block‑chain protocol, leaving more to be done in the second stage. However, there is a clear route to that second stage, by replacing PoW with a protocol like PoSAT that has full unpredictabilty and dynamic availability. PoSAT does this using a VDF, and as it happens, Halo 2 is a strong candidate to be used to construct such a VDF. If the arguments in [DKT2021] about the need for proposer unpredictability are persuasive, then this approach defers the complexity of requiring a VDF without losing any security, since Zcash’s PoW is already unpredictable.","breadcrumbs":"Design » Crosslink » Potential Changes to Crosslink » What about making the bc‑block‑producer the bft‑proposer?","id":"116","title":"What about making the bc‑block‑producer the bft‑proposer?"},"117":{"body":"Building on the previous idea, we can try to eliminate the explicit bft‑chain by piggybacking the information it would hold onto a bc‑block (in the header and/or the block contents). In the previous section we merged the concepts of a bft‑proposal and a bc‑block; the parentbft​(P) and P.context_bft fields of a bft‑proposal were moved into the H.parent_bft and H.context_bft fields of a bc‑header respectively. A field H.pubkey was also added to hold the producer’s public key, so that the producer can sign the bft‑block constructed from it using the corresponding private key. This left the concept of a bft‑block intact. Recall that in Crosslink 1, a bft‑block B consists of (P,proofP​) signed by the proposer. So in “Crosslink with proposer = producer”, a bft‑block consists of (H,proofH​) signed by the producer. What if a bc‑block H were to “inline” its parent and context bft‑blocks rather than referring to them? I.e. a bc‑block H with H.parent_bft referring to (H′,proofH′​) signed for H′.pubkey, would instead literally include (either in the header or the coinbase transaction) (H′,proofH′​) signed for H′.pubkey — and similarly for H.context_bft. It would still be necessary to have the message type that the proposer/producer previously used to submit a notarized bft‑block. (It cannot be merged with a bc‑block announcement: the producer of a new block is not in general the producer of its parent, and their incentives may differ; also we cannot wait until a new block is found before publishing the previous notarization.) It would also still be necessary for Crosslink nodes to keep track of notarizations that have not made it into any bc‑block. Nevertheless, this is a potential simplification. Note that unless notarization proofs are particularly short and constant-length, it would not be appropriate to include them in the bc‑block headers, and so they would have to go into the coinbase transaction or another similar variable-length data structure. In that case we would still have an indirection to obtain the bft‑block information; it would just be merged with the indirection to obtain a coinbase transaction (or similar) — which is already needed in order to check validity of the bc‑block. As discussed under Recording more info about the bft‑chain in bc‑blocks above, we might in any case want to record information about other proposed and notarized bft‑blocks, and the data structure needed for this would necessarily be variable-length. The complexity burden of doing so would be shared between these two changes. It would be possible to save some space in headers (while keeping them fixed-length), by inlining only one of H.parent_bft and H.context_bft in the header and keeping the other as a hash. As discussed under “What are the caveats?” above, the only reason for the two bft‑blocks referred to by these fields to be different, is that the bc‑block producer has observed a violation of Final Agreement in Πbft​. In that case, we can include an inlining of the other H.∗_bft block, and any other information needed to prove that a violation of Final Agreement has occurred, in a variable-length overflow structure. Pros: No additional mechanism or messages are needed to obtain bft‑blocks given their hashes. It could be a performance/latency improvement to not have to separately fetch bft‑blocks. Con: Additional complexity of the variable-length overflow mechanism suggested above, if it is used. Assumes that notarization proofs are not too large.","breadcrumbs":"Design » Crosslink » Potential Changes to Crosslink » Do we need an explicit bft‑chain at all?","id":"117","title":"Do we need an explicit bft‑chain at all?"},"118":{"body":"A potential simplification can be obtained by combining the following two ideas: Str4d suggested that the snapshot of each bft‑block should descend from the snapshot of its parent bft‑block. Nate suggested that each bc‑block H should descend from snapshot(LF(H)). Str4d’s suggestion can be written as: Linearity rule: snapshot(B⌈bft1​)⪯bc​snapshot(B). Notice that this implies the existing Increasing Score rule in Crosslink 1, because score necessarily increases within a bc‑valid‑chain. Therefore it would in practice be a replacement for the Increasing Score rule . It does not imply the Increasing Tip Score rule discussed above , and in fact it could make sense to enforce both the Linearity rule and the Increasing Tip Score rule . The Linearity rule implies that it is no longer possible for a bft‑valid‑chain to snapshot a bc‑chain that rolls back relative to the previous snapshot. This makes it unnecessary to sanitize LOGfin​: the sequence of snapshots considered by san-ctx is linear, and so the “sanitization” would just return the transactions in the last snapshot. To remove the need to sanitize LOGbda​ as well, we need a further modification to Πbc​. Recall that in Crosslink 1 we define: fin(H)bda-ctx(H,μ)​:=:=​[snapshot(B) for B⪯bft​LF(H⌈bcσ​)]san-ctx(fin(H)∣∣[H⌈bcμ​]).​ The Linearity rule ensures that fin(H) is a linear sequence of snapshots, but for fin(H)∣∣[H⌈bcμ​] to be linear, we also need last(fin(H))=snapshot(LF(H⌈bcσ​))⪯bc​H⌈bcμ​. In order for this to hold for any choice of μ with 0≤μ≤σ, we require the strongest version of this condition with μ=σ, i.e. snapshot(LF(H⌈bcσ​))⪯bc​H⌈bcσ​. Since we can only enforce that this holds for H⌈bcσ​ by enforcing that it holds for an arbitrary bc‑valid‑block H, the rule becomes: Last Final Snapshot rule: snapshot(LF(H))⪯bc​H. This is exactly Nate’s suggestion discussed in Questions about Crosslink . In that document we argued against this rule, but that argument was made in the context of a protocol without the Linearity rule (and originally, even without the Increasing Score rule ). Combining the Linearity rule and Last Final Snapshot rule , on the other hand, completely eliminates the need for sanitization. This could be a huge simplification — and potentially safer, since it would avoid breaking assumptions that may be made by existing Zcash node implementations and other consumers of the Zcash block chain. To spell out the resulting simplifications to the definitions of LOGfin,it​ and \\LOG^t_{\\bda,\\mu,i}, we would just have: \\begin{array}{rcl} \\LOG_{\\fin,i}^t &:=& \\snapshotlf{\\ch_i^t \\trunc_{\\bc}^\\sigma} \\ \\LOG_{\\bda,\\mu,i}^t &:=& \\ch_i^t \\trunc_{\\bc}^\\mu \\end{array} Here it is no longer necessary to define LOGfin,it​ and \\LOG^t_{\\bda,\\mu,i} as sequences of transactions, since the final and bounded-available chains are both just bc‑valid‑chains. The definition of finality-depth in the Finality depth rule becomes much simpler: finality-depth(H):=height(H)−height(snapshot(LF(H⌈bcσ​))) As before, either finality-depth(H)≤L or is-stalled-block(H). Avoiding sanitization also means that the bug we described in Snap‑and‑Chat, that could prevent spending outputs from a snapshotted chain after a σ-block rollback, cannot occur by construction. That is, the changes in Πbc​ contexual validity relative to Πorigbc​ are not needed any more. This almost seems too simple, and indeed we should be skeptical, because the security analysis essentially has to be redone. The reason why Snap‑and‑Chat didn’t take this approach is that it requires a more complicated argument to show that it is reasonable to believe in the safety assumptions of Πbc​ whenever it is reasonable to believe in the corresponding assumptions for Πorigbc​. This is because the ... We will need to do some work to show that the changes are benign.","breadcrumbs":"Design » Crosslink » Potential Changes to Crosslink » Linearity and Last Final Snapshot rules","id":"118","title":"Linearity and Last Final Snapshot rules"},"119":{"body":"The key observation needed for this analysis is that neither the Linearity rule nor the Last Final Snapshot rule affect the evolution of Πbc​ unless we are in a situation where its Prefix Consistency or Prefix Agreement properties would be violated. This implies that any safety property that we can prove given Prefix Consistency plus ****.","breadcrumbs":"Design » Crosslink » Potential Changes to Crosslink » Security Analysis","id":"119","title":"Security Analysis"},"12":{"body":"Miners who provide Proof‑of‑Work security will necessarily see some reduction in their block rewards, since this proposal maintains the same issuance schedule and supply cap of ZEC while also spending some rewards on finality. Important Note: The proportion and details of how much mining rewards will be impacted, and conversely how much finality/PoS providers will earn, are not yet specified in this proposal.","breadcrumbs":"Introduction » Trailing Finality Layer in a Nutshell » Miners","id":"12","title":"Miners"},"120":{"body":"","breadcrumbs":"References » References","id":"120","title":"References"},"121":{"body":"Ebb-and-Flow Protocols: A Resolution of the Availability-Finality Dilemma","breadcrumbs":"References » Ebb-and-Flow Protocols","id":"121","title":"Ebb-and-Flow Protocols"},"122":{"body":"","breadcrumbs":"Version History » Version History","id":"122","title":"Version History"},"123":{"body":"This version updated the Crosslink construction and its security analysis to version 2.","breadcrumbs":"Version History » 0.2.0 - Crosslink 2","id":"123","title":"0.2.0 - Crosslink 2"},"124":{"body":"Issue Tracking This version introduced the Crosslink construction by linking to an external write-up. It transitioned away from the implicit design presented at Zcon4 to a rigorously defined hybrid construction .","breadcrumbs":"Version History » 0.1.0 - Introducing Crosslink","id":"124","title":"0.1.0 - Introducing Crosslink"},"125":{"body":"Prior to the initial release of this book, a design sketch was presented in Interactive Design of a Zcash Trailing Finality Layer (Hybrid PoW+PoS) by Nate Wilcox: Zcon4 .","breadcrumbs":"Version History » Prehistory - Zcon4 Presentation","id":"125","title":"Prehistory - Zcon4 Presentation"},"13":{"body":"This design is appealing as a safer first step in transitioning the Zcash protocol for multiple reasons:","breadcrumbs":"Introduction » Trailing Finality Layer in a Nutshell » Why is this a Good Approach to a PoS Transition?","id":"13","title":"Why is this a Good Approach to a PoS Transition?"},"14":{"body":"This transition would enable PoS mechanisms, including the ability to operate PoS infratructure and delegate ZEC towards those providers to earn protocol rewards. While all PoS transitions would accomplish this, this approach does so in a conservative manner: it introduces these mechanics while striving to minimize the impact on existing use cases and protocol security. In a sense, we can think of this approach as enabling the Zcash community to “dip our toes in the PoS waters” rather than diving in. If the results pan out well, it gives us confidence for further transitions. If we discover challenges, flaws, or risks, we anticipate their impact will be more limited since this is a more cautious transition step.","breadcrumbs":"Introduction » Trailing Finality Layer in a Nutshell » It Enables Proof-of-Stake Mechanisms Conservatively","id":"14","title":"It Enables Proof-of-Stake Mechanisms Conservatively"},"15":{"body":"In many cases, existing products, services, and tools can continue using the mainnet chain with no changes to code assuming they rely on existing consensus nodes. We view this as a major benefit which allows Zcash’s existing user network effect to continue safely unperturbed. There will be certain narrow exceptional areas if those products, services, or tools need to be precise in areas where the protocol has changed, such as mining/staking reward calculations, transaction formats (in particular any new PoS-related fields or logic), or chain rollback logic.","breadcrumbs":"Introduction » Trailing Finality Layer in a Nutshell » Minimal Use-Case Disruption","id":"15","title":"Minimal Use-Case Disruption"},"16":{"body":"By conceptualizing the TFL as a distinct “layer” or subprotocol, the consensus rules can be described in terms of two consensus subprotocols , one embodying most of the current consensus logic of Zcash and another the TFL. These protocols interact through a hybrid construction . See Design at a Glance to learn more about these distinct subprotocols. Reasoning about the whole protocol can leverage analysis and understanding of each subprotocol and the hybrid construction somewhat independently due to this modular design. Note that although this design is modular, the hybrid construction may require modifications to the [PoW] and/or [PoS] subprotocols to protect safety and liveness properties. Nevertheless, the modularity still improves analysis and reasoning compared to a monolithic design. Finally, since one subprotocol is very similar to the existing Zcash NU5/NU6 consensus protocol, this lessens risk that the consensus properties within that subprotocol compromise current NU5/NU6 properties.","breadcrumbs":"Introduction » Trailing Finality Layer in a Nutshell » Modular Design","id":"16","title":"Modular Design"},"17":{"body":"In addition to the other benefits of protocol design modularity, we anticipate actual implementations can realize this modularity in code. This can help makes implementations more robust, easier to maintain, and more interoperable. For example, we can envision a standardized interface between PoW & TFL consensus components, enabling different development teams to provide these different components and for “full node” packagers to mix and match them. This is somewhat reminiscent of Ethereum’s execution/consensus layer separation which we believe has shown great success in implementation team and product diversity.","breadcrumbs":"Introduction » Trailing Finality Layer in a Nutshell » Modular Implementation","id":"17","title":"Modular Implementation"},"18":{"body":"In the rest of the introductory section of this book, we describe the status and next steps for the TFL proposal, provide a motivation for finality, and suggestions for getting involved.","breadcrumbs":"Introduction » Trailing Finality Layer in a Nutshell » Cracking the Nutshell","id":"18","title":"Cracking the Nutshell"},"19":{"body":"This is an early and incomplete protocol design proposal. It has not been well vetted for feasibility and safety. It has not had broad review from the Zcash community, so its status on any Zcash roadmap is undetermined.","breadcrumbs":"Introduction » Status and Next Steps » Status and Next Steps","id":"19","title":"Status and Next Steps"},"2":{"body":"There are multiple developer orgs working on different proposed features for Zcash. Some of these involve multiple large distinct upgrade steps, and several of these steps depend on other such steps. This could be represented as a directed acyclic graph. We have begun referring to this space of possible future improvements as the Zcash Tech-Tree , taking inspiration from an analogous concept in gaming. [1] We envision a proof-of-stake transition path as one of the potential paths within this tech-tree which is the primary protocol focus of this proposal. An example visualization of this Zcash Tech-Tree might look like this:","breadcrumbs":"Introduction » A Path to Proof-of-Stake Zcash » The Zcash Tech-Tree","id":"2","title":"The Zcash Tech-Tree"},"20":{"body":"","breadcrumbs":"Introduction » Status and Next Steps » Current Components","id":"20","title":"Current Components"},"21":{"body":"This book is intended to become both a high-level overview and introduction to TFL , and a full specification.","breadcrumbs":"Introduction » Status and Next Steps » This Book","id":"21","title":"This Book"},"22":{"body":"The current heart of the design work is an in-progress hybrid consensus protocol construction called Crosslink (definition) . This is defined in the Crosslink chapter .","breadcrumbs":"Introduction » Status and Next Steps » Crosslink","id":"22","title":"Crosslink"},"23":{"body":"We've begun creating a simulator called simtfl which we will use to model security and abstract performance concerns. Its development is tracked at https://github.com/zcash/simtfl .","breadcrumbs":"Introduction » Status and Next Steps » simtfl","id":"23","title":"simtfl"},"24":{"body":"PoS subprotocol selection, Issuance and supply mechanics, such as how much ZEC stakers may earn, Integrated Zcash transaction semantics, A transition plan from current Zcash mainnet to this protocol design, ZIP s specifying the above to the level of specificity required by ZIPs, Security and safety analyses, Economic analyses. This list may be incomplete, and as the design matures the need for major new components may be revealed.","breadcrumbs":"Introduction » Status and Next Steps » Major Missing Components","id":"24","title":"Major Missing Components"},"25":{"body":"This design proposal is being developed by Electric Coin Company and Shielded Labs as the first major milestone in our focus of deploying Proof-of-Stake to the Zcash protocol. Our rough near-term plan for this proposal is as follows: Complete the Crosslink description. Complete core security arguments for Crosslink . Define the Major Missing Components above, including considerations such as issuance mechanics and Proof-of-Stake mechanisms. Complete auxillary security arguments and analyses, such as specific attack scenarios, game-theoretic security, and so forth. Mature simtfl to analyze all cases of interest. Follow the general Zcash process for proposal/review/refinement, including proposing one or more ZIP s. Follow the general Zcash governance process for proposal review and refinement. If accepted, productionize the proposal in ECC products and collaborate with other implementors who implement the proposal. Celebrate when and if the proposal is activated on Mainnet. 🎉 The fine-grained day-to-day goals and tasks for this project are present in the Zcash Developers Hub in the TFL-focused DAG . Please also see Get Involved if you are interested in tracking this progress more closely, or in contributing.","breadcrumbs":"Introduction » Status and Next Steps » Next Steps","id":"25","title":"Next Steps"},"26":{"body":"In Zcash currently, consensus relies solely on PoW, which only provides probabilistic finality , rather than assured finality . [1] This style of consensus does not offer a guarantee that any given block may not be rolled back which may invalidate the transactions it contains. Instead, the probability that a block may be rolled back decreases as more blocks are mined subsquently on top of it. [2] Let’s walk through an example of how Zcash’s current PoW with probabilistic finality can impede important use cases. Consider a PoW node which sees this block sequence at time T=0: When should a user, wallet, or other system choose to act based on a transaction in a block? For this example, let’s assume a bridging system may have received a deposit for ZEC in block f and issued a corresponding number of proxy tokens on a different network. At a later time, T=1, this same node may see a longer PoW chain which invalidates some previously seen blocks: The node has observed a longer chain ending at block h', so PoW consensus calls for that new sequence to be treated as consensus. The previously seen blocks f and g are no longer part of the consensus history, and have been rolled back.","breadcrumbs":"Introduction » Motivating Finality » Motivating Finality","id":"26","title":"Motivating Finality"},"27":{"body":"In our example, the bridging system acted in response to a transaction in the original block f at T=0. If the new sequence ending at h' no longer contains the deposit to the bridging system, the integrity of the bridge has been violated [3] ; the associated proxy tokens may have already been used in a complex chain of Defi applications or deposited onto an exchange and sold, which would make any recovery impossible. The proxy tokens on the other network no longer correspond to the correct amount of ZEC on the Zcash network.","breadcrumbs":"Introduction » Motivating Finality » Impact of the Rollback","id":"27","title":"Impact of the Rollback"},"28":{"body":"This example demonstrates how a lack of assured finality can impede many useful real-world scenarios. In practice, systems and services which need greater assurances wait for more block confirmations. This has several drawbacks: It doesn’t remove the vulnerability , it only reduces the likelihood. Different applications/services may require different block depths, making it difficult to compose or chain together different applications/services. Different block depth policies potentially confuse users, i.e. “why do I have to wait one hour for my deposit in this exchange, but only 30 minutes on that exchange?”. It introduces a long delay which inhibits many useful applications. In addition to these user-facing and economic drawbacks, correctly handling rollbacks makes the code for nodes, wallets, and other infrastructure more complex. Worse still, many systems may not have correct behavior for rollbacks at different depths, and since large rollbacks are rarer, these implementation flaws may not surface until there is a large rollback. While a large rollback would be disruptive all by itself, it becomes even worse when previously undiscovered bugs exacerbate the situation.","breadcrumbs":"Introduction » Motivating Finality » Rollback Complications","id":"28","title":"Rollback Complications"},"29":{"body":"Trailing finality extends the existing PoW consensus so that older blocks become final, assuring they cannot be rolled back, and by extension neither can any of the transactions they contain. This directly addresses the first two flaws above: it completely removes the vulnerability, and it ensures all systems that need finality behave consistently with each other. As for delay, trailing finality also introduces delay since final blocks “trail behind” the most recent PoW blocks. This can be an improvement for some applications, depending on their latency requirements. For example, if the delay to finality averages around 10 minutes, then this would enable an improvement for an exchange that requires 60 minutes of PoW blocks for a deposit. On the other hand, it would not be an improvement for an application that needs finality faster than 10 minutes. Finally, implementations can be simplified by relying on the guarantee of finality. For example, a wallet can describe any transaction as pending or final, and does not need to provide difficult and potentially confusing UX (and the supporting database sophistication) for handling rollbacks.","breadcrumbs":"Introduction » Motivating Finality » Trailing Finality Benefits","id":"29","title":"Trailing Finality Benefits"},"3":{"body":"Given that context, we envision a path within the Zcash Tech-Tree for transitioning Zcash to PoS. At the top level we propose that this path should include at least two major milestones: Transitioning from Zcash NU5/NU6 PoW consensus to a PoW/PoS hybrid consensus protocol dubbed PoW+TFL . Transitioning from PoW+TFL to pure PoS protocol. After this transition to pure PoS, there are likely to be future improvements to the PoS protocol, or the consensus protocol more generally. This TFL book focuses almost exclusively on the first step in this sequence. Our primary motivation for proposing (at least) two steps is to minimize disruption to usability, safety, security, and the ecosystem during each step. This book primarily focuses this first step: the transition to PoW+TFL . To understand the specific goals for that, see Design Goals . With this approach, the Zcash Tech Tree with the TFL approach might look something like this:","breadcrumbs":"Introduction » A Path to Proof-of-Stake Zcash » A Proof-of-Stake Transition Path","id":"3","title":"A Proof-of-Stake Transition Path"},"30":{"body":"Throughout this book, when we say finality or final without other qualifiers, we specifically are referring to assured finality or an assured final block. Where we call out probabalistic finality we always use that qualifier. The estimated probability of a rollback relies on a variety of PoW security assumptions, and can be violated in various conditions, such as in mining efficiency breakthroughs, compromises of the PoW challenge algorithm (e.g. hash function collision resistance failure), difficulty-adjustment-algorithm failures, sudden/surprise mining capacity increases, and so on. So the estimated probability can be violated in potential “black swan” events. This discussion simplifies consideration of transaction rollback vs block rollback . When a block is rolled back, it is possible for some of the transactions contained in it to appear in new canonical blocks. The conditions when this can occur vs when it cannot are multifaceted and also subject to malicious influence, so for simplicity we assume all transactions within a rolled-back block are also rolled back.","breadcrumbs":"Introduction » Motivating Finality » Footnotes","id":"30","title":"Footnotes"},"31":{"body":"We welcome contributions! There are a variety of ways to contribute to this project:","breadcrumbs":"Introduction » Get Involved » Get Involved","id":"31","title":"Get Involved"},"32":{"body":"If you have a GitHub account, you can get hands-on via the GitHub repository for this book, including: Ask a Question - all questions welcome from basics to in-depth. Suggest an Improvement to the content: anything from typo fixes to major design change proposals. Report Rendering / Infrastructure Issues in case you're having trouble reading the content, viewing diagrams, rendering on your own computer, etc…","breadcrumbs":"Introduction » Get Involved » GitHub","id":"32","title":"GitHub"},"33":{"body":"The Zcash Forum is a hangout for many Zcash enthusiasts. This is a good spot for more open-ended discussion about this design proposal, alternatives, and other developments in Zcash.","breadcrumbs":"Introduction » Get Involved » Zcash Forum","id":"33","title":"Zcash Forum"},"34":{"body":"You can catch us on the Zcash R&D Discord in the #proof-of-stake channel.","breadcrumbs":"Introduction » Get Involved » Zcash R&D Discord","id":"34","title":"Zcash R&D Discord"},"35":{"body":"The Zcash Arborist Calls are bi-weekly Zcash protocol development calls, where proposals like this are discussed. Feel free to come lurk, ask questions, or provide feedback or suggestions.","breadcrumbs":"Introduction » Get Involved » Zcash Arborist Calls","id":"35","title":"Zcash Arborist Calls"},"36":{"body":"This book relies on the following terminology. We strive to keep these definitions as precise as possible and their definitions may deviate from uses elsewhere. Definitions are sorted alphabetically.","breadcrumbs":"Terminology » Terminology","id":"36","title":"Terminology"},"37":{"body":"Assured Finality : A protocol property that assures that transactions cannot be reverted by that protocol. As with all protocol guarantees, a protocol assumes certain conditions must be met. A transaction may either be final or not: transactions which are not final may not become final, whereas once transactions do achieve finality they retain that property indefinitely (so long as protocol requirements are met). Importantly, it is not feasible for any protocol to prevent reversing final transactions \"out of band\" from the protocol, such as if a sufficiently large and motivated group of users forks the network to include a specific new validity rule reverting transactions. In some cases this might be desirable, for example to mitigate exploitation of a security flaw. We are investigating the implications for governance and how to incorporate such situations into our security model. In any case, for this reason we eschew the term \"absolute finality\" sometimes used in technical discussions about consensus protocols. Consensus Subprotocols : The PoW and PoS subprotocols in PoW+TFL or other hybrid protocols . Crosslink : A hybrid construction consensus protocol striving to implement the TFL design goals. See Status and Next Steps: Current Components for current status. Final : A protocol property of transactions. In this book, this always implies assured finality , in contrast to concepts like \"probabilistic finality\" provided by PoW . Hybrid Consensus : A consensus protocol that integrates more than one consensus subprotocol. PoW+TFL is an instance of a hybrid protocol integrating PoW and PoS protocols. Hybrid Construction : The design component of a hybrid consensus which specifies how to integrate subprotocols and what modifications, if any, those subprotocols need to be safely integrated. Examples include Crosslink and Snap-and-Chat . Liveness : The property of a distributed protocol which ensures that the protocol may progress provided liveness requirements are met. TODO: Fix the definition of Liveness #120 NU5/NU6 : The Zcash consensus protocol as of NU5 or NU6 (which do not differ significantly in terms of the base block chain layer). [1] Objective Validity : A validity property of a protocol history (such as a ledger) which can be computed purely from that history with no other context. Objective validity is needed to define consensus rules that will lead to the same protocol state being eventually agreed on by all nodes. Proof-of-Stake : A PoS protocol achieves consensus on transaction status by taking into account the weighting of staking tokens. PoS protocols exist under a large umbrella and may or may not provide assured finality or other properties this design requires of TFL . Proof‑of‑Work : A PoW protocol uses Nakamoto consensus pioneered by Bitcoin. The PoW subprotocol within PoW+TFL is a different consensus protocol from NU5/NU6 and encompasses more than narrow Nakamoto PoW consensus, including transaction semantics such as for shielded transfers. PoW+TFL : the overall complete, integrated consensus protocol specified in this book. Safety : The property of a distributed protocol that guarantees a participant may safely rely on a consistent local state, provided safety requirements are met. TODO: Provide a rigorous definition of Safety #121 simtfl : a protocol simulator for analyzing TFL security and abstract performance. Development lives at https://github.com/zcash/simtfl . See Status and Next Steps: Current Components for current status. Snap-and-Chat : A hybrid construction consensus protocol introduced in Ebb-and-Flow Protocols . TFL : The Trailing Finality Layer subprotocol within PoW+TFL. This is a new PoS subprotocol which provides assured finality for Zcash. Trailing Finality : A protocol property wherein transactions become final some time after first appearing in PoW blocks. ZIP : a Zcash Improvement Proposal is the protocol development process the Zcash community uses to safely define potential protocol improvements. See https://zips.z.cash . TODO: Clarify the distinctions between pure PoW, the PoW subprotocol, NU6, and fork-choice vs all of transaction semantics. #119","breadcrumbs":"Terminology » Terms","id":"37","title":"Terms"},"38":{"body":"If new consensus changes are deployed to Zcash mainnet prior to PoW+TFL design finalization, this design must be updated to refer to the new delta (e.g. by reanalyzing all changes against NU6 or NU7, etc…)","breadcrumbs":"Terminology » Footnotes","id":"38","title":"Footnotes"},"39":{"body":"This design augments the existing Zcash Proof‑of‑Work (PoW) network with a new consensus layer which provides trailing finality , called the Trailing Finality Layer (TFL) . This layer enables blocks produced via PoW to become final which ensures they may never be rolled back. This enables safer and simpler wallets and other infrastructure, and aids trust-minimized cross-chain bridges. This consensus layer uses a finalizing Proof-of-Stake (PoS) consensus protocol, and enables ZEC holders to earn protocol rewards for contributing to the security of the Zcash network. By integrating a PoS layer with the current PoW Zcash protocol, this design specifies a hybrid consensus protocol . The integration of the current PoW consensus with the TFL produces a new top-level consensus protocol referred to as PoW+TFL . In the following subchapters we introduce the Design at a Glance , then provide an overview of the major components of the design. Following this overview chapter, we proceed into a detailed Protocol Specification (TODO) .","breadcrumbs":"Design » Design Overview","id":"39","title":"Design Overview"},"4":{"body":"One question we've gotten in proposing this approach is why take a two-step process with an intermediate hybrid consensus protocol, rather than a single transition directly to a PoS protocol? Here’s how we think about those trade-offs:","breadcrumbs":"Introduction » A Path to Proof-of-Stake Zcash » Why Two Steps?","id":"4","title":"Why Two Steps?"},"40":{"body":"The PoW+TFL consensus protocol is logically an extension of the Zcash consensus rules to introduce trailing finality . This is achieved by compartmentalizing the top-level PoW+TFL protocol into two consensus subprotocols , one embodying most of the current consensus logic of Zcash and another the TFL. These protocols interact through a hybrid construction , which specifies how the protocols interact, and what changes from \"off-the-shelf\" behavior, if any, need to be imposed on the subprotocols. Each of these components (the two subprotocols and the hybrid construction) are somewhat modular: different subprotocols or hybrid constructions may be combined (with some modification) to produce a candidate PoW+TFL protocol. TODO: Add a protocol component diagram to \"Design at a Glance\" #122","breadcrumbs":"Design » Design at a Glance » Design at a Glance","id":"40","title":"Design at a Glance"},"41":{"body":"The hybrid construction is a major design component of the full consensus protocol which specifies how the subprotocols integrate. So far we have considered three candidates: The implied/loosely defined hybrid construction presented at Zcon4 . The Snap-and-Chat from the Ebb-and-Flow paper . The Crosslink construction. We believe Crosslink is the best candidate, due to its more rigorous specification and security analysis , and due to the issues with Snap-and-Chat described in Notes on Snap-and-Chat .","breadcrumbs":"Design » Design at a Glance » Hybrid Construction","id":"41","title":"Hybrid Construction"},"42":{"body":"The PoW+TFL hybrid consensus consists of two interacting subprotocols: PoW Subprotocol : this subprotocol is very similar to NU5 or NU6 consensus. It is a design goal of the TFL design to minimize changes to this subprotocol. Note: the shorthand \"PoW\" is potentially misleading, because this subprotocol is also responsible for the bulk of all supply and transaction semantic consensus rules. PoS Subprotocol : this is a new subprotocol which provides trailing finality via a finalizing PoS protocol. TODO: Clarify the distinctions between pure PoW, the PoW subprotocol, NU6, and fork-choice vs all of transaction semantics. #119 Note that the hybrid construction may require modification to the \"off-the-shelf\" versions of these subprotocols. In particular Crosslink requires each protocol to refer to the state of the other to provide objective validity .","breadcrumbs":"Design » Design at a Glance » Subprotocols","id":"42","title":"Subprotocols"},"43":{"body":"Here we strive to lay out our high level TFL design goals.","breadcrumbs":"Design » Design Goals » Design Goals","id":"43","title":"Design Goals"},"44":{"body":"Here we lay out ideal goals. As we develop a complete design, we are likely to inevitably encounter trade-offs some of which may preclude achieving the full idealized goals. Wherever possible, we motivate design decisions by these goals, and when goals are impacted by trade-offs we describe that impact and the rationale for the trade-off decision. For example, one ideal user experience goal below is to avoid disruption to existing wallets. However, the Crosslink construction may require wallets to alter their context of valid transactions differently from the current NU5/NU6 consensus protocol.","breadcrumbs":"Design » Design Goals » Goals, Design, and Trade-offs","id":"44","title":"Goals, Design, and Trade-offs"},"45":{"body":"We strive to start our protocol design process from user experience (UX) and use case considerations foremost, since at the end of the day all that matters in a protocol is what user needs it meets and how well. All currently supported wallet user experience should continue to operate seamlessly without change during or after protocol transitions. This covers the use of addresses, payment flow, transfers, ZEC supply cap and issuance rate, backup/restore, and other features users currently rely on. There must be no security or safety degradation due to wallet user behavior introduced by PoS transitions, assuming users follow their current behaviors unchanged and continue to use the same cognitive model of the impacts of their behaviors. This goal encompasses all of security and safety, including privacy and transparency or more explicit disclosures. The protocol should enable users of shielded mobile wallets to delegate ZEC to PoS consensus providers and earn a return on that ZEC coming via ZEC issuance or fees. Doing this may expose users to a risk of loss of delegated ZEC (such as through “slashing fees”). The protocol must guarantee that PoS consensus providers have no discretionary control over such delegated funds (including that they cannot steal those funds). For any hybrid PoW/PoS protocol (including the PoW+TFL protocol we’re proposing), the process and UX of mining remains unchanged except that the return on investment may be altered. This is true both of consensus-level block miners (i.e. mining pools and solo miners) and mining pool participants. Any hybrid PoW/PoS protocol (including PoW+TFL) block explorers will continue to function with the same UX through transitions as far as displaying information about transactions, the mempool, and blocks. Block explorers and other network metrics sites may require UX changes with respect to mining rewards and issuance calculations. Network metrics sites may require UX changes with respect to the p2p protocol or other network-specific information. Users can rely on assured finality with an expected time‑to‑finality below 30 minutes. [1]","breadcrumbs":"Design » Design Goals » User Experience and Use Case Goals","id":"45","title":"User Experience and Use Case Goals"},"46":{"body":"For a full PoS transition, ecosystem developers for products such as consensus nodes, wallets, mining services, chain analytics, and more will certainly need to update their code to support transitions. However, we carve out a few goals as an exception to this for this category of users: Wallet developers should not be required to make any changes through protocol transitions as long as they rely solely on the lightwalletd protocol or a full node API (such as the zcashd RPC interface). For any hybrid PoW/PoS protocol (including PoW+TFL), mining pools and miners should not be required to make any software or protocol changes as long as they rely on zcashd-compatible GetBlockTemplate. One exception to this is software that bakes in assumptions about the block reward schedule, rather than relying on GetBlockTemplate solely.","breadcrumbs":"Design » Design Goals » Developer Experience Goals","id":"46","title":"Developer Experience Goals"},"47":{"body":"Zcash has always had exemplary safety, security, and privacy, and we aim to continue that tradition: For any hybrid PoW/PoS protocol (including PoW+TFL), the cost‑of‑attack for a 1-hour rollback should not be reduced, given a “reasonably rigorous” security argument. For any hybrid PoW/PoS protocol (including PoW+TFL), the cost‑of‑attack to halt the chain should be larger than the 24‑hour revenue of PoW mining rewards, given a “reasonably rigorous” security argument. TODO: Define privacy goals of TFL #118 TODO: Define PoS Subprotocol desiderata which are distinct from Crosslink integration #117","breadcrumbs":"Design » Design Goals » Safety, Security, and Privacy Goals","id":"47","title":"Safety, Security, and Privacy Goals"},"48":{"body":"We want to follow some conservative design heuristics to minimize risk and mistakes: Rely as much as possible on design components that are already proven in production environments. Rely as much as possible on design components with adequate theoretical underpinnings and security analyses. Minimize changes or variations on the above: strive to only alter existing work when necessary for overall design goals. For example, Zcash’s privacy or issuance constraints are likely less common among existing PoS designs.","breadcrumbs":"Design » Design Goals » Design Conservatism Goals","id":"48","title":"Design Conservatism Goals"},"49":{"body":"These are not goals of the TFL design, either to simplify the scope of the initial design (a.k.a. Out-of-Scope Goals ), or because we believe some potential goal should not be supported (a.k.a. Anti-goals ).","breadcrumbs":"Design » Design Goals » Non-goals","id":"49","title":"Non-goals"},"5":{"body":"Pros We already understand the current PoW protocol well, and if we transition to an existing proven PoS protocol, then we could skip the complexity of an intermediate hybrid stage. The node implementation might be simpler. Explaining to people what is happening might be simpler. Something like “Zcash has been PoW since it launched, but on DATE (e.g. at block height X) it will switch to PoS.” Given that the issuance in a given time period is bounded by the supply curve, the full amount that was previously allocated to mining rewards becomes immediately available for staking rewards at the switch-over, rather than having to share this amount between mining and staking during the hybrid stage. Cons If there is any unforeseen show‑stopping problem in the new protocol or the transition process, we’d have to react to a network‑wide issue. It may be more likely to cause ecosystem disruption; unforeseen differences between PoW and PoS might cause various kinds of snags or papercuts throughout the ecosystem, and these would all pop up around the same time, which may lead to a loss of confidence/retention/adoption or at the very least inconvenience many users for some time. Losing miners: since the transition would be all at once, we may lose some number of miners, who are participants and users in the ecosystem. Miners may leave prior to the transition in order to take care of their own needs. If there is some showstopper in the transition, one possible short‑term mitigation would be to fall back on PoW which is well known, but if we’ve lost most miners, that may no longer be viable.","breadcrumbs":"Introduction » A Path to Proof-of-Stake Zcash » Considering Single Transition (vs Hybrid Multi-Step)","id":"5","title":"Considering Single Transition (vs Hybrid Multi-Step)"},"50":{"body":"While these desiderata may be common across the block‑chain consensus design space, they are not specific goals for the initial TFL design. Note that these may be goals for future protocol improvements. Prioritizing minimal time-to-finality over other considerations (such as protocol simplicity, impact on existing use cases, or other goals above). In-protocol liquid staking derivatives. Maximizing the PoS staked-voter count ceiling. For example, Tendermint BFT has a relatively low ceiling of ~hundreds of staked voters, whereas Ethereum’s Gasper supports hundreds of thousands of staked voters. Reducing energy usage. While this would presumably be a goal of a pure PoS transition, it likely cannot be achieved for hybrid PoW/PoS without loss of security.","breadcrumbs":"Design » Design Goals » Out-of-Scope Goals","id":"50","title":"Out-of-Scope Goals"},"51":{"body":"Distinctly from Out-of-Scope Goals we track \"anti-goals\" which are potential goals that we explicitly reject , which are potential goals we aim to not support even in future protocol improvements. We currently have no defined anti-goals.","breadcrumbs":"Design » Design Goals » Anti-Goals","id":"51","title":"Anti-Goals"},"52":{"body":"This requirement comes from a request from a DEX developer . While we have not yet surveyed DEX and Bridge designs, we're relying on this as a good starting point.","breadcrumbs":"Design » Design Goals » Footnotes","id":"52","title":"Footnotes"},"53":{"body":"Crosslink is the proposed hybrid construction for the Trailing Finality Layer. The current version is Crosslink 2.","breadcrumbs":"Design » Crosslink » Crosslink","id":"53","title":"Crosslink"},"54":{"body":"The Arguments for Bounded Availability and Finality Overrides Notes On Snap‑and‑Chat The Crosslink 2 Construction Security Analysis of Crosslink 2 Questions About Crosslink Potential Changes to Crosslink","breadcrumbs":"Design » Crosslink » Contents","id":"54","title":"Contents"},"55":{"body":"This document considers disadvantages of allowing transactions to continue to be included at the chain tip while the gap from the last finalized block becomes unbounded, and our perspective on what should be done instead. This condition is allowed by Ebb‑and‑Flow protocols [NTT2020] . We also argue that it is necessary to allow for the possibility of overriding finalization in order to respond to certain attacks, and that this should be explicitly modelled and subject to a well-defined governance process. This is a rewritten version of this forum post , adapting the main argument to take into account the discussion of “tail‑thrashing attacks” and finalization availability from the Addendum. More details of how bounded availability could be implemented in the context of a Snap‑and‑Chat protocol are in Notes on Snap‑and‑Chat . The proposed changes end up being significant enough to give our construction a new name: “ Crosslink ”, referring to the cross-links between blocks of the BFT and best-chain protocols. Crosslink has evolved somewhat, and now includes other changes not covered in either this document or Notes on Snap‑and‑Chat . The current version is called Crosslink 2 .","breadcrumbs":"Design » Crosslink » The Arguments for Bounded Availability and Finality Overrides » The Arguments for Bounded Availability and Finality Overrides","id":"55","title":"The Arguments for Bounded Availability and Finality Overrides"},"56":{"body":"“Ebb‑and‑Flow”, as described in [NTT2020] ( arXiv version ), is a security model for consensus protocols that provide two transaction logs, one with dynamic availability, and a prefix of it with finality. The paper proposes an instantiation of this security model called a “Snap‑and‑Chat” construction. It composes two consensus subprotocols, a BFT subprotocol and a best-chain subprotocol (it calls this the “longest chain protocol”). The above logs are obtained from the output of these subprotocols in a non-trivial way. This is claimed by the paper to “resolve” the tension between finality and dynamic availability. However, a necessary consequence is that in a situation where the “final” log stalls and the “available” log does not, the “finalization gap” between the finalization point and the chain tip can grow without bound. In particular, this means that transactions that spend funds can remain unfinalized for an arbitrary length of time. In this document, we argue that this is unacceptable, and that it is preferable to sacrifice strict dynamic availability. The main idea behind Ebb‑and‑Flow protocols is a good one, and allowing the chain tip to run ahead of the finalization point does make sense and has practical advantages. However, we also argue that it should not be possible to include transactions that spend funds in blocks that are too far ahead of the finalization point. Note Naive ways of preventing an unbounded finalization gap, such as stopping the chain completely in the case of a finalization stall, turn out to run into serious security problems — at least when the best-chain protocol uses Proof‑of‑Work. We’ll discuss those in detail in the section on tail‑thrashing attacks . Our proposed solution will be to require coinbase-only blocks during a long finalization stall. This solution has the advantage of not complicating the security analysis. We argue that losing strict dynamic availability in favour of “bounded availability” is preferable to the consequences of the unbounded finality gap, if/when a “long finalization stall” occurs. We also argue that it is beneficial to explicitly allow “finality overrides” under the control of a well-documented governance process. Such overrides allow long rollbacks that may be necessary in the case of an exploited security flaw. This is complementary to the argument for bounded availability, because the latter limits the period of user transactions that could be affected. The governance process can impose a limit on the length of this long rollback if desired.","breadcrumbs":"Design » Crosslink » The Arguments for Bounded Availability and Finality Overrides » Background","id":"56","title":"Background"},"57":{"body":"Since partition between nodes sufficient for finalization cannot be prevented, loosely speaking the CAP theorem implies that any consistent protocol (and therefore any protocol with finality) may stall for at least as long as the partition takes to heal. Background That “loosely speaking” is made precise by [LR2020] . Dynamic availability implies that the chain tip will continue to advance, and so the finalization gap increases without bound. Partition is not necessarily the only condition that could cause a finalization stall, it is just the one that most easily proves that this conclusion is impossible to avoid.","breadcrumbs":"Design » Crosslink » The Arguments for Bounded Availability and Finality Overrides » Finality + Dynamic availability + Possibility of Partition implies an unbounded finalization gap","id":"57","title":"Finality + Dynamic availability + Possibility of Partition implies an unbounded finalization gap"},"58":{"body":"Both the available protocol, and the subprotocol that provides finality, will be used in practice — otherwise, one or both of them might as well not exist. There is always a risk that blocks may be rolled back to the finalization point, by definition. Suppose, then, that there is a long finalization stall. The final and available protocols are not separate: there is no duplication of tokens between protocols, but the rules about how to determine best-effort balance and guaranteed balance depend on both protocols, how they are composed, and how the history after the finalization point is interpreted. Discussion The guaranteed minimum balance of a given party is not just the minimum of their balance at the finalization point and their balance at the current tip. It is the minimum balance taken over all possible transaction histories that extend the finalized chain — taking into account that a party’s previously published transactions might be able to be reapplied in a different context without its explicit consent. The extent to which published transactions can be reapplied depends on technical choices that we must make, subject to some constraints (for example, we know that shielded transactions cannot be reapplied after their anchors have been invalidated). It may be desirable to further constrain re-use in order to make guaranteed minimum balances easier to compute. As the finalization gap increases, the negative consequences of rolling back user transactions that spend funds increase. (Coinbase transactions do not spend funds; they are a special case that we will discuss later.) There are several possible —not mutually exclusive— outcomes: Users of the currency start to consider the available protocol increasingly unreliable. Users start to consider a rollback to be untenable, and lobby to prevent it or cry foul if it occurs. Users start to consider finalization increasingly irrelevant. Services that depend on finality become unavailable. There is no free lunch that would allow us to avoid availability problems for services that also depend on finality. Service providers adopt temporary workarounds that may not have had adequate security analysis. Any of these might precipitate a crisis of confidence, and there are reasons to think this effect might be worse than if the chain had switched to a “Stalled Mode” designed to prevent loss of user funds. Any such crisis may have a negative effect on token prices and long-term adoption. Note that adding finalization using an Ebb‑and‑Flow protocol does not by itself increase the probability of a rollback in the available chain, provided the PoW remains as secure against rollbacks of a given length as before. But that is a big proviso. We have a design constraint (motivated by limiting token devaluation and by governance issues) to limit issuance to be no greater than that of the original Zcash protocol up to a given height. Since some of the issuance is likely needed to reward staking, the amount of money available for mining rewards is reduced, which may reduce overall hash rate and security of the PoW. Independently, there may be a temptation for design decisions to rely on finality in a way that reduces security of PoW (“risk compensation”). There is also pressure to reduce the energy usage of PoW, which necessarily reduces the global hash rate, and therefore the cost of performing an attack that depends on the adversary having any given proportion of global hash rate. It could be argued that the issue of availability of services that depend on finality is mainly one of avoiding over-claiming about what is possible. Nevertheless there are also real usability issues if balances as seen by those services can differ significantly and for long periods from balances at the chain tip. Regardless, incorrect assumptions about the extent to which the finalized and available states can differ are likely to be exposed if there is a finalization stall. And those who made the assumptions may (quite reasonably!) not accept “everything is fine, those assumptions were always wrong” as a satisfactory response.","breadcrumbs":"Design » Crosslink » The Arguments for Bounded Availability and Finality Overrides » Problems with allowing spends in an unbounded finalization gap","id":"58","title":"Problems with allowing spends in an unbounded finalization gap"},"59":{"body":"An intuitive notion of “availability” for block‑chain protocols includes the ability to use the protocol as normal to spend funds. So, just to be clear, in a situation where that cannot happen we have lost availability, even if the block chain is advancing. Bounded availability is a weakening of dynamic availability [DKT2020] . It means that we intentionally sacrifice availability when some potentially hazardous operation —a “hazard” for short— would occur too far after the current finalization point. For now, assume for simplicity that our only hazard is spending funds. More generally, the notion of bounded availability can be applied to a wider range of protocols by tailoring the definition of “hazard” to the protocol. Background This talk by Soubhik Deb accompanying [DKT2020] provides a good explanation of the advantages of dynamic availability. We do not define bounded availability formally in this document, but informally, we aim to preserve the ability to securely adapt to large changes in hash rate or total stake.","breadcrumbs":"Design » Crosslink » The Arguments for Bounded Availability and Finality Overrides » What is Bounded Availability?","id":"59","title":"What is Bounded Availability?"},"6":{"body":"Note: TFL is one instance of a multi‑step approach. Pros We can hopefully be less disruptive across the ecosystem so that there are fewer snags and disruptions with each step. If there is a show-stopping flaw in any step, the fall‑back possibility seems more plausible. For example, if there is a show-stopper when transitioning from PoW to PoW+TFL , falling back to pure PoW seems more feasible, since both protocols rely on mainnet PoW infrastructure, so those participants will be present in either case. Retaining miners during a hybrid phase: while it is true that a hybrid protocol will lower miner revenue (since we aim to maintain the issuance schedule constraints), there is also more possibility and likelihood of keeping some of these users engaged. For example, they may begin participating in staking services (either as delegators or as infrastructure operators). If that is successful, then they’re also more likely to remain engaged in the subsequent transition to pure PoS. This general approach was demonstrated successfully by Ethereum, which is the largest or second largest cryptocurrency network for several important metrics (e.g. market cap, fees paid, user and developer activity, …). So we know this can be done well without disruptions. Cons The intermediate hybrid step will be a more novel and less well understood protocol. (It will necessarily be fairly different from Ethereum’s Beacon chain era.) Consensus nodes will be more complex, involving logic for both sub‑protocols as well as their integration. (Ideally this complexity can be modularized so that the nodes are easier to maintain and improve.) This may be more complicated to explain to current and potential new users. Something like “Zcash launched as PoW, and on $DATE (block height X) it will transition to a hybrid system, then later to a pure PoS system.” The available issuance must be shared between mining and staking rewards during the hybrid stage. The security of the PoW layer and of the PoS layer during this stage is partially dependent on the funds allocated to issuance for each protocol, and it is not yet clear to what extent splitting rewards would affect overall security.","breadcrumbs":"Introduction » A Path to Proof-of-Stake Zcash » Considering Hybrid Multi-Step approach (vs Single-Step):","id":"6","title":"Considering Hybrid Multi-Step approach (vs Single-Step):"},"60":{"body":"[NTT2020] calls the dynamically available block‑chain protocol Πlc​ that provides input to the rest of the contruction, the “longest chain” protocol. There are two reasons to avoid this terminology: In Bitcoin, Zcash, and most other PoW-based protocols, what is actually used by each node is not its longest observed chain, but its observed consensus-valid chain with most accumulated work. In Zcash this is called the node’s “ best valid block chain ”, which we shorten to “best chain”. As footnote 2 on page 3 of [NTT2020] says, that paper does not require Πlc​ to be a “longest chain” protocol anyway. Historical note The error in conflating the “longest chain” with the observed consensus-valid chain with most accumulated work, originates in the Bitcoin whitepaper. [ Nakamoto2008 , page 3] We will use the term “best‑chain protocol” instead. Note that this corresponds roughly to Πlc​ in the Snap‑and‑Chat construction, although the Crosslink 2 protocol that we propose will end up having other significant differences from Snap-and-Chat.","breadcrumbs":"Design » Crosslink » The Arguments for Bounded Availability and Finality Overrides » Terminology note","id":"60","title":"Terminology note"},"61":{"body":"We have not yet decided how to block hazards during a long finalization stall. We could do so directly, or by stopping block production in the more-available protocol. For reasons explained in the section on tail‑thrashing attacks below, it’s desirable not to stop block production. And so it’s consistent to have bounded availability together with another liveness property —which can be defined similarly to dynamic availability— that says the more-available protocol’s chain is still advancing. This is what we will aim for. We will call this method of blocking hazards, without stopping block production, “going into Stalled Mode”. Historical note This concept of Stalled Mode is very similar to a feature that was discussed early in the development of Zcash , but never fully designed or implemented. (After originally being called “Stalled Mode”, it was at some point renamed to “Emergency Mode”, but then the latter term was used for something else .) For Zcash, we propose that the main restriction of Stalled Mode should be to require coinbase-only blocks. This achieves a similar effect, for our purposes, as actually stalling the more-available protocol’s chain. Since funds cannot be spent in coinbase-only blocks, the vast majority of attacks that we are worried about would not be exploitable in this state. Info It is possible that a security flaw could affect coinbase transactions. We might want to turn off shielded coinbase for Stalled Mode blocks in order to reduce the chance of that. Also, mining rewards cannot be spent in a coinbase-only block; in particular, mining pools cannot distribute rewards. So there is a risk that an unscrupulous mining pool might try to do a rug-pull after mining of non-coinbase-only blocks resumes, if there were a very long finalization stall. But this approach works at least in the short term, and probably for long enough to allow manual intervention into the finalization protocol, or governance processes if needed. An analogy for the effect of this on availability that may be familiar to many people, is that it works like video streaming. All video streaming services use a buffer to paper over short-term interruptions or slow-downs of network access. In most cases, this buffer is bounded. This allows the video to be watched uninterrupted and at a constant rate in most circumstances. But if there is a longer-term network failure or insufficient sustained bandwidth, the playback will unavoidably stall. In our case, block production does not literally stall, but it’s the same as far as users’ ability to perform “hazardous” operations is concerned.","breadcrumbs":"Design » Crosslink » The Arguments for Bounded Availability and Finality Overrides » How to block hazards","id":"61","title":"How to block hazards"},"62":{"body":"So, why do we advocate this over: A protocol that only provides dynamic availability; A protocol that only provides finality; An unmodified Ebb‑and‑Flow protocol? The reason to reject option 1 is straightforward: finality is a valuable security property that is necessary for some use cases. If a protocol only provides finality (option 2), then short-term availability is directly tied to finalization. It may be possible to make finalization stalls sufficiently rare or short-lived that this is tolerable. But that is more likely to be possible if and when there is a well-established staking ecosystem. Before that ecosystem is established, the protocol may be particularly vulnerable to stalls. Furthermore, it’s difficult to get to such a protocol from a pure PoW system like current Zcash. We argued in the previous section that allowing hazards in an unbounded finalization gap is bad. Option 3 entails an unbounded finalization gap that will allow hazards. However, that isn’t sufficient to argue that bounded availability is better. Perhaps there are no good solutions! What are we gaining from a bounded availability approach that would justify the complexity of a hybrid protocol without obtaining strict dynamic availability? The argument goes like this: It is likely that a high proportion of the situations in which a sustained finalization stall happens will require human intervention. If the finality protocol were going to recover without intervention, there is no reason to think that it wouldn’t do so in a relatively short time. When human intervention is required, the fact that the chain tip is still proceeding apace (in protocols with strict dynamic availability) makes restarting the finality protocol harder, for many potential causes of a finalization stall. This may be less difficult when only “non-hazardous” transactions are present— in particular, when only coinbase transactions (which are subject to fairly strict rules in Zcash and other Bitcoin-derived chains) are present. This argument carries even more force when the protocol also allows “finality overrides”, as discussed later in the Complementarity section. Nothing about bounded availability prevents us from working hard to design a system that makes finalization stalls as infrequent and short-lived as possible, just as we would for any other option that provides finality. We want to optimistically minimize the finalization gap under good conditions, because this improves the usability of services that depend on finality. This argues against protocols that try to maintain a fixed gap, and motivates letting the gap vary. In practice, the likelihood of short finalization stalls is high enough that heuristically retaining availability in those situations is useful. The argument that it is difficult to completely prevent finalization stalls is supported by experience on Ethereum in May 2023 , when there were two stalls within 24 hours, one for about 25 minutes and one for about 64 minutes . This experience is consistent with our argument: Neither stall required short-term human intervention, and the network did in fact recover from them quickly. The stalls were caused by a resource exhaustion problem in the Prysm consensus client when handling attestations. It’s plausible to think that if this bug had been more serious, or possibly if Prysm clients had made up more of the network, then it would have required a hotfix release (and/or a significant proportion of nodes switching to another client) in order to resolve the stall. So this lines up with the hypothesis that longer stalls are likely to require manual intervention. A bounded availability protocol would very likely have resulted in either a shorter or no interruption in availability. If, say, the availability bound were set to be roughly an hour, then the first finalization stall would have been “papered over” and the second would have resulted in only a short loss of availability. Retaining short-term availability does not result in a risk compensation hazard: A finalization stall is still very visible, and directly affects applications relying on finality. Precisely because of the availability bound, it is obvious that it could affect all applications if it lasted long enough. A potential philosophical objection to lack of strict dynamic availability is that it creates a centralization risk to availability. That is, it becomes more likely that a coalition of validators can deliberately cause a denial of service. This objection may be more prevalent among people who would object to adding a finality layer or PoS at all.","breadcrumbs":"Design » Crosslink » The Arguments for Bounded Availability and Finality Overrides » Why is this better?","id":"62","title":"Why is this better?"},"63":{"body":"Consensus protocols sometimes fail. Potential causes of failure include: A design problem with the finality layer that causes a stall, or allows a stall to be provoked. A balance violation or spend authorization flaw that is being exploited or is sufficiently likely to be exploited. An implementation bug in a widely used node implementation that causes many nodes to diverge from consensus. In these situations, overriding finality may be better than any other alternative. An example is a balance violation flaw due to a 64-bit integer overflow that was exploited on Bitcoin mainnet on 15th August 2010. The response was to roll back the chain to before the exploitation, which is widely considered to have been the right decision. The time between the exploit (at block height 74638) and the forked chain overtaking the exploited chain (at block height 74691) was 53 blocks, or around 9 hours. Of course, Bitcoin used and still uses a pure‑PoW consensus. But the applicability of the example does not depend on that: the flaw was independent of the consensus mechanism. Another example of a situation that prompted this kind of override was the DAO recursion exploit on the Ethereum main chain in June 2016 . The response to this was the forced balance adjustment hard fork on 20th July 2016 commonly known as the DAO fork . Although this adjustment was not implemented as a rollback, and although Ethereum was using PoW at the time and did not make any formal finality guarantees, it did override transfers that would heuristically have been considered final at the fork height. Again, this flaw was independent of the consensus mechanism. The DAO fork was of course much more controversial than the Bitcoin fork, and a substantial minority of mining nodes split off to form Ethereum Classic. In any case, the point of this example is that it’s always possible to override finality in response to an exceptional situation, and that a chain’s community may decide to do so. The fact that Ethereum 2.0 now does claim a finality guarantee, would not in practice prevent a similar response in future that would override that guarantee. The question then is whether the procedure to override finality should be formalized or ad hoc. We argue that it should be formalized, including specifying the governance process to be used. This makes security analysis — of the consensus protocol per se, of the governance process, and of their interaction — much more feasible. Arguably a complete security analysis is not possible at all without it. It also front‑loads arguing about what procedure should be followed, and so it is more likely that stakeholders will agree to follow the process in any time‑critical incident.","breadcrumbs":"Design » Crosslink » The Arguments for Bounded Availability and Finality Overrides » Finality Overrides","id":"63","title":"Finality Overrides"},"64":{"body":"There is another possible way to model a protocol that claims finality but can be overridden in practice. We could say that the protocol after the override is a brand‑new protocol and chain (inheriting balances from the previous one, possibly modulo adjustments such as those that happened in the DAO fork). Although that would allow saying that the finality property has technically not been violated, it does not match how users think about an override situation. They are more likely to think of it as a protocol with finality that can be violated in exceptional cases — and they would reasonably want to know what those cases are and how they will be handled. It also does nothing to help with security analysis of such cases.","breadcrumbs":"Design » Crosslink » The Arguments for Bounded Availability and Finality Overrides » A way of modelling overrides that is insufficient","id":"64","title":"A way of modelling overrides that is insufficient"},"65":{"body":"Finality overrides and bounded availability are complementary in the following way: if a problem is urgent enough, then validators can be asked to stop validating. For genuinely harmful problems, it is likely to be in the interests of enough validators to stop that this causes a finalization stall. If this lasts longer than the availability bound then the protocol will go into Stalled Mode, giving time for the defined governance process to occur and decide what to do. And because the unfinalized consensus chain will contain only a limited period of user transactions that spend funds, the option of a long rollback remains realistically open. If, on the other hand, there is time pressure to make a governance decision about a rollback in order to reduce its length, that may result in a less well-considered decision. A possible objection is that there might be a coalition of validators who ignore the request to stop (possibly including the attacker or validators that an attacker can bribe), in which case the finalization stall would not happen. But that just means that we don’t gain the advantage of more time to make a governance decision; it isn’t actively a disadvantage relative to alternative designs. This outcome can also be thought of as a feature rather than a bug: going into Stalled Mode should be a last resort, and if the argument given for the request to stop failed to convince a sufficient number of validators that it was reason enough to do so, then perhaps it wasn’t a good enough reason. Rationale This resolves one of the main objections to the original Stalled Mode idea that stopped us from implementing it in Zcash. The original proposal was to use a signature with a key held by ECC to trigger Stalled Mode, which would arguably have been too centralized. The Stalled Mode described in this document, on the other hand, can only be entered by consensus of a larger validator set, or if there is an availability failure of the finalization protocol. It is also possible to make the argument that the threshold of stake needed is imposed by technical properties of the finality protocol and by the resources of the attacker, which might not be ideal for the purpose described above. However, we would argue that it does not need to be ideal, and will be in the right ballpark in practice. There’s a caveat related to doing intentional rollbacks when using the Stalled Mode approach, where block production in the more-available protocol continues during a long finalization stall. What happens to incentives of block producers (miners in the case of Proof‑of‑Work), given that they know the consensus chain might be intentionally rolled back? They might reasonably conclude that it is less valuable to produce those blocks, leading to a reduction of hash rate or other violations of the security assumptions of Πlc​. This is actually fairly easy to solve. We have the governance procedures say that if we do an intentional rollback, the coinbase-only mining rewards will be preserved. I.e. we produce a block or blocks that include those rewards paid to the same addresses (adjusting the consensus to allow them to be created from thin air if necessary), have everyone check it thoroughly, and require the chain to restart from that block. So as long as block producers believe that this governance procedure will be followed and that the chain will eventually recover at a reasonable coin price, they will still have incentive to produce on Πlc​, at least for a time. Rationale Although the community operating the governance procedures has already obtained the security benefit of mining done on the rolled-back chain by the time it creates the new chain, there is a strong incentive not to renege on the agreement with miners, because the same situation may happen again.","breadcrumbs":"Design » Crosslink » The Arguments for Bounded Availability and Finality Overrides » Complementarity","id":"65","title":"Complementarity"},"66":{"body":"Earlier we said that there were two possible approaches to preventing hazards during a long finalization stall: a) go into a Stalled Mode that directly disallows hazardous transactions (for example, by requiring Πlc​ blocks to be coinbase-only in Zcash); b) temporarily cause the more-available chain to stall. This section describes an important class of potential attacks on approach b) that are difficult to resolve. They are based on the fact that when the unfinalized chain stalls, an adversary has more time to find blocks, and this might violate security assumptions of the more-available protocol. For instance, if the more-available protocol is PoW-based, then its security in the steady state is predicated on the fact that an adversary with a given proportion of hash power has only a limited time to use that power, before the rest of the network finds another block. Background For an analysis of the concrete security of Nakamoto-like protocols, see [DKT+2020] and [GKR2020] . These papers confirm the intuition that the “private attack” —in which an adversary races privately against the rest of the network to construct a forking chain— is optimal, obtaining the same tight security bound independently using different techniques. During a chain stall, the adversary no longer has a limited time to construct a forking chain. If, say, the adversary has 10% hash power, then it can on average find a block in 10 block times. And so in 100 block times it can create a 10-block fork. It may in fact be worse than this: once miners know that a finalization stall is happening, their incentive to continue mining is reduced, since they know that there is a greater chance that their blocks might be rolled back. So we would expect the global hash rate to fall —even before the finality gap bound is hit— and then the adversary would have a greater proportion of hash rate. Info Even in a pure Ebb‑and‑Flow protocol, a finalization stall could cause miners to infer that their blocks are more likely to be rolled back, but the fact that the chain is continuing would make that more difficult to exploit. This issue with the global hash rate is mostly specific to the more-available protocol being PoW: if it were PoS, then its validators might as well continue proposing blocks because it is cheap to do so. There might be other attacks when the more-available protocol is PoS; we haven’t spent much time analyzing that case. The problem is that the more-available chain does not necessarily just halt during a chain stall. In fact, for a finality gap bound of k blocks, an adversary could cause the k-block “tail” of the chain as seen by any given node to “thrash” between different chains. We will call this a tail‑thrashing attack . If a protocol allowed such attacks then it would be a regression relative to the security we would normally expect from an otherwise similar PoW-based protocol. It only occurs during a finalization stall, but note that we cannot exclude the possibility of an adversary being able to provoke a finalization stall. Note that in the Snap‑and‑Chat construction, snapshots of Πlc​ are used as input to the BFT protocol. That implies that the tail‑thrashing problem could also affect the input to that protocol, which would be bad (not least for security analysis of availability, which seems somewhat intractable in that case). Also, when restarting Πlc​, we would need to take account of the fact that the adversary has had an arbitrary length of time to build long chains from every block that we could potentially restart from. It could be possible to invalidate those chains by requiring blocks after the restart to be dependent on fresh randomness, but that sounds quite tricky (especially given that we want to restart without manual intervention if possible), and there may be other attacks we haven’t thought of. This motivates using approach a) instead. Note that we have still glossed over precisely how consensus rules would change to enforce a). This will be covered later in The Crosslink 2 Construction , but first we will discuss other issues with Snap-and-Chat.","breadcrumbs":"Design » Crosslink » The Arguments for Bounded Availability and Finality Overrides » Tail-thrashing attacks","id":"66","title":"Tail-thrashing attacks"},"67":{"body":"The discussion in The Argument for Bounded Availability and Finality Overrides is at an abstract level, applying to any Ebb‑and‑Flow-like protocol. This document considers specifics of the Snap‑and‑Chat construction proposed in [NTT2020] ( arXiv version ). Terminology We are trying to be precise in this document about use of the terms “Ebb‑and‑Flow”, which is the security model and goal introduced in [NTT2020] , vs “Snap‑and‑Chat”, which is the construction proposed in the same paper to achieve that goal. There are other ways to design an Ebb‑and‑Flow protocol that don’t run into the difficulties described in this section (or that run into different difficulties).","breadcrumbs":"Design » Crosslink » Notes On Snap‑and‑Chat » Notes on Snap‑and‑Chat","id":"67","title":"Notes on Snap‑and‑Chat"},"68":{"body":"A general problem with the Snap‑and‑Chat construction is that it does not follow, from enforcement of the original consensus rules on blocks produced in Πlc​, that the properties they are intended to enforce hold for the LOGfin,it​ or LOGda,it​ ledgers. Less obviously, the converse also does not follow: enforcing unmodified consensus rules on Πlc​ blocks is both too lax and too strict. Recall from the paper how LOGfin,it​ and LOGda,it​ are constructed (starting at the end of page 8 of [NTT2020] ): Ledger extraction: Finally, how honest nodes compute LOGfin,it​ and LOGda,it​ from Chit​ and chit​ is illustrated in Figure 6. Recall that Chit​ is an ordering of snapshots, i.e., a chain of chains of LC blocks. First, Chit​ is flattened, i.e. the chains of blocks are concatenated as ordered to arrive at a single sequence of LC blocks. Then, all but the first occurrence of each block are removed (sanitized) to arrive at the finalized ledger LOGfin,it​ of LC blocks. To form the available ledger LOGda,it​, chit​, which is a sequence of LC blocks, is appended to LOGfin,it​ and the result again sanitized. This says that LOGfin,it​ and LOGda,it​ are sequences of transactions, not sequences of blocks. Therefore, consensus rules defined at the block level are not applicable. Zcash-specific Most of these rules are Proof‑of‑Work‑related checks that can be safely ignored at this level. Some are related to the hashBlockCommitments field intended for use by the FlyClient protocol. It is not at all clear how to make FlyClient (or other uses of this commitment) work with the Snap‑and‑Chat construction. In particular, the hashEarliest{Sapling,Orchard}Root, hashLatest{Sapling,Orchard}Root, and n{Sapling,Orchard}TxCount fields don’t make sense in this context since they could only reflect the values in chit​, which have no relation in general to those for any subrange of transactions in LOGda,it​. This problem occurs as a result of sanitization and so will be avoided by Crosslink 2, which does not need sanitization. Since LOGda,it​ does not have blocks, it is not well-defined whether it has “coinbase‑only blocks” when in Stalled Mode. That by itself is not so much of a problem because it would be sufficient for it to have only coinbase transactions in that mode.","breadcrumbs":"Design » Crosslink » Notes On Snap‑and‑Chat » Effect on consensus","id":"68","title":"Effect on consensus"},"69":{"body":"The issuance schedule of Zcash was designed under the assumption that blocks only come from a single chain, and that the difficulty adjustment algorithm keeps the rate of block mining roughly constant over time. For Snap‑and‑Chat, if there is a rollback longer than σ blocks in Πlc​, additional coinbase transactions from the rolled-back chain will be included in LOGfin​. We can argue that this will happen rarely enough not to cause any significant problem for the overall issuance schedule. However, it does mean that issuance is less predictable, because the block subsidies will be computed according to their depth in the Πlc​ chain on which they were mined. So it would no longer be the case that coinbase transactions issue a deterministic, non-increasing sequence of block subsidies. (Again, this problem will be avoided by Crosslink 2.)","breadcrumbs":"Design » Crosslink » Notes On Snap‑and‑Chat » Effect on issuance","id":"69","title":"Effect on issuance"},"7":{"body":"See Wikipedia’s Technology Tree - History section for details.","breadcrumbs":"Introduction » A Path to Proof-of-Stake Zcash » Footnotes","id":"7","title":"Footnotes"},"70":{"body":"The order of transactions in any particular chit​ is not in general preserved in either LOGfin​ or LOGda​. This is considered in the paper (middle of the left column on page 10) but it is very easy to miss it: Thus, snapshots taken by different nodes or at different times can conflict. However, Πbft​ is still safe and thus orders these snapshots linearly. Any transactions invalidated by conflicts are sanitized during ledger extraction. That is, a transaction from one snapshot might double-spend an output already spent in a different transaction of a different snaphot earlier in the flattening order. If it is omitted, then later transactions could depend on the outputs of the omitted one. The paper is saying that each transaction is only included if (in Bitcoin and Zcash terminology) it satisfies contextual checks for double-spending and existence of inputs at the point in the ledger where it would be added. Since nullifiers for shielded spends are public, it is possible to do this even for shielded transactions. Each node i will construct commitment trees in the order given by LOGda,it​. Info This means that if LOGfin,i​ is extended by a block that is not the next block in LOGda,i​ after the finalization point (and that has different note commitments), then all shielded transactions from that point onward in the previous LOGda,i​ will be invalidated. It could be possible to do better at the expense of a more complicated note commitment tree structure. In any case, this situation is expected to be rare, because it can only occur if there is a rollback of more than σ blocks in the Πlc​ consensus chain or a failure of BFT safety.","breadcrumbs":"Design » Crosslink » Notes On Snap‑and‑Chat » Effect on transaction ordering","id":"70","title":"Effect on transaction ordering"},"71":{"body":"There are two possible ways to interpret how LOG{fin,da}​ are constructed in Snap‑and‑Chat: Concatenate the transactions from each final BFT block snapshot of an LC chain, and sanitize the resulting transaction sequence by including each transaction iff it is contextually valid. Concatenate the blocks from each final BFT block snapshot of an LC chain, remove duplicate blocks , and only then sanitize the resulting transaction sequence by including each transaction iff it is contextually valid. These are equivalent in the setting considered by [NTT2020] , but the argument for their equivalence is not obvious. We definitely want them to be equivalent: in practice there will be many duplicate blocks from chain prefixes in the input to sanitization, and so a literal implementation of the first variant would have to recheck all duplicate transactions for contextual validity. That would have at least O(n2) complexity (more likely O(n2logn)) in the length n of the block chain, because the length of each final snapshot grows with n. Suppose that, in a particular Πlc​, the only reasons for a transaction to be contextually invalid are double-spends and missing inputs. In that case the argument for equivalence is: If a transaction is omitted due to a double-spend, then any subsequent time it is checked, that input will still have been double-spent. If a transaction is omitted due to a missing input, this can only be because an earlier transaction in the input to sanitization was omitted. So the structure of omitted transactions forms a DAG in which parent links must be to earlier omitted transactions. The roots of the DAG are at double-spending transactions, which cannot be reinstated. A child cannot be reinstated until its parents have been reinstated. Therefore, no transactions are reinstated. Note that any other reason for transactions to be contextually invalid might interfere with this argument. Therefore, strictly speaking Snap‑and‑Chat should require of Πlc​ that there is no such other reason. This does not seem to be explicitly stated anywhere in [NTT2020] . Zcash-specific In Zcash a transaction can also be contextually invalid because it has expired, or because it has a missing anchor. Expiry can be handled by extending the above argument as follows: If a transaction is omitted due to having expired, then any subsequent time it is checked, it will still be expired. It is not obvious how to extend it to handle missing anchors, because it is technically possible for a duplicate transaction that was invalid because of a missing anchor to become valid in a subsequent block. That situation would require careful manipulation of the commitment trees, but there does not seem to be anything preventing it from being provoked intentionally. The argument that was used above for missing inputs does not work here, because there is no corresponding DAG formed by the transactions with missing anchors: the same commitment treestate can be produced by two unrelated transactions.","breadcrumbs":"Design » Crosslink » Notes On Snap‑and‑Chat » Subtlety in the definition of sanitization","id":"71","title":"Subtlety in the definition of sanitization"},"72":{"body":"Transactions in chit​ need to be able to spend outputs that are not necessarily from any previous transaction in chit​. This is because, from the point of view of a user of node i at time t, the block chain includes all transactions in LOGda,it​. All of the transactions after the finalization point are guaranteed to also be in chit​, but the ones before the finalization point (i.e. in LOGfin,it​) are not, because they could be from some other chju​ for u≤t and j=i (intuitively, from some long chain fork that was once considered confirmed by enough nodes). Info Honest nodes only ever vote for confirmed snapshots, that is, prefixes of their best Πlc​ chain truncated by the confirmation depth σ. Obviously the whole point of having the BFT protocol is that chain forks longer than σ can occur in Πlc​ — otherwise we'd just use Πlc​ and have done. So it is not that we expect this case to be common, but if it happens then it will never fix itself: the consensus chain in Πlc​ will continue on without ever including the transactions from LOGfint​ that were obtained from a snapshot of another fork. A user must be able to spend outputs for which they hold the spending key from any finalized transaction, otherwise there would be no point to the finalization. The authors of [NTT2020] probably just missed this: the paper only has evidence that they simulated their construction, rather than implementing it for Bitcoin or any other concrete block chain as Πlc​. Let’s try to repair it. Suppose that node j is trying to determine whether chit​ is a consensus-valid chain, which is necessary for deterministic consensus in Πlc​. It cannot decide whether to allow transactions in chit​ to spend outputs not in the history of chit​ on the basis of its own finalized view LOGfin,jt​, because LOGfin,it​ and LOGfin,jt​ are not in general the same. Of course, we hope that LOGfin,it​ and LOGfin,jt​ are consistent, i.e. one is a prefix of the other. But even if they are consistent, they are not necessarily the same length. In particular, if LOGfin,jt​ is shorter than LOGfin,it​, then node j does not have enough information to fill in the gap — and so it may incorrectly view a transaction in chit​ as spending an output that does not exist, when actually it does exist in LOGfin,it​∖LOGfin,jt​. Conversely if LOGfin,jt​ were longer and node j were to allow spending an output in LOGfin,jt​∖LOGfin,it​, that would be using information that is not necessarily available to other nodes, and so node j could diverge from consensus. Consensus validity of the block at the tip of chit​ can only be a deterministic function of the block itself and its ancestors in chit​. It is crucial to be able to eventually spend outputs from the finalized chain. We are forced to conclude that the chain chit​ must include the information needed to calculate LOGfin,is​ for some s not too far behind t. That is, Πlc​ must be modified to ensure that this is the case. This leads us to strengthen the required properties of an Ebb‑and‑Flow protocol to include another property, “finalization availability”.","breadcrumbs":"Design » Crosslink » Notes On Snap‑and‑Chat » Spending finalized outputs","id":"72","title":"Spending finalized outputs"},"73":{"body":"In the absence of security flaws and under the security assumptions required by the finality layer, the finalization point will not be seen by any honest node to roll back. However, that does not imply that all nodes will see the same finalized height — which is impossible given network delays and unreliable messaging. Both in order to optimize the availability of applications that require finality, and in order to solve the technical issue of spending finalized outputs described in the previous section, we need to consider availability of the information needed to finalize the chain up to a particular point. Note that in Bitcoin-like consensus protocols, we don’t generally consider it to be an availability flaw that a block header only commits to the previous block hash and to the Merkle tree of transactions in the block, rather than including them directly. These commitments allow nodes to check that they have the correct information, which can then be requested separately. Suppose, then, that each block header in Πlc​ commits to the L ast F inal BFT block known by the Πlc​ block producer. For an LC block chain with block H at its tip, we will refer to this commitment as LF(H). We refer to the parent block of H as H⌈bc1​ (this is a special case of a notation that will be defined in The Crosslink 2 Construction ). Consensus rule: Extension We require, as a consensus rule, that if H is not the genesis block header, then this BFT block either descends from or is the same as the final BFT block committed to by the Πlc​ block’s parent. That is, LF(H⌈bc1​)⪯bft​LF(H). This Extension rule will be preserved into Crosslink 2. The Extension rule does not prevent the BFT chain from rolling back, if the security assumptions of Πbft​ were violated. However, it means that if a node i does not observe a rollback in Πlc​ at confirmation depth σ, then it will also not observe any instability in LOGfin,i​, even if the security assumptions of Πbft​ are violated. This property holds by construction , and in fact regardless of Πbft​. Info In the Snap‑and‑Chat construction, we also have BFT block proposals committing to Πlc​ snapshots (top of right column of [ NTT2020 , page 7]): In addition, chit​ is used as side information in Πbft​ to boycott the finalization of invalid snapshots proposed by the adversary. This does not cause any circularity, because each protocol only commits to earlier blocks of the other. In fact, BFT validators have to listen to transmission of Πlc​ block headers anyway, so that could be also the protocol over which they get the information needed to make and broadcast their own signatures or proposals. (A possible reason not to broadcast individual signatures to all nodes is that with large numbers of validators, the proof that a sufficient proportion of validators/stake has signed can use an aggregate signature, which could be much smaller. Also, Πlc​ nodes only need to know about successful BFT block proposals.) Now suppose that, in a Snap‑and‑Chat protocol, the BFT consensus finalizes a Πlc​ snapshot that does not extend the snapshot in the previous block (which can happen if either Πbft​ is unsafe, or Πlc​ suffers a rollback longer than σ blocks). In that case we will initially not be able to spend outputs from the old snapshot in the new chain. But eventually for some node i that sees the header H at the tip of its best chain at time t, LF(H) will be such that from then on (i.e. at time u≥t), LOGfin,iu​ includes the output that we want to spend. This assumes liveness of Πlc​ and safety of Πbft​. That is, including a reference to a recent final BFT block in Πlc​ block headers both incentivizes nodes to propagate this information, and can be used to solve the “spending finalized outputs” problem. Optionally, we could incentivize the block producer to include the latest information it has, for example by burning part of the block reward or by giving the producer some limited mining advantage that depends on how many Πlc​ blocks back the finalization information is. This raises the question of how we measure how far ahead a given block is relative to the finalization information it provides. As we said before, LOGfin,it​ is a sequence of transactions, not blocks. The transactions will in general be in a different order, and also some transactions from chit​ may have been omitted from LOGfin,it​ (and even LOGda,it​) because they were not contextually valid. Info In Crosslink 2 , we will sidestep this problem by avoiding the need for sanitization — that is, LOGfin,it​ will correspond exactly to a chain of blocks that is a prefix of chit​. Actually we use the notation finit​ to reflect the fact that it is a bc‑chain, not a sequence of bc‑transactions. This invariant is maintained statefully on each node i: any rollback past finit​ will be ignored. If a new finit​ would conflict with the old one, the node will refuse to use it. This allows each node to straightforwardly measure how many blocks chit​ is ahead of finit​ as the difference in heights. Since this document is intended to explain the development of Crosslink from Snap‑and‑Chat, here we describe the more complicated approach that we originally came up with for Crosslink 1 — which also serves to motivate the simplification in Crosslink 2. Assume that a block unambiguously specifies its ancestor chain. For a block H, define:tailhead(H):=last-common-ancestor(snapshot(LF(H)),H)finality-depth(H):=height(H)−height(tailhead(H)) Here LF(H) is the BFT block we are providing information for, and snapshot(LF(H)) is the corresponding Πlc​ snapshot. For a node i that sees LF(H) as the most recent final BFT block at time t, LOGfin,it​ will definitely contain transactions from blocks up to tailhead(H), but usually will not contain subsequent transactions on H’s fork. Info Strictly speaking, it is possible that a previous BFT block took a snapshot H′ that is between tailhead(H) and H. This can only happen if there have been at least two rollbacks longer than σ blocks (i.e. we went more than σ blocks down H’s fork from tailhead(H), then reorged to more than σ blocks down snapshot(LF(H))’s fork, then reorged again to H’s fork). In that case, the finalized ledger would already have the non-conflicting transactions from blocks between tailhead(H) and H′ — and it could be argued that the correct definition of finality depth in such cases is the depth of H′ relative to H, not of tailhead(H) relative to H. However, The definition above is simpler and easier to compute. The effect of overestimating the finality depth in such corner cases would only cause us to enforce Stalled Mode slightly sooner, which seems fine (and even desirable) in any situation where there have been at least two rollbacks longer than σ blocks. By the way, the “tailhead” of a tailed animal is the area where the posterior of the tail joins the rump (also called the “dock” in some animals). We could alternatively just rely on the fact that some proportion of block producers are honest and will include the latest information they have. However, it turns out that having a definition of finality depth will also be useful to enforce going into Stalled Mode. Specifically, if we accept the above definition of finality depth, then the security property we want is Definition Bounded hazard-freeness for a finality gap bound of L blocks: There is never, for any node i at time t, observed to be a more-available ledger LOGda,it​ with a hazardous transaction that comes from block H of chit​ such that finality-depth(H)>L. Info This assumes that transactions in the non-finalized suffix LOGda,it​∖LOGfin,it​ come from blocks in chit​. In Snap‑and‑Chat they do by definition, but ideally we wouldn’t depend on that. The difficulty in finding a more general security definition is due to the ledgers in an Ebb‑and‑Flow protocol being specified as sequences of transactions, so that a depth in the ledger would have only a very indirect correspondence to time. We could instead base a definition on timestamps, but that could run into difficulties in ensuring timestamp accuracy. Another possibility would be to count the number of coinbase transactions in LOGda,it​∖LOGfin,it​ before the hazardous transaction. This would still be somewhat ad hoc (it depends on the fact that coinbase transactions happen once per block and cannot conflict with any other distinct transaction). In any case, if finality-depth sometimes overestimates the depth, that cannot weaken this security definition. Note that a node that is validating a chain chit​ must fetch all the chains referenced by BFT blocks reachable from it (back to an ancestor that it has seen before). In theory, there could be a partition that causes there to be multiple disjoint snapshots that get added to the BFT chain in quick succession. However, in practice we expect such long rollbacks to be rare if Πlc​ is meeting its security goals. Going into Stalled Mode if there is a long finalization stall helps to reduce the cost of validation when the stall resolves. That is, if there is a partition and nodes build on several long chains, then in unmodified Snap‑and‑Chat, it could be necessary to validate an arbitrary number of transactions on each chain when the stall resolves. Having only coinbase transactions after a certain point in each chain would significantly reduce the concrete validation costs in this situation. Nodes should not simply trust that the BFT blocks are correct; they should check validator signatures (or aggregate signatures) and finalization rules. Similarly, Πlc​ snapshots should not be trusted just because they are referenced by BFT blocks; they should be fully validated, including the proofs-of-work. It is also possible for a snapshot reference to include the subsequent σ block headers, which are guaranteed to be available for a confirmed snapshot. Having all nodes validate the proofs-of-work in these headers is likely to significantly increase the work that an attacker would need to perform to cause disruption under a partial failure of either Πbft​ or Πlc​’s security properties. Info Note that [NTT2020] (bottom of right column, page 9) makes a safety assumption about Πlc​ in order to prove the consistency of LOGfin​ with the output of Πlc​: As indicated by Algorithm 1, a snapshot of the output of Πlc​ becomes final as part of a BFT block only if that snapshot is seen as confirmed by at least one honest node. However, since Πlc​ is safe [i.e., does not roll back further than the confirmation depth σ], the fact that one honest node sees that snapshot as confirmed implies that every honest node sees the same snapshot as confirmed. We claim that, while this may be a reasonable assumption to make for parts of the security analysis, in practice we should always require any adversary to do the relevant amount of Proof‑of‑Work to construct block headers that are plausibly confirmed. This is useful even though we cannot require, for every possible attack, that it had those headers at the time they should originally have appeared.","breadcrumbs":"Design » Crosslink » Notes On Snap‑and‑Chat » Finalization Availability","id":"73","title":"Finalization Availability"},"74":{"body":"The following idea for enforcing finalization availability and a bound on the finality gap was originally conceived before we had switched to advocating the Stalled Mode approach. It’s simpler to explain first in that variant. Suppose that for an L-block availability bound, we required each block header to include the information necessary for a node to finalize to L blocks back. This would automatically enforce a chain stall after the availability bound without any further explicit check, because it would be impossible to produce a block after the bound. Note that if full nodes have access to the BFT chain, knowing LF(H) is sufficient to tell whether the correct version of any given BFT block in LF(H)’s ancestor chain has been obtained. Suppose that the finality gap bound is L blocks. Having already defined finality-depth, the necessary Πlc​ consensus rule is attractively simple: Consensus rule For every Πlc​ block H, finality-depth(H)≤L. To adapt this approach to enforce Stalled Mode instead of stalling the chain, we can allow the alternative of producing a block that follows the Stalled Mode restrictions: Consensus rule For every Πlc​ block H, either finality-depth(H)≤L, or H follows the Stalled Mode restrictions. Note that Stalled Mode will be exited automatically as soon as the finalization point catches up to within L blocks (if it does without an intentional rollback). Typically, after recovery from whatever was causing the finalization stall, the validators will be able to obtain consensus on the same chain as LOGda​, and so there will be no rollback (or at least not a long one) of LOGda​. Info An earlier iteration of this idea required the finalization information to be included in Πlc​ block headers. This is not necessary when we assume that full nodes have access to the BFT chain and can obtain arbitrary BFT blocks. This also sidesteps any need to relax the rule in order to bound the size of Πlc​ block headers. Πlc​ block producers are still incentivized to make the relevant BFT blocks available, because without them the above consensus rule cannot be checked, and so their Πlc​ blocks would not be accepted. There is, however, a potential denial-of-service attack by claiming the existence of a BFT block that is very far ahead of the actual BFT chain tip. This attack is not very serious as long as nodes limit the number of BFT blocks they will attempt to obtain in parallel before having checked validator signatures.","breadcrumbs":"Design » Crosslink » Notes On Snap‑and‑Chat » Enforcing Finalization Availability and Stalled Mode","id":"74","title":"Enforcing Finalization Availability and Stalled Mode"},"75":{"body":"Consider Lemma 5: Moreover, for a BFT block to become final in the view of an honest node i under (A2∗​,Z2​), at least one vote from an honest node is required, and honest nodes only vote for a BFT block if they view the referenced LC block as confirmed. The stated assumptions are: (A2​(β),Z2​) formalizes the model of P2 , a synchronous network under dynamic participation, with respect to a bound β on the fraction of awake nodes that are adversarial: At all times, A2​ is required to deliver all messages sent between honest nodes in at most Δ slots. At all times, A2​ determines which honest nodes are awake/asleep and when, subject to the constraint that at all times at most fraction β of awake nodes are adversarial and at least one honest node is awake. (A2∗​,Z2​) is defined as (A2​(21​,Z2​)). Now consider this statement and figure: Even if Πbft​ is unsafe (Figure 9c), finalization of a snapshot requires at least one honest vote, and thus only valid snapshots become finalized. Figure 9 of [NTT2020] This argument is technically correct but has to be interpreted with care. It only applies when the number of malicious nodes f is such that n/3<f<n/2. What we are trying to do with Crosslink is to ensure that a similar conclusion holds even if Πbft​ is completely subverted, i.e. the adversary has 100% of validators (but only < 50% of Πlc​ hash rate).","breadcrumbs":"Design » Crosslink » Notes On Snap‑and‑Chat » Comment on security assumptions","id":"75","title":"Comment on security assumptions"},"76":{"body":"We are now ready to give a description of a protocol that takes into account the issues described in Notes on Snap‑and‑Chat , and that implements bounded availability . We call this the “Crosslink” construction; more precisely the version described here is “Crosslink 2”. This description will attempt to be self-contained, but [NTT2020] ( arXiv version ) is useful background on the general model of Ebb-and-Flow protocols.","breadcrumbs":"Design » Crosslink » The Crosslink 2 Construction » The Crosslink 2 Construction","id":"76","title":"The Crosslink 2 Construction"},"77":{"body":"“∗” is a metavariable for the name of a protocol. We also use it as a wildcard in protocol names of a particular type, for example “∗bc” for the name of some best‑chain protocol. Protocols are referred to as Π∗​ for a name “∗”. Where it is useful to avoid ambiguity, when referring to a concept defined by Π∗​ we prefix it with “∗‑”. We do not take synchrony or partial synchrony as an implicit assumption of the communication model; that is, unless otherwise specified , messages between protocol participants can be arbitrarily delayed or dropped. A given message is received at most once, and messages are nonmalleably authenticated as originating from a given sender whenever needed by the applicable protocol. Particular subprotocols may require a stronger model. Background For an overview of communication models used to analyze distributed protocols, see this blog post by Ittai Abraham . Discussion of incorrect applications of the GST formalization of partial synchrony to continuously operating protocols. The original context for the definition of the partially synchronous model in [DLS1988] was for “one‑shot” Byzantine Agreement — called “the consensus problem” in that paper. The following argument is used to justify assuming that all messages from the Global Stabilization Time onward are delivered within the upper time bound Δ: Therefore, we impose an additional constraint: For each execution there is a global stabilization time (GST), unknown to the processors, such that the message system respects the upper bound Δ from time GST onward. This constraint might at first seem too strong: In realistic situations, the upper bound cannot reasonably be expected to hold forever after GST, but perhaps only for a limited time. However, any good solution to the consensus problem in this model would have an upper bound L on the amount of time after GST required for consensus to be reached; in this case it is not really necessary that the bound Δ hold forever after time GST, but only up to time GST +L. We find it technically convenient to avoid explicit mention of the interval length L in the model, but will instead present the appropriate upper bounds on time for each of our algorithms. Several subsequent authors applying the partially synchronous model to block chains appear to have forgotten or neglected this context. In particular, the argument depends on the protocol completing soon after GST. Obviously a block‑chain protocol does not satisfy this assumption; it is not a “one‑shot” consensus problem. This assumption could be removed, but some authors of papers about block‑chain protocols have taken it to be an essential aspect of modelling partial synchrony. I believe this is contrary to the intent of [DLS1988] : Instead of requiring that the consensus problem be solvable in the GST model, we might think of separating the correctness conditions into safety and termination properties. The safety conditions are that no two correct processors should ever reach disagreement, and that no correct processor should ever make a decision that is contrary to the specified validity conditions. The termination property is just that each correct processor should eventually make a decision. Then we might require an algorithm to satisfy the safety conditions no matter how asynchronously the message system behaves, that is, even if Δ does not hold eventually. On the other hand, we might only require termination in case Δ holds eventually. It is easy to see that these safety and termination conditions are [for the consensus problem] equivalent to our GST condition: If an algorithm solves the consensus problem when Δ holds from time GST onward, then that algorithm cannot possibly violate a safety property even if the message system is completely asynchronous. This is because safety violations must occur at some finite point in time, and there would be some continuation of the violating execution in which Δ eventually holds. This argument is correct as stated, i.e. for the one‑shot consensus problem. Subtly, essentially the same argument can be adapted to protocols with safety properties that need to be satisfied continuously. However, it cannot correctly be applied to liveness properties of non‑terminating protocols. The authors (Cynthia Dwork, Nancy Lynch, and Larry Stockmeyer) would certainly have known this: notice how they carefully distinguish “the GST model” from “partial synchrony”. They cannot plausibly have intended this GST formalization to be applied unmodified to analyze liveness in such protocols, which seems to be common in the block‑chain literature, including in the Ebb-and-Flow paper [NTT2020] and the Streamlet paper [CS2020] . The Ebb-and-Flow paper acknowledges the issue by saying “Although in reality, multiple such periods of (a‑)synchrony could alternate, we follow the long‑standing practice in the BFT literature and study only a single such transition.” This is not adequate: “long‑standing practice” notwithstanding, it is not valid in general to infer that properties holding for the first transition to synchrony also apply to subsequent transitions (where the protocol can be in states that would not occur initially), and it is plausible that this inference could fail for real protocols. The Streamlet paper also refers to “periods of synchrony” which indicates awareness of the issue, but then it uses the unmodified GST model in the proofs. Informally, to solve this issue it is necessary to also prove that existing progress is maintained during periods of asynchrony, and that during such periods the protocol remains in states where it will be able to take advantage of a future period of synchrony to make further progress. This provides further motivation to avoid taking the GST formalization of partial synchrony as a basic assumption. Note that the recent result [CGSW2024] does not contradict anything we say here. Although the GST and Unknown Latency models are “equally demanding” in the sense of existence of protocols that satisfy a given goal, this result does not show that the models are equivalent for any specific protocol. In particular the requirements of the “clock‑slowing” technique fail in practice for any protocol involving Proof‑of‑Work. A ∗‑execution is the complete set of events (message sends/receives and decisions by protocol participants) that occur in a particular run of Π∗​ from its initiation up to a given time. A prefix of a ∗‑execution is also a ∗‑execution. Since executions always start from protocol initiation, a strict suffix of a ∗‑execution is not a ∗‑execution. Times are modelled as values of a totally ordered type Time with minimum value 0. For convenience, we consider all protocol executions to start at time 0. Remark Although protocols may be nondeterministic, an execution fixes the events that occur and times at which they occur, for the purpose of modeling. For simplicity, we assume that all events occur at global times in a total ordering. This assumption is not realistic in an asynchronous communication model, but it is not essential to the design or analysis and could be removed: we could use a partial happens-before ordering on events in place of a total ordering on times. A “∗‑node” is a participant in Π∗​ (the protocol may be implicit). A ∗‑node is “honest at time t” in a given execution iff it has followed the protocol up to and including time t in that execution. A time series on type U is a function S⦂Time→U assigning a value of U to each time in an execution. By convention, we will write the time t as a superscript: St=S(t). A ∗‑chain is a nonempty sequence of ∗‑blocks, starting at the “genesis block” O∗​, in which each subsequent block refers to its preceding or “parent block” by a collision‑resistant hash. The “tip” of a ∗‑chain is its last element. For convenience, we conflate ∗‑blocks with ∗‑chains; that is, we identify a chain with the block at its tip. This is justified because, assuming that the hash function used for parent links is collision‑resistant, there is exactly one ∗‑chain corresponding to a ∗‑block; and conversely there is exactly one ∗‑block at the tip of a ∗‑chain. If C is a ∗‑chain, C⌈∗k​ means C with the last k blocks pruned, except that if len(C)≤k, the result is the genesis ∗‑chain consisting only of O∗​. The block at depth k∈N in a ∗‑chain C is defined to be the tip of C⌈∗k​. Thus the block at depth k in a chain is the last one that cannot be affected by a rollback of length k (this also applies when len(C)≤k because the genesis ∗‑chain cannot roll back). Terminology Our usage of “depth” is different from [NTT2020] , which uses “depth” to refer to what Bitcoin and Zcash call “height”. It also differs by 1 from the convention for confirmation depths in zcashd, where the tip is considered to be at depth 1, rather than 0. For ∗‑blocks B and C: The notation B⪯∗​C means that the ∗‑chain with tip B is a prefix of the one with tip C. This includes the case B=C. The notation B⪯⪰∗​C means that either B⪯∗​C or C⪯∗​B. That is, “one of B and C is a prefix of the other”. This also includes the case B=C. The notation B/⪯⪰∗​C means that both B⪯∗​C and C⪯∗​B. That is, “neither of B and C is a prefix of the other”. A function S⦂I→∗-block is ∗‑linear iff for every t,u⦂I where t≤u we have S(t)⪯∗​S(u). (This definition can be applied to time series where I=Time, or to sequences of ∗‑blocks where values of I are indices.) Lemma: Linear prefix If A⪯∗​C and B⪯∗​C then A⪯⪰∗​B. Proof: The chain of ancestors of C is ∗-linear, and A, B are both on that chain. The notation [f(X) for X⪯∗​Y] means the sequence of f(X) for each ∗‑block X in chain order from genesis up to and including Y. (X is a bound variable within this construct.) TODO: remove this if not used: We use log⪯log′ (without a subscript on ⪯) to mean that the transaction ledger log is a prefix of log′. Similarly to ⪯⪰∗​ above, log⪯⪰log′ means that either log⪯log′ or log′⪯log; that is, “one of log and log′ is a prefix of the other”.","breadcrumbs":"Design » Crosslink » The Crosslink 2 Construction » Conventions","id":"77","title":"Conventions"},"78":{"body":"In the simplest case, a block‑chain protocol Π∗​ provides a single “view” that, for a given ∗‑execution, provides each ∗‑node with a time series on ∗‑chains. More generally a protocol may define several “views” that provide each ∗‑node with time series on potentially different chain types. We model a ∗‑view as a function V⦂Node×Time→∗-chain. By convention, we will write the node index i as a subscript and the time t as a superscript: Vit​=V(i,t). Definition: Agreement on a view An execution of Π has Agreement on the view V⦂Node×Time→∗-chain iff for all times t, u and all Π nodes i, j (potentially the same) such that i is honest at time t and j is honest at time u, we have Vit​⪯⪰∗​Vju​.","breadcrumbs":"Design » Crosslink » The Crosslink 2 Construction » Views","id":"78","title":"Views"},"79":{"body":"As in Snap‑and‑Chat, we depend on a BFT protocol Πorigbft​, and a best‑chain protocol Πorigbc​. Info See this terminology note for why we do not call Πorigbc​ a “longest‑chain” protocol. We modify Πorigbft​ (resp. Πorigbc​) to give Πbft​ (resp. Πbc​) by adding structural elements, changing validity rules, and changing the specified behaviour of honest nodes. A Crosslink 2 node must participate in both Πbft​ and Πbc​; that is, it must maintain a view of the state of each protocol. Acting in more specific roles such as bft‑proposer, bft‑validator, or bc‑block‑producer is optional, but we assume that all such actors are Crosslink 2 nodes.","breadcrumbs":"Design » Crosslink » The Crosslink 2 Construction » Subprotocols","id":"79","title":"Subprotocols"},"8":{"body":"The hybrid PoW/PoS consensus protocol proposed in this book is similar to today’s Zcash NU5/NU6 consensus protocol with the addition of a Trailing Finality Layer : TODO: Add network topology / software subcomponent diagram #124 The Zcash Trailing Finality Layer refers to a new subprotocol of a new hybrid PoW/PoS protocol, which we refer to as PoW+TFL . This subprotocol introduces assured finality for the Zcash block chain, ensuring that final blocks (and the transactions within them) may never be rolled back. We use the term “layer” because we can understand this design as introducing a new layer to the Zcash network, making only minimal changes to the existing network and consensus protocol. This modular separation is present in the consensus rules, the network protocol, and the code architecture.","breadcrumbs":"Introduction » Trailing Finality Layer in a Nutshell » Trailing Finality Layer in a Nutshell","id":"8","title":"Trailing Finality Layer in a Nutshell"},"80":{"body":"A ∗bft‑node’s view includes a set of ∗bft‑block chains each rooted at a fixed genesis ∗bft‑block O∗bft​. There is a ∗bft‑block‑validity rule (specified below), which depends only on the content of the block and its ancestors. A non‑genesis block can only be ∗bft‑block‑valid if its parent is ∗bft‑block‑valid. A ∗bft‑valid‑chain is a chain of ∗bft‑block‑valid blocks. Execution proceeds in a sequence of epochs. In each epoch, an honest proposer for that epoch may make a ∗bft‑proposal. A ∗bft‑proposal refers to a parent ∗bft‑block, and specifies the proposal’s epoch. The content of a proposal is signed by the proposer using a strongly unforgeable signature scheme. We consider the proposal to include this signature. There is a ∗bft‑proposal‑validity rule, depending only on the content of the proposal and its parent block, and the validity of the proposer’s signature. We extend the ⌈k notation to ∗bft‑proposals in the obvious way: if k≥1, P is a ∗bft‑proposal and B its parent ∗bft‑block, then P⌈∗bftk​:=B⌈∗bftk−1​. Terminology We will shorten “∗bft‑block‑valid ∗bft‑block” to “∗bft‑valid‑block”, and “∗bft‑proposal‑valid ∗bft‑proposal” to “∗bft‑valid‑proposal”. For each epoch, there is a fixed number of voting units distributed between the ∗bft‑nodes, which they use to vote for a ∗bft‑proposal. We say that a voting unit has been cast for a ∗bft‑proposal P at a given time in a ∗bft‑execution, if and only if P is ∗bft‑proposal‑valid and a ballot for P authenticated by the holder of the voting unit exists at that time. Using knowledge of ballots cast for a ∗bft‑proposal P that collectively satisfy a notarization rule at a given time in a ∗bft‑execution, and only with such knowledge, it is possible to obtain a valid ∗bft‑notarization‑proof proofP​. The notarization rule must require at least a two‑thirds absolute supermajority of voting units in P’s epoch to have been cast for P. It may also require other conditions. A voting unit is cast non‑honestly for an epoch’s proposal iff: it is cast other than by the holder of the unit (due to key compromise or any flaw in the voting protocol, for example); or it is double‑cast (i.e. there are at least two ballots casting it for distinct proposals); or the holder of the unit following the conditions for honest voting in Π∗bft​, according to its view, should not have cast that vote. Note that a unit should be considered to be cast non-honestly in the case of key compromise, because it is then effectively under the control of an adversary. The key compromise may or may not be attributable to another flaw in the protocol, but such a flaw would not be a break of the consensus mechanism per se. Definition: One‑third bound on non‑honest voting An execution of Πbft​ has the one‑third bound on non‑honest voting property iff for every epoch, strictly fewer than one third of the total voting units for that epoch are ever cast non‑honestly. Info It may be the case that a ballot cast for P is not in honest view when it is used to create a notarization proof for P. Since we are not assuming synchrony, it may also be the case that such a ballot is in honest view but that any given node has not received it (and perhaps will never receive it). There may be multiple distinct ballots or distinct ballot messages attempting to cast a given voting unit for the same proposal; this is undesirable for bandwidth usage, but it is not necessary to consider it to be non‑honest behaviour for the purpose of security analysis, as long as such ballots are not double‑counted toward the two‑thirds threshold. Security caveat The one‑third bound on non‑honest voting property considers all ballots cast in the entire execution. In particular, it is possible that a validator’s key is compromised and then used to cast its voting units for a proposal of an epoch long finished. If the number of voting units cast non-honestly for any epoch ever reaches one third of the total voting units for that epoch during an execution, then the one‑third bound on non‑honest voting property is violated for that execution. Therefore, validator keys of honest nodes must remain secret indefinitely. Whenever a key is rotated, the old key must be securely deleted. For further discussion and potential improvements, see tfl-book issue #140 . A ∗bft‑block consists of (P,proofP​) re‑signed by the same proposer using a strongly unforgeable signature scheme. It is ∗bft‑block‑valid iff: P is ∗bft‑proposal‑valid; and proofP​ is a valid proof that some subset of ballots cast for P are sufficient to satisfy the notarization rule; and the proposer’s outer signature on (P,proofP​) is valid. A ∗bft‑proposal’s parent reference hashes the entire parent ∗bft‑block, i.e. proposal, proof, and outer signature. Info Neither proofP​ nor the proposer’s outer signature is unique for a given P. The proposer’s outer signature is however third‑party nonmalleable, by definition of a strongly unforgeable signature scheme. An “honest ∗bft‑proposal” is a ∗bft‑proposal made for a given epoch by a proposer who is honest in that epoch. Such a proposer will only create one proposal and only sign at most once for each epoch, and so there will be at most one “honestly submitted” ∗bft‑block for each epoch. It is possible for there to be multiple ∗bft‑valid‑blocks for the same proposal, with different notarization proofs and/or outer signatures, if the proposer is not honest. However, the property that there will be at most one “honestly submitted” ∗bft‑block for each epoch is important for liveness, even though we cannot guarantee that any particular proposer for an epoch is honest. TODO: check that we are correctly using this in the liveness analysis. There is an efficiently computable function ∗bft-last-final⦂∗bft-block→∗bft-block∪{⊥}. For a ∗bft‑block‑valid input block C, this function outputs the last ancestor of C that is final in the context of C. Info The chain of ancestors is unambiguously determined because a ∗bft‑proposal’s parent reference hashes the entire parent ∗bft‑block; each ∗bft‑block commits to a proposal; and the parent hashes are collision‑resistant. This holds despite the caveat mentioned above that there may be multiple ∗bft‑valid‑blocks for the same proposal. ∗bft-last-final must satisfy all of the following: ∗bft-last-final(C)=⊥⟺C is not ∗bft‑block‑valid. If C is ∗bft‑block‑valid, then: ∗bft-last-final(C)⪯∗bft​C (and therefore it must also be ∗bft‑block‑valid); for all ∗bft‑valid‑blocks D such that C⪯∗bft​D, ∗bft-last-final(C)⪯∗bft​∗bft-last-final(D). ∗bft-last-final(O∗bft​)=O∗bft​. Info It is correct to talk about the “last final block” of a given chain (that is, each ∗bft‑valid-block C unambiguously determines a ∗bft‑valid-block ∗bft-last-final(C)), but it is not correct to refer to a given ∗bft‑block as objectively “∗bft‑final”. A particular BFT protocol might need adaptations to fit it into this model for Πorigbft​, before we apply the Crosslink 2 modifications to obtain Πbft​. Any such adaptions are necessarily protocol-specific. In particular: origbft‑proposal‑validity should correspond to the strongest property of an origbft‑proposal that is objectively and feasibly verifiable from the content of the proposal and its parent origbft‑block at the time the proposal is made. It must include verification of the proposer’s signature. origbft‑block‑validity should correspond to the strongest property of an origbft‑block that is objectively and feasibly verifiable from the content of the block and its ancestors at the time the block is added to an origbft‑chain. It should typically include all of the relevant checks from origbft‑proposal‑validity that apply to the created block (or equivalent checks). It must also include verification of the notarization proof and the proposer’s outer signature. If a node observes an origbft‑valid block C, then it should be infeasible for an adversary to cause a rollback in that node’s view past origbft-last-final(C), and the view of the chain up to origbft-last-final(C) should agree with that of all other honest nodes. This is formalized in the next section. Safety of Π∗bft​ The intuition behind the following safety property is that: For Π∗bft​ to be safe, it should never be the case that two honest nodes observe (at any time) ∗bft‑blocks B and B′ respectively that they each consider final in some context, but B⪯⪰∗bft​B′ does not hold. By definition, an honest node observes a ∗bft‑block to be final in the context of another ∗bft‑block C, iff B⪯∗bft​∗bft-last-final(C). We say that a ∗bft‑block is “in honest view” if a party observes it at some time at which that party is honest. Definition: Final Agreement An execution of Π∗bft​ has Final Agreement iff for all ∗bft‑valid blocks C in honest view at time t and C′ in honest view at time t′, we have ∗bft-last-final(C)⪯⪰∗bft​∗bft-last-final(C′). Note that it is possible for this property to hold for an execution of a BFT protocol in an asynchronous communication model. As previously mentioned , if the one‑third bound on non‑honest voting property is ever broken at any time in an execution, then it may not be possible to maintain Final Agreement from that point on. Adapting the Streamlet BFT protocol. Streamlet as described in [CS2020] has three possible states of a block in a player’s view: “valid” (but not notarized or final); “notarized” (but not final); “final”. By “valid” the Streamlet paper means just that it satisfies the structural property of being part of a block chain with parent hashes. The role of ∗bft‑block‑validity in our model corresponds roughly to Streamlet’s “notarized”. It turns out that with some straightforward changes relative to Streamlet, we can identify “origbft‑block‑valid” with “notarized” and consider an origbft‑valid‑chain to only consist of notarized blocks. This is not obvious, but is a useful simplification. Here is how the paper defines “notarized”: When a block gains votes from at least 2n/3 distinct players, it becomes notarized. A chain is notarized if its constituent blocks are all notarized. This implies that blocks can be added to chains independently of notarization. However, the paper also says that an honest leader always proposes a block extending from a notarized chain. Therefore, only notarized chains really matter in the protocol. In unmodified Streamlet, the order in which a player sees signatures might cause it to view blocks as notarized out of order. Streamlet’s security analysis is in a synchronous model, and assumes for liveness that any vote will have been received by all players (Streamlet nodes) within two epochs. In Crosslink 2, however, we need origbft‑block‑validity to be an objectively and feasibly verifiable property. We also would prefer reliable message delivery within bounded time not to be a basic assumption of our communication model. (This does not dictate what assumptions about message delivery are made for particular security analyses.) If we did not make a modification to the protocol to take this into account, then some Crosslink 2 nodes might receive a two‑thirds absolute supermajority of voting messages and consider a BFT block to be notarized, while others might never receive enough of those messages. Obviously a proposal cannot include signatures on itself — but the block formed from it can include proofs about the proposal and signatures. We can therefore say that when a proposal gains a two‑thirds absolute supermajority of signatures, a block is created from it that contains a proof (such as an aggregate signature) that it had such a supermajority. For example, we can have the proposer itself make this proof once it has enough votes, sign the resulting (P,proofP​) to create a block, then submit that block in a separate message. (The proposer has most incentive to do this in order to gain whatever reward attaches to a successful proposal; it can outsource the proving task if needed.) Then the origbft‑block‑validity rule can require a valid supermajority proof, which is objectively and feasibly verifiable. Players that see an origbft‑valid‑block can immediately consider it notarized. Note that for the liveness analysis to be unaffected, we need to assume that the combined latency of messages, of collecting and aggregating signatures, and of block submission is such that all adapted‑Streamlet nodes will receive a notarized block corresponding to a given proposal (rather than just all of the votes for the proposal) within two epochs. Alternatively we could re‑do the timing analysis. With this change, “origbft‑block‑valid” and “notarized” do not need to be distinguished. Streamlet’s finality rule is: If in any notarized chain, there are three adjacent blocks with consecutive epoch numbers, the prefix of the chain up to the second of the three blocks is considered final. When a block becomes final, all of its prefix must be final too. We can straightforwardly express this as an origbft-last-final function of a context block C, as required by the model: For an origbft‑valid‑block C, origbft-last-final(C) is the last origbft‑valid‑block B⪯origbft​C such that either B=Oorigbft​ or B is the second block of a group of three adjacent blocks with consecutive epoch numbers. Note that “When a block becomes final, all of its prefix must be final too.” is implicit in the model.","breadcrumbs":"Design » Crosslink » The Crosslink 2 Construction » Model for BFT protocols (Π{origbft,bft})","id":"80","title":"Model for BFT protocols (Π{origbft,bft})"},"81":{"body":"A node’s view in Π∗bc​ includes a set of ∗bc‑block chains each rooted at a fixed genesis ∗bc‑block O∗bc​. There is a ∗bc‑block‑validity rule (often described as a collection of “consensus rules”), depending only on the content of the block and its ancestors. A non‑genesis block can only be ∗bc‑block‑valid if its parent is ∗bc‑block‑valid. By “∗bc‑valid‑chain” we mean a chain of ∗bc‑block‑valid blocks. Terminology The terminology commonly used in the block‑chain community does not distinguish between rules that are part of the consensus protocol proper, and rules required for validity of the economic computation supported by the block chain. Where it is necessary to distinguish, the former can be called “L0” consensus rules, and the latter “L1” consensus rules. The definition of ∗bc‑block‑validity is such that it is hard for a block producer to extend a ∗bc‑valid‑chain unless they are selected by a random process that chooses a block producer in proportion to their resources with an approximately known and consistent time distribution, subject to some assumption about the total proportion of resources held by honest nodes. There is a function score⦂∗bc-valid-chain→Score, with < a strict total ordering on Score. An honest node will choose one of the ∗bc‑valid‑chains with highest score as the ∗bc‑best‑chain in its view. Any rule can be specified for breaking ties. The score function is required to satisfy score(H⌈∗bc1​)<score(H) for any non‑genesis ∗bc‑valid‑chain H. Info For a Proof‑of‑Work protocol, the score of a ∗bc‑chain should be its accumulated work. Unless an adversary is able to censor knowledge of other chains from a node’s view, it should be difficult to cause the node to switch to a chain with a last common ancestor more than σ blocks back from the tip of its previous ∗bc‑best‑chain. Let ch be a view such that chit​ is node i’s ∗bc‑best‑chain at time t. (This matches the notation used in [NTT2020] .) We define chi0​ to be Obc​. A ∗bc‑valid‑block is assumed to commit to a collection (usually, a sequence) of ∗bc‑transactions. Unlike in Crosslink 1 or Snap-and-Chat, we do not need to explicitly model ∗bc‑transaction validity or impose any additional constraints on it. The consensus rules applying to ∗bc‑transactions are entirely unchanged, including any rules that depend on ∗bc‑block height or previous ∗bc‑blocks. This is because Crosslink 2 never reorders or selectively “sanitizes” transactions as Snap-and-Chat does. If a bc‑block is included in a Crosslink 2 block chain then its entire parent bc‑block chain is included just as it would have been in Πorigbc​ (only modified by the structural additions described later), so block heights are also preserved. A “coinbase transaction” is a ∗bc‑transaction that only distributes newly issued funds and has no inputs. Define is-coinbase-only-block⦂∗bc-block→boolean so that is-coinbase-only-block(B)=true iff B has exactly one transaction that is a coinbase transaction. Each ∗bc‑block is summarized by a ∗bc‑header that commits to the block. There is a notion of ∗bc‑header‑validity that is necessary, but not sufficient, for validity of the block. We will only make the distinction between ∗bc‑headers and ∗bc‑blocks when it is necessary to avoid ambiguity. Header validity for Proof‑of‑Work protocols. In a Proof‑of‑Work protocol, it is normally possible to check the Proof‑of‑Work of a block using only the header. There is a difficulty adjustment function that determines the target difficulty for a block based on its parent chain. So, checking that the correct difficulty target has been used relies on knowing that the header’s parent chain is valid. Checking header validity before expending further resources on a purported block can be relevant to mitigating denial‑of‑service attacks that attempt to inflate validation cost. Typically, Bitcoin‑derived best chain protocols do not need much adaptation to fit into this model. The model still omits some details that would be important to implementing Crosslink 2, but distracting for this level of abstraction. Safety of Π∗bc​ We make an assumption on executions of Πorigbc​ that we will call Prefix Consistency (introduced in [ PSS2016 , section 3.3] as just “consistency”): Definition: Prefix Consistency An execution of Π∗bc​ has Prefix Consistency at confirmation depth σ, iff for all times t≤u and all nodes i, j (potentially the same) such that i is honest at time t and j is honest at time u, we have that chit​⌈∗bcσ​⪯∗bc​chju​. Explain the confusion in the literature about what variants of this property are called. The literature uses the same name, “common‑prefix property”, for two different properties of very different strength. [ PSS2016 , section 3.3] introduced the stronger variant. That paper first describes the weaker variant, calling it the “common‑prefix property by Garay et al [GKL2015] .” Then it explains what is essentially a bug in that variant, and describes the stronger variant which it just calls “consistency”: The common‑prefix property by Garay et al [GKL2015] , which was already considered and studied by Nakamoto [Nakamoto2008] , requires that in any round r, the record chains of any two honest players i, j agree on all, but potentially the last T, records. We note that this property (even in combination with the other two desiderata [of Chain Growth and Chain Quality]) provides quite weak guarantees: even if any two honest parties perfectly agree on the chains, the chain could be completely different on, say, even rounds and odd rounds. We here consider a stronger notion of consistency which additionally stipulates players should be consistent with their “future selves”. Let consistentT(view)=1 iff for all rounds r≤r′, and all players i, j (potentially the same) such that i is honest at viewr and j is honest at viewr′, we have that the prefixes of Cir​(view) and Cjr′​(view) consisting of the first ℓ=∣Cir​(view)∣−T records are identical. Unfortunately, [GKL2020] , which is a revised version of [GKL2015] , switches to the stronger variant without changing the name . (The eprint version history may be useful; the change was made in version 20181013:200033 , page 17.) Note that [GKL2020] uses an adaptive‑corruption model, “meaning that the adversary is allowed to take control of parties on the fly”, and so their wording in Definition 3: ... for any pair of honest players P1​, P2​ adopting the chains C1​, C2​ at rounds r1​≤r2​ in viewΠ,A,Zt,n​ respectively, it holds that C1⌈k​⪯C2​. is intended to mean the same as our ... for all times t≤u and all nodes i, j (potentially the same) such that i is honest at time t and j is honest at time u, we have that chit​⌈∗bcσ​⪯∗bc​chju​. The latter is closer to [PSS2016] . Incidentally, this property does not seem to be mentioned in [Nakamoto2008] , contrary to the [PSS2016] authors’ assertion. Maybe implicitly, but it’s a stretch. Discussion of [GKL2020]’s communication model and network partition. When Prefix Consistency is taken to hold of typical PoW-based block‑chain protocols like Bitcoin (as it often is), this implies that, in the relevant executions, the network of honest nodes is never partitioned — unless any partition lasts only for a short length of time relative to σ block times. If node i is on one side of a full partition and node j on the other, then after node i’s best chain has been extended by more than σ blocks, chit​⌈∗bcσ​ will contain information that has no way to get to node j. And even if the partition is incomplete, we cannot guarantee that the Prefix Consistency property will hold for any given pair of nodes. It might be possible to maintain Prefix Consistency if the honest nodes on one side of the partition knew that they should not continue building on their chain until the partition has healed, but it is unclear how that would be done in general without resorting to a BFT protocol (as opposed to in specific cases like a single node being unable to connect to the rest of the network). Certainly there is no mechanism to explicitly detect and respond to partitions in protocols derived from Bitcoin. And yet, [GKL2020] claims to prove Prefix Consistency from other assumptions. So we know that those assumptions must also rule out a long partition between honest nodes. In fact the required assumption is implicit in the communication model: A synchronous network cannot be partitioned. A partially synchronous network —that is, providing reliable delivery with bounded but unknown delay— cannot be partitioned for longer than the delay. We might be concerned that these implicit assumptions are stronger than we would like. In practice, the peer‑to‑peer network protocol of Bitcoin and Zcash attempts to flood blocks to all nodes. This protocol might have weaknesses, but it is not intended to (and plausibly does not) depend on all messages being received. (Incidentally, Streamlet also implicitly floods messages to all nodes.) Also, Streamlet and many other BFT protocols do not assume for safety that the network is not partitioned. That is, BFT protocols can be safe in a fully asynchronous communication model with unreliable messaging. That is why we avoid taking synchrony or partial synchrony as an implicit assumption of the communication model, or else we could end up with a protocol with weaker safety properties than Πorigbft​ alone. This leaves the question of whether the Prefix Consistency property is still too strong, even if we do not rely on it for the analysis of safety when Πbft​ has not been subverted. In particular, if a particular node h is not well-connected to the rest of the network, then that will inevitably affect node h’s security, but should not affect other honest nodes’ security. Fortunately, it is not the case that disconnecting a single node h from the network causes the security assumption to be voided. The solution is to view h as not honest in that case (even though it would follow the protocol if it could). This achieves the desired effect within the model, because other nodes can no longer rely on h’s honest input. Although viewing h as potentially adversarial might seem conservative from the point of view of other nodes, bear in mind that an adversary could censor an arbitrary subset of incoming and outgoing messages from the node, and this may be best modelled by considering it to be effectively controlled by the adversary. Prefix Consistency compares the σ-truncated chain of some node i with the untruncated chain of node j. For our analysis of safety of the derived ledgers, we will also need to make an assumption on executions of Πorigbc​ that at any given time t, any two honest nodes i and j agree on their confirmed prefixes — with only the caveat that one may have observed more of the chain than the other. That is: Definition: Prefix Agreement An execution of Π∗bc​ has Prefix Agreement at confirmation depth σ iff it has Agreement on the view (i,t)↦chit​⌈∗bcσ​. Why are this property, and Prefix Consistency above, stated as unconditional properties of protocol executions , rather than as probabilistic assumptions? Our security arguments that depend on these properties will all be of the form “in an execution where ⟨safety properties⟩ are not violated, ⟨undesirable thing⟩ cannot happen”. It is not necessary to involve probability in arguments of this form. Any probabilistic reasoning can be done separately. In particular, if a statement of this form holds, and ⟨safety properties⟩ are violated with probability at most p under certain conditions, then it immediately follows that under those conditions ⟨undesirable thing⟩ happens with probability at most p. Furthermore, ⟨undesirable thing⟩ can only happen after ⟨safety properties⟩ have been violated, because the execution up to that point has been an execution in which ⟨safety properties⟩ are not violated. With few exceptions, involving probability in a security argument is best done only to account for nondeterministic choices in the protocol itself. This is opinionated advice, but a lot of security proofs would likely be simpler if inherently probabilistic arguments were more distinctly separated from unconditional ones. In the case of the Prefix Agreement property, an alternative approach would be to prove that Prefix Agreement holds with some probability given Prefix Consistency and some other chain properties. This is what [NTT2020] does in its Theorem 2, which essentially says that under certain conditions Prefix Agreement holds except with probability e−Ω(σ​). The conclusions that can be obtained from this approach are necessarily probabilistic, and depending on the techniques used, the proof may not be tight; that is, the proof may obtain a bound on the probability of failure that is (either asymptotically or concretely) higher than needed. This is the case for [ NTT2020 , Theorem 2]; footnote 10 in that paper points out that the expression for the probability can be asymptotically improved: Using the recursive bootstrapping argument developed in [ DKT+2020 , Section 4.2], it is possible to bring the error probability e−Ω(σ​) as close to an exponential decay as possible. In this context, for any ϵ>0, it is possible to find constants Aϵ​, aϵ​ such that Πlc​(p) is secure after C (max(GST,GAT)) with confirmation time Tlc​=σ except with probability Aϵ​e−aϵ​σ1−ϵ. (Here p is the probability that any given node gets to produce a block in any given time slot.) In fact none of the proofs of security properties for Snap‑and‑Chat depend on the particular expression e−Ω(σ​); for example in the proofs of Lemma 5 and Theorem 1, this probability just “passes through” the proof from the premisses to the conclusion, because the argument is not probabilistic. The same will be true of our safety arguments. Talking about what is possible in particular executions has further advantages: It sidesteps the issue of how to interpret results in the GST model of partial synchrony, when we do not know what C (max(GST,GAT)) is. See also the critique of applying the GST model to block‑chain protocols under “Discussion of [GKL2020]’s communication model and network partition” above. (This is not an inherent problem with analyzing the protocol in the partially synchronous setting, but only with inappropriate use of the GST model of that setting.) We do not require Πbc​ to be a Nakamoto‑style Proof‑of‑Work block chain protocol. Some other kind of protocol could potentially satisfy Prefix Consistency and Prefix Agreement. It is not clear whether a e−Ω(σ​) probability of failure would be concretely adequate. That would depend on the value of σ and the constant hidden by the Ω notation. The asymptotic property using Ω tells us whether a sufficiently large σ could be chosen, but we are more interested in what needs to be assumed for a given concrete choice of σ. If a violation of a required safety property occurs in a given execution , then the safety argument for Crosslink that depended on the property fails for that execution, regardless of what the probability of that occurrence was. This approach therefore more precisely models the consequences of such violations. Why, intuitively, should we believe that Prefix Agreement and Prefix Consistency for a large enough confirmation depth hold with high probability for executions of a PoW‑based best‑chain protocol? Roughly speaking, the intuition behind both properties is as follows: Honest nodes are collectively able to find blocks faster than an adversary, and communication between honest nodes is sufficiently reliable that they act as a combined network racing against that adversary. Then by the argument in [Nakamoto2008] , modified by [GP2020] to correct an error in the concrete analysis, a private mining attack that attempts to cause a σ‑block rollback will, with high probability, fail for large enough σ. A private mining attack is optimal by the argument in [DKT+2020] . Any further analysis of the conditions under which these properties hold should be done in the context of a particular Π∗bc​. Why is the quantification in Prefix Agreement over two different times t and t′ ? This strengthens the security property, relative to quantifying over a single time. The question can then be split into several parts: What does the strengthened property mean, intuitively? Consider the full tree of ∗bc‑valid-blocks that honest nodes have considered to be part of their ∗bc‑best-chain at any times during the execution. This property holds iff, when we strip off all branches of length up to and including σ blocks, the resulting tree is ∗bc‑linear. Why is the strengthening needed? Suppose that time were split into periods such that honest nodes agreed on one chain in odd periods, and a completely different chain in even periods. This would obviously not satisfy the intent, but it would satisfy a version of the property that did not quantify over different times t and u. Why should we expect the strengthened property to hold? If node j were far ahead, i.e. u≫t, then it is obvious that chit​⌈∗bcσ​⪯∗bc​chju​⌈∗bcσ​ should hold. Conversely, if node i were far ahead then it is obvious that chju​⌈∗bcσ​⪯∗bc​chit​⌈∗bcσ​ should hold. The case where t=u is the same as quantifying over a single time. By considering intermediate cases where t and u converge from the extremes or where they diverge from being equal, you should be able to convince yourself that the property holds for any relative values of t and u, in executions of a reasonable best‑chain protocol.","breadcrumbs":"Design » Crosslink » The Crosslink 2 Construction » Model for best-chain protocols (Π{origbc,bc})","id":"81","title":"Model for best-chain protocols (Π{origbc,bc})"},"82":{"body":"","breadcrumbs":"Design » Crosslink » The Crosslink 2 Construction » Definition of Crosslink 2","id":"82","title":"Definition of Crosslink 2"},"83":{"body":"Crosslink 2 is parameterized by a bc‑confirmation‑depth σ∈N+ (as in Snap‑and‑Chat), and also a finalization gap bound L∈N+ with L significantly greater than σ. Each node i always uses the fixed confirmation depth σ to obtain its view of the finalized chain finit​⦂bc-valid-chain. Unlike in Snap‑and‑Chat or Crosslink 1, this is just a block chain; because we do not need sanitization, there is no need to express it as a log of transactions rather than blocks. Each node i chooses a potentially different bc‑confirmation‑depth μ∈N where 0<μ≤σ to obtain its view of the bounded‑available ledger at time t, (baμ​)it​⦂bc-valid-chain. (We make the restriction μ≤σ because there is no reason to choose a larger μ.) Security caveat Choosing μ<σ is at the node’s own risk and may increase the risk of rollback attacks against (baμ​)it​ (it does not affect finit​). Using small values of μ is not recommended. The default should be μ=σ.","breadcrumbs":"Design » Crosslink » The Crosslink 2 Construction » Parameters","id":"83","title":"Parameters"},"84":{"body":"Consider, roughly speaking, the number of bc‑blocks that are not yet finalized at time t (a more precise definition will be given in the section on Πbc​ changes from Πorigbc​ ). We call this the “finality gap” at time t. Under an assumption about the distribution of bc‑block intervals, if this gap stays roughly constant then it corresponds to the approximate time that transactions take to be finalized after being included in a bc‑block (if they are finalized at all) just prior to time t. As explained in detail by The Arguments for Bounded Availability and Finality Overrides , if this bound exceeds a threshold L, then it likely signals an exceptional or emergency condition, in which it is undesirable to keep accepting user transactions that spend funds into new bc‑blocks. In practice, L should be at least 2σ. The condition that the network enters in such cases will be called “Stalled Mode”. For a given higher‑level transaction protocol, we can define a policy for which bc‑blocks will be accepted in Stalled Mode. This will be modelled by a predicate is-stalled-block⦂bc-block→boolean. A bc‑block for which is-stalled-block returns true is called a “stalled block”. Caution A bc‑block producer is only constrained to produce stalled blocks while, roughly speaking, its view of the finalization point is not advancing. In particular an adversary that has subverted the BFT protocol in a way that does not keep the finalization point from advancing, can always avoid being constrained by Stalled Mode. The desired properties of stalled blocks and a possible Stalled Mode policy for Zcash are discussed in the How to block hazards section of The Arguments for Bounded Availability and Finality Overrides . In practice a node's view of the finalized chain, finit​, is likely to lag only a few blocks behind bai,σt​ (depending on the latency overhead imposed by Πbft​), unless the chain has entered Stalled Mode. So when μ=σ, the main factor influencing the choice of a given application to use finit​ or (baμ​)it​ is not the average latency, but rather the desired behaviour in the case of a finalization stall: i.e. stall immediately, or keep processing user transactions until L blocks have passed.","breadcrumbs":"Design » Crosslink » The Crosslink 2 Construction » Stalled Mode","id":"84","title":"Stalled Mode"},"85":{"body":"Each bc‑header has, in addition to origbc‑header fields, a context_bft field that commits to a bft‑block. Each bft‑proposal has, in addition to origbft‑proposal fields, a headers_bc field containing a sequence of exactly σ bc‑headers (zero‑indexed, deepest first). Each non‑genesis bft‑block has, in addition to origbft‑block fields, a headers_bc field containing a sequence of exactly σ bc‑headers (zero-indexed, deepest first). The genesis bft‑block has headers_bc=∅. For a bft‑block or bft‑proposal B, define LF(H)snapshot(B)​:={Obc​,B.headers_bc[0]⌈bc1​,​if B.headers_bc=∅otherwise.​​ For a bc‑block H, define snapshot(B)LF(H)candidate(H)​:=bft-last-final(H.context_bft):=last-common-ancestor(snapshot(LF(H)),H⌈bcσ​)​ When H is the tip of a node’s bc‑best‑chain, candidate(H) will give the candidate finalization point, subject to a condition described below that prevents local rollbacks. Use of the headers_bc field, and its relation to the ch field in Snap‑and‑Chat. For a bft‑proposal or bft‑block B, the role of the bc‑chain snapshot referenced by B.headers_bc[0]⌈bc1​ is comparable to the Πlc​ snapshot referenced by B.ch in the Snap‑and‑Chat construction from [NTT2020] . The motivation for the additional headers is to demonstrate, to any party that sees a bft‑proposal (resp. bft‑block), that the snapshot had been confirmed when the proposal (resp. the block’s proposal) was made. Typically, a node that is validating an honest bft‑proposal or bft‑block will have seen at least the snapshotted bc‑block (and possibly some of the subsequent bc‑blocks in the headers_bc chain) before. For this not to be the case, the validator’s bc‑best‑chain would have to be more than σ bc‑blocks behind the honest proposer’s bc‑best‑chain at a given time, which would violate the Prefix Consistency property of Πbc​. If the headers do not connect to any bc‑valid‑chain known to the validator, then the validator should be suspicious that the proposer might not be honest. It can assign a lower priority to validating the proposal in this case, or simply drop it. The latter option could drop a valid proposal, but this does not in practice cause a problem as long as a sufficient number of validators are properly synced (so that Prefix Consistency holds for them). If the headers do connect to a known bc‑valid‑chain, it could still be the case that the whole header chain up to and including B.headers_bc[σ−1] is not a bc‑valid‑chain. Therefore, to limit denial‑of‑service attacks the validator should first check the Proofs‑of‑Work and difficulty adjustment —which it can do locally using only the headers— before attempting to download and validate any bc‑blocks that it has not already seen. This is why we include the full headers rather than just the block hashes. Nodes may “trim” (i.e. not explicitly store) headers in a bft‑block that overlap with those referred to by its ancestor bft‑block(s). Why is a distinguished value needed for the headers_bc field in the genesis bft‑block? It would be conceptually nice for Obft​.headers[0]⌈bc1​ to refer to Obc​, as well as Obc​.context_bft being Obft​ so that bft-last-final(Obc​.context_bft)=Obft​. That reflects the fact that we know “from the start” that neither genesis block can be rolled back. This is not literally implementable using block hashes because it would involve a hash cycle, but we achieve the same effect by defining a snapshot function that allows us to “patch” snapshot(Obft​) to be Obc​. We do it this way around rather than “patching” the link from a bc‑block to a bft‑block, because the genesis bft‑block already needs a special case since there are not σ bc‑headers available. Why is the context_bft field needed? Why not use a final_bft field to refer directly to the last final bft‑block before the context block? The finality of some bft‑block is only defined in the context of another bft‑block. One possible design would be for a bc‑block to have both final_bft and context_bft fields, so that the finality of final_bft could be checked objectively in the context of context_bft. However, specifying just the context block is sufficient information to determine its last final ancestor. There would never be any need to give a context block and a final ancestor that is not the last one. The bft-last-final function can be computed efficiently for typical BFT protocols. Therefore, having just the context_bft field is sufficient.","breadcrumbs":"Design » Crosslink » The Crosslink 2 Construction » Structural additions","id":"85","title":"Structural additions"},"86":{"body":"Each node i keeps track of a “locally finalized” bc‑chain finit​ at time t. Each node’s locally finalized bc‑chain starts at fini0​=Obc​. However, this chain state should not be exposed to clients of the node until it has synced . Definition: Local finalization linearity Node i has Local finalization linearity up to time t iff the time series of bc‑blocks finir≤t​ is bc‑linear . When node i’s bc‑best‑chain view is updated from chis​ to chit​, the node’s finit​ will become candidate(chit​) if and only if this is a descendant of finis​. Otherwise finit​ will stay at finis​. This guarantees Local finalization linearity by construction. If when making this update, candidate(chit​)/⪯⪰finis​ (i.e. candidate(chit​) and finis​ are on different forks), then the node should record a finalization safety hazard. This can only happen if global safety assumptions are violated. Note that Local finalization linearity on each node is not sufficient for Assured Finality , but it is necessary. This can be expressed by the following state update algorithm, where s is the time of the last update and t is the time of the current update: update-fin(mut fini​,chit​,s):let N=candidate(chit​)if N⪰finis​:finit​←Nelse:finit​←finis​if N⪯finis​:record finalization safety hazard A safety hazard record should include chit​ and the history of fini​ updates including and since the last one that was an ancestor of N. Lemma: Local fin‑depth In any execution of Crosslink 2, for any node i that is honest at time t, there exists a time r≤t such that finit​⪯chir​⌈bcσ​. Proof: By the definition of candidate we have candidate(chiu​)⪯chiu​⌈bcσ​ for all times u. Let r≤t be the last time at which finit​ changed, or the genesis time 0 if it has never changed. Then for r>0 we have finit​=finir​=candidate(chir​)⪯chir​⌈bcσ​, and for r=0 we have finit​=finir​=Obc​⪯chir​⌈bcσ​ (because chi0​=Obc​, and truncating Obc​ always yields Obc​). Why does fini need to be maintained using local state? When a node’s view of the bc‑best‑chain reorgs to a different fork (even if the reorg is shorter than σ blocks), it may be the case that candidate(chit​) rolls back. If Final Agreement of Πbft​ holds up to time t, the new snapshot should in that case be an ancestor of the old one. If all is well then this snapshot will subsequently roll forward along the same path. However, we do not want applications using the node to see the temporary rollback. Assured Finality is our main safety goal for Crosslink 2. It is essentially the same goal as Final Agreement but applied to nodes’ locally finalized bc‑chains; intuitively it means that honest nodes never see conflicting locally finalized chains. We intend to prove that this goal holds under reasonable assumptions about either Πorigbft​ or Πorigbc​. Definition: Assured Finality An execution of Crosslink 2 has Assured Finality iff for all times t, u and all nodes i, j (potentially the same) such that i is honest at time t and j is honest at time u, we have finit​⪯⪰bc​finju​. Note that if an execution of Crosslink 2 has Assured Finality , then all nodes that are honest for that execution have Local finalization linearity . That is because the restriction of Assured Finality to the case i=j is equivalent to Local finalization linearity for node i up to any time at which node i is honest. Why do we need to use candidate(H) rather than snapshot(LF(H))? This ensures that the candidate is at least σ‑confirmed. In practice candidate(H) will rarely differ from snapshot(LF(H)), but using the former patches over a potential gap in the safety proof. The Last Final Snapshot rule specified later will guarantee that snapshot(LF(H))⪯bc​H, and this ensures that snapshot(LF(H))⪯⪰bc​H⌈bcσ​. However, the depth of snapshot(LF(H)) relative to H is not guaranteed to be ≥σ. For the proof we will need candidate(H)⪯bc​H⌈bcσ​, so that we can use the Local fin‑depth lemma together with Prefix Agreement of Πbc​ at confirmation depth σ to prove Assured Finality . An alternative would be to change the Last Final Snapshot rule to directly require snapshot(LF(H))⪯bc​H⌈bcσ​. TODO: Choose between these options based on what works well for the security proofs and finalization latency.","breadcrumbs":"Design » Crosslink » The Crosslink 2 Construction » Locally finalized chain","id":"86","title":"Locally finalized chain"},"87":{"body":"Define the locally bounded‑available chain on node i for bc‑confirmation‑depth μ, as (baμ​)it​={chit​⌈bcμ​,finit​,​if finit​⪯chit​⌈bcμ​otherwise.​ Like the locally finalized bc‑chain, this chain state should not be exposed to clients of the node until it has synced . Theorem: Ledger prefix property For any node i that is honest at time t, and any confirmation depth μ, finit​⪯(baμ​)it​. Proof: By construction of (baμ​)it​. Lemma: Local ba‑depth In any execution of Crosslink 2, for any confirmation depth μ≤σ and any node i that is honest at time t, there exists a time r≤t such that (baμ​)it​⪯bc​chir​⌈bcμ​. Proof: Either (baμ​)it​=finit​, in which case the result follows by the Local fin‑depth lemma since μ≤σ, or (baμ​)it​=chit​⌈bcμ​ in which case it follows trivially with r=t. Our security goal for baμ​ will be Agreement on baμ​ as already defined. Why is the ‘otherwise’ case in the definition of (baμ)it necessary? Assume for this discussion that Πbc​ uses PoW. Depending on the value of σ, the timestamps of bc‑blocks, and the difficulty adjustment rule, it can be the case that if fini​ switches to a different fork, the difficulty on that fork is greater than on the chain of the previous snapshot. Then, the new bc‑chain could reach a higher score than the previous chain in fewer than σ blocks from the fork point, and so chit​⌈bcμ​ might not be a descendant of finit​ (which is more likely if μ=σ). This can occur even when all safety assumptions are satisfied. For Zcash’s difficulty adjustment algorithm , the difficulty of each block is adjusted based on the median timestamps and difficulty target thresholds over a range of PoWAveragingWindow=17 blocks, where each median is taken over PoWMedianBlockSpan=11 blocks. Other damping factors and clamps are applied in order to prevent instability and to reduce the influence that adversarially chosen timestamps can have on difficulty adjustment. This makes it unlikely that an adversary could gain a significant advantage by manipulating the difficulty adjustment. So it is safe to use finit​ in this case: even though it does not have μ confirmations relative to chit​, it does have at least the required amount of work “on top” of it. Defining (baμ​)it​ this way also has the advantage of making the proof of the Ledger prefix property trivial.","breadcrumbs":"Design » Crosslink » The Crosslink 2 Construction » Locally bounded‑available chain","id":"87","title":"Locally bounded‑available chain"},"88":{"body":"It is recommended that node implementations “bake in” a checkpointed bft‑block Bcheckpoint​ to each released version, and that node i should only expose finit​ and (baμ​)it​ to its clients once it is “synced”, that is: Bcheckpoint​⪯bft​LF(chit​); and snapshot(Bcheckpoint​)⪯bc​finit​; and the timestamp of finit​ is within some threshold of the current time.","breadcrumbs":"Design » Crosslink » The Crosslink 2 Construction » Syncing and checkpoints","id":"88","title":"Syncing and checkpoints"},"89":{"body":"Πbft proposal and block validity Genesis bft‑block rule: Obft​ is bft‑block‑valid. A bft‑proposal (resp. non‑genesis bft‑block) B is bft‑proposal‑valid (resp. bft‑block‑valid) iff all of the following hold: Inherited origbft rules: The corresponding origbft‑proposal‑validity (resp. origbft‑block‑validity) rules hold for B. Linearity rule: snapshot(B⌈bft1​)⪯bc​snapshot(B). Tail Confirmation rule: B.headers_bc form the σ‑block tail of a bc‑valid‑chain. The “corresponding validity rules” are assumed to include the Parent rule that B’s parent is bft‑valid. Note: origbft‑block‑validity rules may be different to origbft‑proposal‑validity rules. For example, in adapted Streamlet, a origbft‑block needs evidence that it was voted for by a supermajority, and an origbft‑proposal doesn’t. Such differences also apply to bft‑block‑validity vs bft‑proposal‑validity. Why have validity rules been separated from the honest voting condition below? The reason to separate the validity rules from the honest voting condition, is that the validity rules are objective: they don’t depend on an observer’s view of the bc‑best‑chain. Therefore, they can be checked independently of validator signatures. Even a proposal voted for by 100% of validators will not be considered bft‑proposal‑valid by other nodes unless it satisfies the above rules. If more than two thirds of voting units are cast for an invalid proposal, something is seriously and visibly wrong; in any case, the block will not be accepted as a bft‑valid‑block. Importantly, a purportedly valid bft‑block will not be recognized as such by any honest Crosslink 2 node even if it includes a valid notarization proof, if it does not meet other bft‑block‑validity rules. This is essential to making the finalized chain safe against a flaw in Πbft​ or its security assumptions (even, say, a complete break of the validator signature algorithm), as long as Πbc​ remains safe. What does the Linearity rule do? This rule is key to combining simplicity with strong security properties in Crosslink 2. It essentially says that, in a given bft‑valid‑chain, the snapshots pointed to by blocks in that chain cannot roll back. This allows the informal safety argument for Crosslink 2 to be rather intuitive. Informally, if Πbft​ has Final Agreement , then all nodes see only one consistent bft‑linear chain (restricting to bft‑blocks that are final in the context of some bft‑block in honest view ). Within such a bft‑chain, the Linearity rule ensures by construction that the sequence of referenced bc‑chain snapshots is bc‑linear . This implies Assured Finality , without needing to assume any safety property of Πbc​. We will also be able to prove safety of the finalized snapshots based only on safety of Πbc​ (for a confirmation depth of σ), without needing to assume any safety property of Πbft​. Informally, that is because each node sees each candidate final snapshot at a given time as a σ-confirmed prefix of its bc‑best‑chain at that time (this can be proven based on the Last Final Snapshot rule and the fact that a snapshot includes σ subsequent headers), and Prefix Agreement implies that honest nodes agree on this prefix. We will leave a more detailed argument until after we have presented the Πbc​ changes from Πorigbc​ . The Linearity rule replaces the “Increasing Score rule” used in Crosslink 1. The Increasing Score rule required that each snapshot in a bft‑valid‑chain either be the same snapshot, or a higher-scoring snapshot to that of its parent block. Since scores strictly increase within a bc‑valid‑chain, the Linearity rule implies the Increasing Score rule. It retains the same or stronger positive effects: It prevents potential attacks that rely on proposing a bc‑valid‑chain that forks from a much earlier block. This is necessary because the difficulty (or stake threshold) at that point could have been much lower. It limits the extent of disruption an adversary can feasibly cause to the bounded‑available chain chi​, even if it has subverted Πbft​. Informally, because the finalized chain is a Πbc​ chain, its safety is no worse than Πbc​ alone for a rollback of any depth. It ensures that either progress is made (the snapshot advances relative to that of the parent bft‑block), or there is no further validation that needs to be done for the snapshot because it was already validated. Note that the adversary could take advantage of an “accidental” fork and start its attack from the base of that fork, so that not all of this work is done by it alone. This is also possible in the case of a standard “private mining” attack, and is not so much of a problem in practice because accidental forks are expected to be short. In any case, σ should be chosen to take it into account. The Linearity rule is also critical to removing the need for one of the most complex elements of Snap‑and‑Chat and Crosslink 1, “sanitization”. In those protocols, because bc‑chain snapshots could be unrelated to each other, it was necessary to sanitize the chain formed from these snapshots to remove transactions that were contextually invalid (e.g. because they double‑spend). The negative consequences of this are described in Notes on Snap‑and‑Chat ; avoiding it is much simpler. The linearity property is intentionally always relative to the snapshot of the parent bft‑block, even if it is not final in the context of the current bft‑block. This is because the rule needs to hold if and when it becomes final in the context of some descendant bft‑block. TODO: PoS Desideratum: we want leader selection with good security / performance properties that will be relevant to this rule. (Suggested: PoSAT .) Why does the Linearity rule allow keeping the same snapshot as the parent? This is necessary in order to preserve liveness of Πbft​ relative to Πorigbft​. Liveness of Πorigbft​ might require honest proposers to make proposals at a minimum rate. That requirement could be consistently violated if it were not always possible to make a valid proposal. But given that it is allowed to repeat the same snapshot as in the parent bft‑block, neither the Linearity rule nor the Tail Confirmation rule can prevent making a valid proposal — and all other rules of Πbft​ affecting the ability to make valid proposals are the same as in Πorigbft​. (In principle, changes to voting in Πbft​ could also affect its liveness; we’ll discuss that in the liveness proof later.) For example, Streamlet requires three notarized blocks in consecutive epochs in order to finalize a block [ CS2020 , section 1.1]. Its proof of liveness depends on the assumption that in each epoch for which the leader is honest, that leader will make a proposal, and that during a “period of synchrony” this proposal will be received by every node [ CS2020 , section 3.6]. This argument can also be extended to adapted‑Streamlet. We could alternatively have allowed to always make a “null” proposal, rather than to always make a proposal with the same snapshot as the parent. We prefer the latter because the former would require specifying the rules for null proposals in Πorigbft​. As a clarification, no BFT protocol that uses leader election can require a proposal in each epoch, because the leader might be dishonest. The above issue concerns liveness of the protocol when assumptions about the attacker’s share of bft‑validators or stake are met, so that it can be assumed that sufficiently long periods with enough honest leaders to make progress (5 consecutive epochs in the case of Streamlet), will occur with significant probability. Πbft block finality in context The finality rule for bft‑blocks in a given context is unchanged from origbft‑finality. That is, bft-last-final is defined in the same way as origbft-last-final (modulo referring to bft‑block‑validity and Obft​). Πbft honest proposal An honest proposer of a bft‑proposal P chooses P.headers_bc as the σ‑block tail of its bc‑best‑chain, provided that it is consistent with the Linearity rule . If it would not be consistent with that rule, it sets P.headers_bc to the same headers_bc field as P’s parent bft‑block. It does not make proposals until its bc‑best‑chain is at least σ+1 blocks long. Why σ + 1? If the length were less than σ+1 blocks, it would be impossible to construct the headers_bc field of the proposal. Note that when the length of the proposer’s bc‑best‑chain is exactly σ+1 blocks, the snapshot must be of Obc​. But this does not violate the Linearity rule , because Obc​ matches the previous snapshot by Obft​. How is it possible that the Linearity rule would not be satisfied by choosing headers from an honest proposer’s bc‑best‑chain? As in the answer to Why is the ‘otherwise’ case in the definition of (baμ​)it​ necessary? above, after a reorg on the bc‑chain, the σ-confirmed block on the new chain might not be a descendant of the σ-confirmed block on the old chain, which could break the Linearity rule . Πbft honest voting An honest validator considering a proposal P, first updates its view of both subprotocols with the bc‑headers given in P.headers_bc, downloading bc‑blocks for these headers and checking their bc‑block‑validity. For each downloaded bc‑block, the bft‑chain referenced by its context_bft field might need to be validated if it has not been seen before. Wait what, how much validation is that? In general the entire referenced bft‑chain needs to be validated, not just the referenced block — and for each bft‑block, the bc‑chain in headers_bc needs to be validated, and so on recursively. If this sounds overwhelming, note that: We should check the requirement that a bft‑valid‑block must have been voted for by a two‑thirds absolute supermajority of validators, and any other non‑recursive bft‑validity rules, first . Before validating a bc‑chain referenced by a headers_bc field, we check that it connects to an already-validated bc‑chain and that the Proofs‑of‑Work are valid. This implies that the amount of bc‑block validation is constrained by how fast the network can find valid Proofs‑of‑Work. The Linearity rule reduces the worst‑case validation effort, by ensuring that only one bc‑chain needs to be validated for any bft‑chain. Assuming safety of Πbft​ and that the adversary does not have an overwhelming advantage in computing the Proof‑of‑Work, this is effectively only one bc‑chain overall with, at most, short side branches. In summary, the order of validation is important to avoid denial‑of‑service — but it already is in Bitcoin and Zcash. After updating its view, the validator will vote for a proposal P only if : Valid proposal criterion: it is bft‑proposal‑valid, and Confirmed best‑chain criterion: snapshot(P) is part of the validator’s bc‑best‑chain at a bc‑confirmation‑depth of at least σ. Blocks in a bc‑best‑chain are by definition bc‑block‑valid. If we’re checking the Confirmed best‑chain criterion , why do we need to have separately checked that the blocks referenced by the headers are bc‑block‑valid? The Confirmed best‑chain criterion is quite subtle. It ensures that snapshot(P)=P.headers_bc[0]⌈bc1​ is bc‑block‑valid and has σ bc‑block‑valid blocks after it in the validator’s bc‑best‑chain. However, it need not be the case that P.headers_bc[σ−1] is part of the validator’s bc‑best‑chain after it updates its view. That is, the chain could fork after snapshot(P). The bft‑proposal‑validity rule must be objective; it can’t depend on what the validator’s bc‑best‑chain is. The validator’s bc‑best‑chain may have been updated to P.headers_bc[σ−1] (if it has the highest score), but it also may not. However, if the validator’s bc‑best‑chain was updated, that makes it more likely that it will be able to vote for the proposal. In any case, if the validator does not check that all of the blocks referenced by the headers are bc‑block‑valid, then its vote may be invalid. How does this compare to Snap‑and‑Chat? Snap‑and‑Chat already had the voting condition: An honest node only votes for a proposed BFT block B if it views B.ch as confirmed. but it did not give the headers potentially needed to update the validator’s view, and it did not require a proposal to be for an objectively confirmed snapshot as a matter of validity. If a Crosslink‑like protocol were to require an objectively confirmed snapshot but without including the bc‑headers in the proposal, then validators would not immediately know which bc‑blocks to download to check its validity. This would increase latency, and would be likely to lead proposers to be more conservative and only propose blocks that they think will already be in at least a two‑thirds absolute supermajority of validators’ best chains. That is, showing P.headers_bc to all of the validators is advantageous to the proposer, because the proposer does not have to guess what blocks the validators might have already seen. It is also advantageous for the protocol goals in general, because it improves the trade‑off between finalization latency and security.","breadcrumbs":"Design » Crosslink » The Crosslink 2 Construction » Πbft changes from Πorigbft","id":"89","title":"Πbft changes from Πorigbft"},"9":{"body":"There are three categories of users this proposed TFL protocol would impact:","breadcrumbs":"Introduction » Trailing Finality Layer in a Nutshell » Why Should Users Care?","id":"9","title":"Why Should Users Care?"},"90":{"body":"Πbc block validity Genesis bc‑block rule: For the genesis bc‑block we must have Obc​.context_bft=Obft​, and therefore bft-last-final(Obc​.context_bft)=Obft​. A bc‑block H is bc‑block‑valid iff all of the following hold: Inherited origbc rules: H satisfies the corresponding origbc‑block‑validity rules. Valid context rule: H.context_bft is bft‑block‑valid. Extension rule: LF(H⌈bc1​)⪯bft​LF(H). Last Final Snapshot rule: snapshot(LF(H))⪯bc​H. Finality depth rule: Define: finality-depth(H):=height(H)−height(snapshot(LF(H))) Then either finality-depth(H)≤L or is-stalled-block(H). Explain the definition of finality‑depth. The finality depth must be objectively defined, since it is used in a consensus rule. Therefore it should measure the height of H relative to snapshot(LF(H)), which is an objectively defined function of H, rather than relative to finit​. (These will only differ for H=chit​ when node i has just reorged, and only then in corner cases.) Note that the Last Final Snapshot rule ensures that it is meaningful to simply use the difference in heights, since snapshot(LF(H))⪯bc​H. Πbc contextual validity The consensus rule changes above are all non-contextual. Modulo these changes, contextual validity in Πbc​ is the same as in Πorigbc​. Πbc honest block production An honest producer of a bc‑block H must follow the consensus rules under Πbc​ block validity above. In particular, it must produce a stalled block if required to do so by the Finality depth rule . To choose H.context_bft, the producer considers a subset of the tips of bft‑valid‑chains in its view: {T:T is bft‑block‑valid and LF(H⌈bc1​)⪯bft​bft-last-final(T)} It chooses one of the longest of these chains, C, breaking ties by maximizing score(snapshot(bft-last-final(C))), and if there is still a tie then by taking C with the smallest hash. The honest block producer then sets H.context_bft to C. Attention An honest bc‑block‑producer must not use information from the BFT protocol, other than the specified consensus rules, to decide which bc‑valid‑chain to follow. The specified consensus rules that depend on Πbft​ have been carefully constructed to preserve safety of Πbc​ relative to Πorigbc​. Imposing any additional constraints could potentially allow an adversary that is able to subvert Πbft​, to influence the evolution of the bc‑best‑chain in ways that are not considered in the safety argument. Why not choose T such that H ⌈1bc . context_bft ⪯bft bft‑last‑final( T )? The effect of this would be to tend to more often follow the last bft‑block seen by the producer of the parent bc‑block, if there is a choice. It is not always possible to do so, though: the resulting set of candidates for C might be empty. Also, it is not clear that giving the parent bc‑block‑producer the chance to “guide” what bft‑block should be chosen next is beneficial, since that producer might be adversarial and the resulting incentives are difficult to reason about. Why choose the longest C , rather than the longest bft‑last‑final( C )? We could have instead chosen C to maximize the length of bft-last-final(C). The rule we chose follows Streamlet, which builds on the longest notarized chain, not the longest finalized chain. This may call for more analysis specific to the chosen BFT protocol. Why this tie‑breaking rule? Choosing the bft‑chain that has the last final snapshot with the highest score, tends to inhibit an adversary’s ability to finalize its own chain if it has a lesser score. (If it has a greater score, then it has already won a hash race and we cannot stop the adversary chain from being finalized.) For discussion of potentially unifying the roles of bc‑block producer and bft‑proposer, see What about making the bc‑block‑producer the bft‑proposer? in Potential changes to Crosslink . At this point we have completed the definition of Crosslink 2. In Security Analysis of Crosslink 2 , we will prove it secure.","breadcrumbs":"Design » Crosslink » The Crosslink 2 Construction » Πbc changes from Πorigbc","id":"90","title":"Πbc changes from Πorigbc"},"91":{"body":"This document analyzes the security of Crosslink 2 in terms of liveness and safety. It assumes that you have read The Crosslink 2 Construction which defines the protocol.","breadcrumbs":"Design » Crosslink » Security Analysis of Crosslink 2 » Security Analysis of Crosslink 2","id":"91","title":"Security Analysis of Crosslink 2"},"92":{"body":"First note that Crosslink 2 intentionally sacrifices availability if there is a long finalization stall. Info This is technically independent of the other changes; you can omit the Finality depth rule and the protocol would still have security advantages over Snap‑and‑Chat, as well as being much simpler and solving its “spending from finalized outputs” issue. In that case the incentives to “pull” the finalization point forward to include new final bft‑blocks would be weaker, but honest bc‑block‑producers would still do it. It would still be a bug if there were any situation in which Πbc​ failed to be live , though, because that would allow tail‑thrashing attacks . Crosslink 2 involves a bidirectional dependence between Πbft​ and Πbc​. The Ebb‑and‑Flow paper [NTT2020] argues in Appendix E (\"Bouncing Attack on Casper FFG\") that it can be more difficult to reason about liveness given a bidirectional dependence between protocols: To ensure consistency among the two tiers [of Casper FFG], the fork choice rule of the blockchain is modified to always respect ‘the justified checkpoint of the greatest height [*]’ [22] . There is thus a bidirectional interaction between the block proposal and the finalization layer: blocks proposed by the blockchain are input to finalization, while justified checkpoints constrain future block proposals. This bidirectional interaction is intricate to reason about and a gateway for liveness attacks. Info [*] The quotation changes this to “[depth]”, but our terminology is consistent with Ethereum here and not with [NTT2020]’s idiosyncratic use of “depth” to mean “block height”. The argument is correct as far as it goes. The main reason why this does not present any great difficulty to proving liveness of Crosslink, is due to a fundamental difference from Casper FFG: in Crosslink 2 the fork‑choice rule of Πbc​ is not modified. Let S be the subset of bc‑blocks {B:is-coinbase-only-block(B) and is-stalled-block(B)}. Assume that S is such that a bc‑block‑producer can always produce a block in S. In that case it is straightforward to convince ourselves that the additional bc‑block‑validity rules are never an obstacle to producing a new bc‑block in S: The changes to the definition of contextual validity do not interfere with liveness, since they do not affect coinbase transactions, and therefore do not affect blocks in S. That is, the bc‑block‑producer can always just omit non‑coinbase transactions that are contextually invalid. The Genesis bc‑block rule doesn’t apply to new bc‑blocks. The Valid context rule , Extension rule , and Last Final Snapshot rule are always satisfiable by referencing the same context bft‑block as the parent bc‑block. The Finality depth rule always allows the option of producing a stalled block, and therefore does not affect blocks in S. The instructions to an honest bc‑block‑producer allow it to produce a block in S. Therefore, Πbc​ remains live under the same conditions as Πorigbc​. The additional bft‑proposal‑validity, bft‑block‑validity, bft‑finality, and honest voting rules are also not an obstacle to making, voting for, or finalizing bft‑proposals: Because Πbc​ is live, there will always be some point in time at which a fresh valid bc‑header chain that satisfies both the Linearity rule and the Tail confirmation rule exists for use in the P.headers_bc field. If no fresh valid bc‑header chain is available, the Linearity rule and Tail confirmation rule allow an honest bft‑proposer to choose headers_bc to be the same as in the previous bft‑block. So, if liveness of Πorigbft​ depends on an honest proposer always being able to make a proposal (as it does in adapted‑Streamlet for example), then this requirement will not be violated. The changes to voting are only requiring a vote to be for a proposal that could have been honestly proposed. The bft‑finality rules are unchanged from origbft‑finality. Therefore, Πbft​ remains live under the same conditions as Πorigbft​. The only other possibility for a liveness issue in Crosslink 2 would be if the change to the constructions of fini​ or (baμ​)i​ could cause either of them to stall, even when Πbft​ and Πbc​ are both still live. However, liveness of Πbft​ and the Linearity rule together imply that at each point in time, provided there are sufficient honest bft‑proposers/validators, eventually a new bft‑block with a higher-scoring snapshot will become final in the context of the longest bft‑valid‑chain. TODO: make that more precise. Because of the Extension rule , this new bft‑block must be a descendent of the previous final bft‑block in the context visible to bc‑block‑producers. Therefore, the new finalized chain will extend the old finalized chain. Finally, we need to show that Stalled Mode is only triggered when it should be; that is, when the assumptions needed for liveness of Πbft​ are violated. Informally, that is the case because, as long as there are sufficient honest bc‑block‑producers and sufficient honest bft‑proposers/validators, the finalization point implied by the context_bft field at the tip of the bc‑best chain in any node’s view will advance fast enough for the finalization gap bound L not to be hit. This depends on the value of L relative to σ, the network delay, the hash rate of honest bc‑block‑producers, the number of honest bft‑proposers and the proportion of voting units they hold, and other details of the BFT protocol. TODO: more detailed argument needed, especially for the dependence on L.","breadcrumbs":"Design » Crosslink » Security Analysis of Crosslink 2 » Liveness argument","id":"92","title":"Liveness argument"},"93":{"body":"TODO: Not updated for Crosslink 2 below this point.","breadcrumbs":"Design » Crosslink » Security Analysis of Crosslink 2 » Safety argument","id":"93","title":"Safety argument"},"94":{"body":"Recall the definition of Assured Finality . Definition: Assured Finality An execution of Crosslink 2 has Assured Finality iff for all times t, u and all nodes i, j (potentially the same) such that i is honest at time t and j is honest at time u, we have finit​⪯⪰bc​finju​. First we prove that Assured Finality is implied by Prefix Agreement of Πbc​. Definition: Prefix Agreement An execution of Π∗bc​ has Prefix Agreement at confirmation depth σ, iff for all times t, u and all nodes i, j (potentially the same) such that i is honest at time t and j is honest at time u, we have chit​⌈∗bcσ​⪯⪰∗bc​chju​⌈∗bcσ​. Safety Theorem: Prefix Agreement of Πbc implies Assured Finality In an execution of Crosslink 2 for which the Πbc​ subprotocol has Prefix Agreement at confirmation depth σ, that execution has Assured Finality . Proof: Suppose that we have times t, u and nodes i, j (potentially the same) such that i is honest at time t and j is honest at time u. Then, by the Local fin-depth lemma applied to each of node i and node j, there exist times r at which node i is honest and r′ at which node j is honest, such that finit​⪯bc​chir​⌈bcσ​ and finju​⪯bc​chjr′​⌈bcσ​. By Prefix Agreement at confirmation depth σ , we have chir​⌈bcσ​⪯⪰bc​chjr′​⌈bcσ​. Wlog due to symmetry, suppose chir​⌈bcσ​⪯bc​chjr′​⌈bcσ​. Then finit​⪯bc​chjr′​⌈bcσ​ (by transitivity of ⪯bc​) and finju​⪯bc​chjr′​⌈bcσ​ (as above), so finit​⪯⪰bc​finju​ by the Linear prefix lemma . Then we prove that Assured Finality is also implied by Final Agreement of Πbft​. Definition: Final Agreement An execution of Π∗bft​ has Final Agreement iff for all ∗bft‑valid blocks C in honest view at time t and C′ in honest view at time t′, we have ∗bft-last-final(C)⪯⪰∗bft​∗bft-last-final(C′). Safety Theorem: Final Agreement of Πbft implies Assured Finality In an execution of Crosslink 2 for which the Πbft​ subprotocol has Final Agreement , that execution has Assured Finality . Proof: Suppose that we have times t, u and nodes i, j (potentially the same) such that i is honest at time t and j is honest at time u. Then, by the Local fin-depth lemma applied to each of node i and node j, there exist times r at which node i is honest and r′ at which node j is honest, such that finit​⪯bc​chir​⌈bcσ​ and finju​⪯bc​chjr′​⌈bcσ​. By Prefix Agreement at confirmation depth σ , we have chir​⌈bcσ​⪯⪰bc​chjr′​⌈bcσ​. Wlog due to symmetry, suppose chir​⌈bcσ​⪯bc​chjr′​⌈bcσ​. Then finit​⪯bc​chjr′​⌈bcσ​ (by transitivity of ⪯bc​) and finju​⪯bc​chjr′​⌈bcσ​ (as above), so finit​⪯⪰bc​finju​ by the Linear prefix lemma . Safety Theorem: Prefix Consistency of Πbc implies Prefix Consistency of ba By the Local ba-depth lemma , we have: In any execution of Crosslink 2, for any confirmation depth μ≤σ and any node i that is honest at time t, there exists a time r≤t such that (baμ​)it​⪯bc​chir​⌈bcμ​. Renaming t to r and σ to μ in the definition of Prefix Consistency gives: An execution of Πbc​ has Prefix Consistency at confirmation depth μ, iff for all times r≤u and all nodes i, j (potentially the same) such that i is honest at time r and j is honest at time u, we have that chir​⌈bcμ​⪯bc​chju​. Since any node i that is honest at time t is also honest at time r≤t, and by transitivity of ⪯bc​, we therefore have: In any execution of Crosslink 2 that has Prefix Consistency at confirmation depth μ, for all times t≤u and all nodes i, j (potentially the same) such that i is honest at time t and j is honest at time u, we have that (baμ​)it​⪯bc​chju​. ---- The Extension rule ensures that, informally, if a given node i’s view of its bc‑best‑chain at a depth of σ blocks does not roll back, then neither does its view of the bft‑final block referenced by its bc‑best‑chain, and therefore neither does its view of LOGfin,it​. This does not by itself imply that all nodes are seeing the “same” confirmed bc‑best‑chain (up to propagation timing), or the same LOGfin,it​. If the network is partitioned and Πbft​ is subverted, it could be that the nodes on each side of the partition follow a different fork, and the adversary arranges for each node’s view of the last final bft‑block to be consistent with the fork it is on. It can potentially do this if it has more than one third of validators, because if the validators are partitioned in the same way as other nodes, it can vote with an additional one third of them on each side of the fork. This is, if you think about it, unavoidable. Πbc​ doesn’t include the mechanisms needed to maintain finality under partition; it takes a different position on the CAP trilemma. In order to maintain finality under partition, we need Πbft​ not to be subverted (and to actually work!) So what is the strongest security property we can realistically get? It is stronger than what Snap‑and‑Chat provides. Snap‑and‑Chat is unsafe even without a partition if Πbft​ is subverted. Ideally we would have a protocol with safety that is only limited by attacks “like” the unavoidable attack described above — which also applies to Πbc​ used on its own.","breadcrumbs":"Design » Crosslink » Security Analysis of Crosslink 2 » Discussion","id":"94","title":"Discussion"},"95":{"body":"In order to capture the intuition that it is hard in practice to cause a consistent partition of the kind described in the previous section, we will need to assume that the Prefix Agreement safety property holds for the relevant executions of Πbc​. The structural and consensus modifications to Πbc​ relative to Πorigbc​ seem unlikely to have any significant effect on this property, given that we proved above that they do not affect liveness. ==TODO: that is a handwave; we should be able to do better, as we do for Πbft​ below.== So, to the extent that it is reasonable to assume that Prefix Agreement holds for executions of Πorigbc​ under some conditions, it should also be reasonable to assume it holds for executions of Πbc​ under the same conditions. Recall that LF(H):=bft-last-final(H.context_bft). Prefix Lemma If H1​, H2​ are bc‑valid blocks with H1​⪯bc​H2​, then LF(H1​)⪯bft​LF(H2​). Proof: Using the Extension rule , by induction on the distance between H1​ and H2​. Using the Prefix Lemma once for each direction, we can transfer the Prefix Agreement property to the referenced bft‑blocks: Prefix Agreement Lemma In an execution of Πbc​ that has Prefix Agreement at confirmation depth σ, for all times t, u and all nodes i, j (potentially the same) such that i is honest at time t and j is honest at time u, we have LF(chit​⌈bcσ​)⪯⪰bft​LF(chju​⌈bcσ​). Let chain-txns(ch) be the sequence of transactions in the given chain ch, starting from genesis. Recall that san-ctxsan-ctx(S)finfin(H)fin-ctx(H)ba-ctx(H,μ)LOGfin,it​LOGba,μ,it​​⦂:=⦂:=:=:=:=:=​[bc-chain]→bc-contextsanitize(concat([chain-txns(C) for C in S]))bc-block→[bc-chain][snapshot(B) for B⪯bft​LF(H⌈bcσ​)]san-ctx(fin(H))san-ctx(fin(H)∣∣[H⌈bcμ​])context-txns(fin-ctx(chit​))context-txns(ba-ctx(chit​,μ))​ Because fin takes the form fin(H):=[f(X) for X⪯bft​g(H)], we have that g(H)⪯bft​g(H′)⟹fin(H)⪯fin(H′). (This would be true for any well‑typed f and g.) By this observation and the Prefix Agreement Lemma , we also have that, in an execution of Crosslink 2 where Πbc​ has the Prefix Agreement safety property at confirmation depth σ, for all times t, u and all nodes i, j (potentially the same) such that i is honest at time t and j is honest at time u, fin(chit​)⪯⪰fin(chju​). Because sanitize only considers previous state, context-txns ∘ san-ctx must be a prefix-preserving map; that is, if S1​⪯S2​ then context-txns(san-ctx(S1​))⪯context-txns(san-ctx(S2​)). Therefore: Theorem: LOGfin Safety (from Prefix Agreement of Πbc) In an execution of Crosslink 2 where Πbc​ has Prefix Agreement at confirmation depth σ, for all times t, u and all nodes i, j (potentially the same) such that i is honest at time t and j is honest at time u, LOGfin,it​⪯⪰LOGfin,ju​. Notice that this does not depend on any safety property of Πbft​, and is an elementary proof. ([ NTT2020 , Theorem 2] is a much more complicated proof that takes nearly 3 pages, not counting the reliance on results from [PS2017] .) In addition, just as in Snap‑and‑Chat, safety of LOGfin​ can be inferred from safety of Πbft​, which follows from safety of Πorigbft​. We prove this based on the Final Agreement property for executions of Πorigbft​: Definition: Final Agreement An execution of Πorigbft​ has the Final Agreement safety property iff for all origbft‑valid blocks C in honest view at time t and C′ in honest view at time t′, we have origbft-last-final(C)⪯⪰origbft​origbft-last-final(C′). The changes in Πbft​ relative to Πorigbft​ only add structural components and tighten bft‑block‑validity and bft‑proposal‑validity rules. So for any legal execution of Πbft​ there is a corresponding legal execution of Πorigbft​, with the structural additions erased and with the same nodes honest at any given time. A safety property, by definition, only asserts that executions not satisfying the property do not occur. Safety properties of Πorigbft​ necessarily do not refer to the added structural components in Πbft​. Therefore, for any safety property of Πorigbft​, including Final Agreement , the corresponding safety property holds for Πbft​. By the definition of fin as above, in an execution of Crosslink 2 where Πbft​ has Final Agreement , for all times t, u and all nodes i, j (potentially the same) such that i is honest at time t and j is honest at time u, fin(chit​)⪯⪰fin(chju​). Therefore, by an argument similar to the one above using the fact that context-txns ∘ san-ctx is a prefix-preserving map: Theorem: LOGfin Safety (from Final Agreement of Πbft or Πorigbft) In an execution of Crosslink 2 where Πbft​ has Final Agreement , for all times t, u and all nodes i, j (potentially the same) such that i is honest at time t and j is honest at time u, LOGfin,it​⪯⪰LOGfin,ju​.","breadcrumbs":"Design » Crosslink » Security Analysis of Crosslink 2 » Proof of safety for LOGfin","id":"95","title":"Proof of safety for LOGfin"},"96":{"body":"From the two LOGfin​ Safety theorems and the Ledger prefix property , we immediately have: Theorem: LOGba does not roll back past the agreed LOGfin Let μi​ be an arbitrary choice of LOGba​ confirmation depth for each node i. Consider an execution of Crosslink 2 where either Πbc​ has Prefix Agreement at confirmation depth σ or Πbft​ has Final Agreement . In such an execution, for all times t, u and all nodes i, j (potentially the same) such that i is honest at time t and j is honest at time u, either LOGfin,it​⪯bc​LOGfin,ju​⪯bc​LOGba,μj​,ju​ or LOGfin,ju​⪯bc​LOGfin,it​⪯bc​LOGba,μi​,it​. Corollary: Under the same conditions, if wlog LOGfin,it​⪯bc​LOGfin,ju​, then LOGfin,it​⪯bc​{LOGba,μi​,it​,LOGba,μj​,ju​}. The above property is not as strong as we would like for practical uses of LOGba​, because it does not say anything about rollbacks up to the finalization point, and such rollbacks may be of unbounded length. (Loosely speaking, the number of non‑Stalled Mode bc‑blocks after the consensus finalization point is bounded by L, but we have also not proven that so far.) As documented in the Model for BFT protocols section of The Crosslink 2 Construction ): For each epoch, there is a fixed number of voting units distributed between the players, which they use to vote for a ∗bft‑proposal. We say that a voting unit has been cast for a ∗bft‑proposal P at a given time in a ∗bft‑execution, if and only if P is ∗bft‑proposal‑valid and a ballot for P authenticated by the holder of the voting unit exists at that time. Using knowledge of ballots cast for a ∗bft‑proposal P that collectively satisfy a notarization rule at a given time in a ∗bft‑execution, and only with such knowledge, it is possible to obtain a valid ∗bft‑notarization‑proof proofP​. The notarization rule must require at least a two‑thirds absolute supermajority of voting units in P’s epoch to have been cast for P. It may also require other conditions. A voting unit is cast non‑honestly for an epoch’s proposal iff: it is cast other than by the holder of the unit (due to key compromise or any flaw in the voting protocol, for example); or it is double‑cast (i.e. there are two ballots casting it for distinct proposals); or the holder of the unit following the conditions for honest voting in Π∗bft​, according to its view, should not have cast that vote. Definition: One‑third bound on non‑honest voting An execution of Πbft​ has the one‑third bound on non‑honest voting property iff for every epoch, strictly fewer than one third of the total voting units for that epoch are ever cast non‑honestly. Theorem: On bft‑valid blocks for a given epoch in honest view By a well known argument often used to prove safety of BFT protocols, in an execution of Crosslink 2 where Πbft​ has the one‑third bound on non‑honest voting property (and assuming soundness of notarization proofs), any bft‑valid block for a given epoch in honest view must commit to the same proposal. Proof (adapted from [ CS2020 , Lemma 1]): Suppose there are two bft‑proposals P and P′, both for epoch e, such that P is committed to by some bft‑block‑valid block B, and P′ is committed to by some bft‑block‑valid block B′. This implies that B and B′ have valid notarization proofs. Let the number of voting units for epoch e be ne​. Assuming soundness of the notarization proofs, it must be that at least 2ne​/3 voting units for epoch e, denoted as the set S, were cast for P, and at least 2ne​/3 voting units for epoch e, denoted as the set S′, were cast for P′. Since there are ne​ voting units for epoch e, S∩S′ must have size at least ne​/3. In an execution of Crosslink 2 where Πbft​ has the one‑third bound on non‑honest voting property, S∩S′ must therefore include at least one voting unit that was cast honestly. Since a voting unit for epoch e that is cast honestly is not double-cast, it must be that P=P′. Info In the case of a network partition, votes may not be seen on both/all sides of the partition. Therefore, it is not necessarily the case that honest nodes can directly see double‑voting. The above argument does not depend on being able to do so. Therefore, in an execution of Crosslink 2 for which Πbft​ has the one‑third bound on non‑honest voting property, for each epoch e there will be at most one bft‑proposal‑valid proposal Pe​, and at least one third of honestly cast voting units must have been cast for it. Let Ie​ be the (necessarily nonempty) set of nodes that cast these honest votes; then, snapshot(Pe​)⪯bc​chite,i​​⌈bcσ​ for all i∈Ie​ at the times te,i​ of their votes in epoch e. (For simplicity, we assume that for each honest node i there is only one time te,i​ at which it obtains a successful check for the voting condition in epoch e, which it uses for any votes that it casts in that epoch.) Let B be any bft‑block for epoch e such that B⪯bft​bft-last-final(C), where C is some bft‑block‑valid block. Since B⪯bft​C, B is bft‑block‑valid. So by the argument above, B commits to the only bft‑proposal‑valid proposal Pe​ for epoch e, and snapshot(B)=snapshot(Pe​) was voted for in that epoch by a nonempty subset of honest nodes Ie​. Let H be any bc‑valid block. We have by definition: fin(H)​==​[snapshot(B) for B⪯bft​LF(H⌈bcσ​)][snapshot(B) for B⪯bft​bft-last-final(H⌈bcσ​.context_bft)]​ So, taking C=H⌈bcσ​.context_bft, each snapshot(B) for B of epoch e in the result of fin(H) satisfies snapshot(B)⪯bc​chite,i​​⌈bcσ​ for all i in some nonempty honest set of nodes Ie​. For an execution of Crosslink 2 in which Πbc​ has the Prefix Consistency property at confirmation depth σ, for every epoch e, for every such (i,te,i​), for every node j that is honest at any time u≥te,i​, we have chite,i​​⌈bcσ​⪯bc​chju​. Let te​=min{te,i​:i∈Ie​}. Then, by transitivity of ⪯bc​: Theorem: On snapshots in LOGfin In an execution of Crosslink 2 where Πbft​ has the one‑third bound on non‑honest voting property and Πbc​ has the Prefix Consistency property at confirmation depth σ, every bc‑chain snapshot(B) in fin(chit​) (and therefore every snapshot that contributes to LOGfin,it​) is, at any time u≥te​, in the bc‑best‑chain of every node j that is honest at time u (where B commits to Pe​ at epoch e and te​ is the time of the first honest vote for Pe​). A similar (weaker) statement holds if we replace u≥te​ with u≥t, since the time of the first honest vote for P necessarily precedes the time at which the signed (P,proofP​) is submitted as a bft‑block, which necessarily precedes t. (Whether or not the notarization proof depends on the first honest vote for B’s proposal Pe​, it must depend on some honest vote for that proposal that was not made earlier than te​.) Furthermore, we have ba-ctx(H,μ)LOGba,μ,it​​==​san-ctx(fin(H)∣∣[H⌈bcμ​])context-txns(ba-ctx(chit​,μ))​ So in an execution of Crosslink 2 where Πbc​ has the Prefix Consistency property at confirmation depth μ, if node i is honest at time t then H⌈bcμ​ is also, at any time u≥t, in the bc‑best‑chain of every node j that is honest at time u. If an execution of Πbc​ has the Prefix Consistency property at confirmation depth μ≤σ, then it necessarily also has it at confirmation depth σ. Therefore we have: Theorem: On snapshots in LOGba In an execution of Crosslink 2 where Πbft​ has the one‑third bound on non‑honest voting property and Πbc​ has the Prefix Consistency property at confirmation depth μ≤σ, every bc‑chain snapshot in fin(chit​)∣∣[chit​⌈bcμ​] (and therefore every snapshot that contributes to LOGba,μ,it​) is, at any time u≥t, in the bc‑best‑chain of every node j that is honest at time u. Sketch: we also need the sequence of snapshots output from fin to only be extended in the view of any node. In that case we can infer that the node does not observe a rollback in LOG_ba. Recall that in the proof of safety for LOGfin​, we showed that in an execution of Crosslink 2 where Πbft​ (or Πorigbft​) has Final Agreement , for all times t, u and all nodes i, j (potentially the same) such that i is honest at time t and j is honest at time u, fin(chit​)⪯⪰fin(chju​). What we want to show is that, under some conditions on executions, ...","breadcrumbs":"Design » Crosslink » Security Analysis of Crosslink 2 » Proof of safety for LOGba","id":"96","title":"Proof of safety for LOGba"},"97":{"body":"","breadcrumbs":"Design » Crosslink » Security Analysis of Crosslink 2 » Disadvantages of Crosslink","id":"97","title":"Disadvantages of Crosslink"},"98":{"body":"Unlike Snap‑and‑Chat, Crosslink 2 requires structural and consensus rule changes to both Πbc​ and Πbft​. On the other hand, several of those changes are arguably necessary to fix a show‑stopping bug in Snap‑and‑Chat (not being able to spend some finalized outputs).","breadcrumbs":"Design » Crosslink » Security Analysis of Crosslink 2 » More invasive changes","id":"98","title":"More invasive changes"},"99":{"body":"For a given choice of σ, the finalization latency is higher. The snapshot of the BFT chain used to obtain LOGfin,μ,it​ is obtained from the block at depth μ on node i’s best Πbc​ chain, which will on average lead to a finalized view that is about μ+1+σ blocks back (in Πbc​), rather than \\sigma_{\\sac}} blocks in Snap‑and‑Chat. This is essentially the cost of ensuring that safety is given by the stronger of the safety of Πbc​ (at μ confirmations) and the safety of Πbft​. On the other hand, the relative increase in expected finalization latency is only \\frac{\\mu + 1 + \\sigma}{\\sigma_{\\sac}}}, i.e. at most slightly more than a factor of 2 for the case \\mu = \\sigma = \\sigma_{\\sac}}.","breadcrumbs":"Design » Crosslink » Security Analysis of Crosslink 2 » Finalization latency","id":"99","title":"Finalization 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book introduces and specifies a Trailing Finality Layer for the Zcash network. This is version 0.1.0 of the book. This design augments the existing Zcash Proof‑of‑Work ( PoW ) network with a new consensus layer which provides trailing finality . This layer enables transactions included via PoW to become final which assures that they cannot be reverted by the protocol. This enables safer and simpler wallets and other infrastructure, and aids trust-minimized cross-chain bridges. This consensus layer uses Proof-of-Stake consensus, and enables ZEC holders to earn protocol rewards for contributing to the security of the Zcash network. By integrating a PoS layer with the current PoW Zcash protocol, this design specifies a hybrid consensus protocol dubbed PoW+TFL . The rest of this introductory chapter is aimed at a general audience interested in the context of this proposal within Zcash development, status and next steps, motivations, a primer on finality, and tips to get involved.","breadcrumbs":"Introduction » A Trailing Finality Layer for Zcash","id":"0","title":"A Trailing Finality Layer for Zcash"},"1":{"body":"The TFL design provides a possible first step in transitioning Zcash to a PoS protocol. Here we describe how a transition to PoS relates to “the Zcash roadmap” and how TFL fits into one approach to a PoS transition.","breadcrumbs":"Introduction » A Path to Proof-of-Stake Zcash » A Path to Proof-of-Stake Zcash","id":"1","title":"A Path to Proof-of-Stake Zcash"},"10":{"body":"Existing ZEC users who are primarily concerned with storing or receiving ZEC, whether private or transparent may benefit from this change in the short or medium term, because it may help lower delay times for some services, such as exchange deposits. As exchanges come to rely on the new finality guarantee, they can often reduce their deposit wait times. Other services with similar confirmation-depth-based wait times can be improved in a similar way to lower these wait times. Other than this improvement, these users should notice no other changes. In the longer term, providing finality will be useful in establishing trust‑minimized bridges to other block‑chain protocols. We anticipate this can enable better connecting ZEC to the Defi ecosystem, and with the introduction of Zcash Shielded Assets, this can enable other assets to connect to the Zcash shielded pool.","breadcrumbs":"Introduction » Trailing Finality Layer in a Nutshell » Current ZEC Users","id":"10","title":"Current ZEC Users"},"100":{"body":"See the Liveness section above.","breadcrumbs":"Design » Crosslink » Security Analysis of Crosslink 2 » More involved liveness argument","id":"100","title":"More involved liveness argument"},"101":{"body":"Warning In order to show that Crosslink 2 is at a local optimum in the security/complexity trade‑off space, for each rule we show attacks on safety and/or liveness that could be performed if that rule were omitted or simplified. Edit: some rules, e.g. the Linearity rule , only contribute heuristically to security in the analysis so far.","breadcrumbs":"Design » Crosslink » Security Analysis of Crosslink 2 » Every rule in Crosslink 2 is individually necessary","id":"101","title":"Every rule in Crosslink 2 is individually necessary"},"102":{"body":"","breadcrumbs":"Design » Crosslink » Questions About Crosslink » Questions about Crosslink","id":"102","title":"Questions about Crosslink"},"103":{"body":"Rationale: Can we outright prevent rollbacks > σ from ever appearing in Chit​ ? Info This document analyzes the effect of this rule on its own . For the effect in combination with an additional Linearity rule , see the Linearity and Last Final Snapshot rules section of Potential changes for Crosslink 2 . Daira-Emma: In a variant of Crosslink with this rule, an adversary’s strategy would be to keep the context_bft fields in its blocks as C such that B=bft-last-final(C) when the attack starts, and then fork from the bc‑best‑chain that extends snapshot(B). If its private chain falls behind the public bc‑best‑chain, it resets, just like in a conventional private mining attack. Note that the proposed rule does not prevent the adversary’s private chain from just staying at the same context_bft block. The reason is that Crosslink does not change the fork-choice rule of Πbc​. That is, even if the adversary’s chain has a context_bft that is far behind the current bft‑block, it is still allowed to become the bc‑best‑chain. (Eventually the adversary’s chain using this strategy will hit the finality gap bound of L blocks. But that must be significantly greater than σ, to avoid availability problems. So it does not prevent the adversary from performing a rollback longer than σ blocks before they hit the bound. Also, going into Stalled Mode for new blocks does not prevent the attacker’s chain from having included harmful transactions before that point.) It is possible to change the fork-choice rule, for example so that the bc‑best‑chain for a node i is required to extend snapshot(B) where B is the last final block for any bft-chain in node i’s view. This would break the current safety and liveness arguments for Crosslink. But for the sake of argument, suppose we did it anyway. The adversary’s strategy would change slightly: it resets if either its private chain falls behind the public bc‑best‑chain, or its private chain is invalidated because it forks before snapshot(B) for some new last final block B of a bft-chain. During the attack, it also attempts to impede progress of the BFT protocol as far as possible. In that case, the proposed rule still does not preclude a rollback of more than σ blocks, for several reasons: In general we can’t say anything about how many bc‑blocks are mined in any given interval, so it could be the case that more than σ blocks are mined on both the honest chain and the adversary’s chain before it would be realistically possible to go through even a single round of the BFT protocol. Nor can we say anything about how quickly those blocks are finalized, unless we enforce it, which we don’t. (In Crosslink we do enforce a finalization gap bound L, but as explained above L must be significantly greater than σ, so that doesn’t really help.) In particular, the adversary could be suppressing publication of final bft‑blocks, or attacking the liveness of the BFT protocol in other ways. An attack against BFT liveness is potentially easier than an attack against BFT safety, and it would be difficult to characterize exactly how much this rule gains you in terms of security given that (at best) it’s dependent on that. snapshot(LF(H)) will typically be at least σ+1 blocks back from H. The argument for that goes: None of the block hashes in LF(H).headers_bc can point to H because that would be a hash cycle. In a typical case where no block withholding and no other rollback (not caused by the adversary) occurs on the honestly mined chain, the proposer of the last final block before a context bft‑block that H can point to will have, at the latest, H⌈bc1​ as tip(B). Under these conditions, snapshot(LF(H)) will point, at the latest, to σ blocks before H⌈bc1​, i.e. σ+1 blocks before H. This means that by the time snapshot(LF(H)) could catch up to H, on average σ+1 block times will have occurred. So, roughly speaking, the rule that snapshot(LF(H))⪯bc​H does not usefully constrain the adversary until after σ+1 block times. Wlog let’s assume Πbc​ uses PoW: if σ is chosen reasonably, then being able to do a σ+1-block rollback at all probably requires having somewhere close to 50% of network hash rate. And so in σ+1 block times the adversary has a significant chance of being able to do the required rollback before the suggested rule “kicks in”. (It also has however much additional latency is added by the BFT protocol, which is simultaneously under attack to maximize this latency.) All that said, does the suggested rule help? First we have to ask whether it introduces any weaknesses. One potential issue is that the rule cuts both ways: if an adversarial rollback of more than σ blocks does occur, then the adversary can make a proposal that will “lock in” its success. But it can be argued that this is intended behaviour anyway: the adversary has a confirmed chain, and is entitled to propose to finalize it. What happens if Πbft​ is broken? Edit: the answer below refers to Crosslink before the Increasing Score rule was added. This could definitely be a problem. If either an adversary has a two‑thirds supermajority of validators, or Πbft​ is completely broken (e.g. by a bug in the implementation of the validation signature scheme), then they can add a final block to the bft-chain with a headers_bc field that does not satisfy the honest voting condition. They do not need this to be bft-valid (although greater vulnerability to bugs in the implementation of bft-validity are also an issue). Then they can use the proposed rule to prevent an arbitrary bc-valid-chain from being extended (e.g. by mining σ+1 blocks from Obc​ at the genesis difficulty, then getting that chain into a final bft‑block). [Edit: they would no longer be able to do this using a chain mined at a much lower difficulty, because of the Increasing Score rule.] In Crosslink, this is very carefully avoided. The only similar rule that depends on snapshot(LF(H)) and that could potentially affect liveness of Πbc​ is the Finality depth rule. But that rule always allows the alternative of producing a Stalled Mode block on the current bc‑best‑chain — and honest block producers will do so. Therefore, the effect of trying to exploit even a catastrophic break in Πbft​ in order to cause a rollback of LOGfin​, as long as Πbc​ has not also been broken, is to go into Stalled Mode. This does not mean that a break of Πbft​ is not a problem for Crosslink. In particular, an adversary that can violate safety of Πbft​ can violate safety of LOGba​ (and of LOGfin​ if there is also a σ‑block rollback in some node’s bc‑best‑chain). The difference is that a safety violation of Πbft​ can be directly observed by nodes without any chance of false positives , which is not necessarily the case for all possible attacks against Πbft​. (The attack described above does not violate safety of Πbft​; it just adds a final bft‑block with a suspiciously long Πbc​ rollback. It could alternatively have added a block with a less suspiciously long rollback, say exactly σ+1 blocks. That is, in pursuit of preventing an attack against Πbc​, we have enabled attacks against Πbft​ to achieve the same effect — precisely what Crosslink is designed to prevent.) This raises an interesting idea: if any node sees a rollback in the chain of final bft‑blocks, it could provide objective evidence of that rollback in the form of a “bft-final-vee”: two final bft‑blocks with the same parent. Similarly, if any node sees more than one third of stake vote for conflicting blocks in a given epoch, then the assumption bounding the adversary’s stake must have been violated. This evidence can be posted in a transaction to the bc-chain. In that case, any node that is synced to the bc-chain can see that the bft-chain suffered a violation of safety or of a safety assumption, without needing to have seen that violation itself. This can be generalized to allow other proofs of flaws in Πbft​. Optionally, a bc-chain that has posted such a proof could be latched into Stalled Mode until manual intervention can occur. (Obviously we need to make sure that this cannot be abused for denial-of-service.) This is now described in Potential changes to Crosslink . Okay, but is it a good idea to make that change to the fork-choice rule anyway? Probably not. I don’t know how to repair the safety and liveness arguments. The change was that the bc‑best‑chain for a node i would be required to extend snapshot(B) where B is the last final bft‑block in node i’s view. From the point of view of any modular analysis that treats Πbft​ as potentially subverted, we cannot say anything useful about snapshot(B). It seems as though any repair would have to assume much more about the BFT protocol than is desirable. In general, changes to fork‑choice rules are tricky; it was a fork-choice rule problem that allowed the liveness attack against Casper FFG described in [ NTT2020 , Appendix E]. What if validators who see that a long rollback occurred, refuse to vote for it? Yep that is allowed. The rule is “An honest validator will only vote for a proposal P if ...” (not if‑and‑only‑if). If an honest validator sees a “good” reason not to vote for a proposal, including reasons based on out‑of‑band information, they should not. The Complementarity argument made in The Argument for Bounded Availability and Finality Overrides actually depends on this. Obviously, it may affect BFT liveness (and that’s okay). The only reason why we don’t make this part of the voting condition is that it’s a stateful rule. A new validator could come along and wouldn’t have the state needed to enforce it. Perhaps that could be fixed.","breadcrumbs":"Design » Crosslink » Questions About Crosslink » Why don’t we have a bc‑block‑validity rule snapshot(LF(H)) ⪯bc H ?","id":"103","title":"Why don’t we have a bc‑block‑validity rule snapshot(LF(H)) ⪯bc H ?"},"104":{"body":"This page documents suggestions that have not had the same attention to security analysis as the Crosslink 2 construction . Some of them are broken. Some of them also increase the complexity of the protocol (while some simplify it or have a mixed effect on complexity), and so we need to consider the security/complexity trade‑off of each suggestion before we could include it. This page is out-of-date This page has not yet been updated for the changes from Crosslink 1 to Crosslink 2.","breadcrumbs":"Design » Crosslink » Potential Changes to Crosslink » Potential Changes to Crosslink","id":"104","title":"Potential Changes to Crosslink"},"105":{"body":"","breadcrumbs":"Design » Crosslink » Potential Changes to Crosslink » Attempts to improve safety or to simplify the protocol","id":"105","title":"Attempts to improve safety or to simplify the protocol"},"106":{"body":"We can allow honest bc‑block‑producers to record information about every proposed and notarized bft‑block, rather than just the one in the context_bft field. Duplicate information that has already been given in an ancestor bc‑block would be omitted. This would automatically expose the following shenanigans to public view (as long as enough bc‑block‑proposers are honest, which is already assumed): any attempt to double‑propose in the same epoch; any successful attempt to double‑notarize. We could also expose attempts to double‑vote. Note that double‑proposal and double‑voting could be a sign that a proposer or validator’s private key is compromised, rather than that it belongs to the adversary per se. However, the security analysis must treat such a proposer/validator as non‑honest in any case.","breadcrumbs":"Design » Crosslink » Potential Changes to Crosslink » [Recommended] Recording more info about the bft‑chain in bc‑blocks","id":"106","title":"[Recommended] Recording more info about the bft‑chain in bc‑blocks"},"107":{"body":"The current Increasing Score rule concerns the score of the snapshot: Increasing Snapshot Score rule: Either score(snapshot(B⌈bft1​))<score(snapshot(B)) or snapshot(B⌈bft1​)=snapshot(B). We could instead require the score of tip(B)=B.headers_bc[σ−1] to increase: Increasing Tip Score rule: Either score(tip(B⌈bft1​))<score(tip(B)) or tip(B⌈bft1​)=tip(B). Pros: This more directly reflects the fork‑choice rule in Πbc​. In Crosslink 1, an honest bft‑proposer uses its bc‑best‑chain tip with the highest score provided that it is consistent with the Increasing Snapshot Score rule . This change removes the caveat, simplifying honest bft‑proposer behaviour. As a result of removing that caveat, we always know about an honest bft‑proposer’s bc‑best‑chain. Con: The score of the snapshot would not necessarily increase. As a result, it is technically possible that the new snapshot can be an ancestor of a previous snapshot. Whether this can actually happen depends on the value of σ and the difficulty adjustment rule. This causes no particular harm other than adding a corner case in ledger sanitization, but is inelegant. This con is removed if we either use the Combined Increasing Score rule variant described below, or we also apply the “ Making bc‑rollbacks more difficult ” change in the next section. Apart from the above con, the original motivations for the Increasing Snapshot Score rule also apply to the Increasing Tip Score rule . In particular, it still prevents potential attacks that rely on proposing a bc‑valid‑chain that forks from a much earlier block; it still limits the extent of disruption an adversary can feasibly cause to \\LOG_{\\bda}}; it still always allows a proposal to be made, which may be needed to preserve liveness of Πbft​ relative to Πorigbft​; it still prevents potential validation cost DoS attacks due to switching between snapshots with the same score. If we switch to using the Increasing Tip Score rule, then it would be more consistent for block producers to also change the tie‑breaking rule for choosing context_bft to use the tip score, i.e. score(tip(bft-last-final(C))). A variation on this suggestion effectively keeps both the Increasing Snapshot Score rule and the Increasing Tip Score rule : Combined Increasing Score rule: Either (score(snapshot(B⌈bft1​))<score(snapshot(B)) and score(tip(B⌈bft1​))<score(tip(B))), or tip(B⌈bft1​)=tip(B). Note that if tip(B⌈bft1​)=tip(B), both scores are necessarily equal. This variation does not simplify honest bft‑proposer behaviour.","breadcrumbs":"Design » Crosslink » Potential Changes to Crosslink » Changing the Increasing Score rule to require the score of the tip (rather than the score of the snapshot) to increase","id":"107","title":"Changing the Increasing Score rule to require the score of the tip (rather than the score of the snapshot) to increase"},"108":{"body":"Basic idea: Detect the case where the bc‑snapshot is rolling back, and impose a longer confirmation depth to switch to the new bc‑chain. Also temporarily stall finalization of the existing bc‑chain until the conflict has been resolved. Let baseline_snapshot be the Crosslink 1 definition of snapshot, i.e. baseline_snapshot(B)={Obc​,B.headers_bc[0]⌈bc1​,​if B.headers_bc=∅otherwise.​ When snapshot(B⌈bft1​)⪯bc​baseline_snapshot(B), we want to go into a mode where we require a longer confirmation depth Σ, say 2σ. Because we don’t know in this situation whether the old bc‑chain or the new bc‑chain will win, we stop finalizing both until a winner is clear. The simplest option is to record the state saying that we are in this mode explicitly, and add a consensus rule requiring it to be correct. That is, add an bc_is_forked field to bft‑proposals and bft‑blocks, and add a bft‑proposal and bft‑block validity rule as follows: Is Forked rule: B.bc_is_forked=(enter_forked(B) or B⌈bft1​.bc_is_forked) and not exit_forked(B) where: enter_forked(B):=snapshot(B⌈bft1​)⪯bc​baseline_snapshot(B) exit_forked(B):=score(tip(B)⌈bcΣ​)>score(snapshot(B⌈bft1​)). It is intentional that exit_forked takes precedence over enter_forked. Then redefine snapshot as follows: snapshot(B)={snapshot(B⌈bft1​),baseline_snapshot(B),​if B.bc_is_forkedotherwise.​ Since Obft​.bc_is_forked=false, the recursion will terminate. Note that there is an interaction between the Increasing Snapshot Score rule and this change: the Increasing Snapshot Score rule should arguably use baseline_snapshot instead of snapshot. The Increasing Tip Score rule , on the other hand, works fine as‑is, and so it makes sense to use both of these changes together. The combination of both changes also fixes the con discussed above for the Increasing Tip Score rule ; it ensures that the score of the snapshot must increase. Pros: If the winning chain is the chain that was first snapshotted, then there ends up being no disruption whatsoever to \\LOG_{\\bda}}. It becomes extremely difficult for an adversary with less than 50% hash power to get the finalized snapshot to switch between two competing chains more than once. Cons: It is potentially easier to cause a temporary finalization stall. An adversary could try to provoke this situation on the honest chain, either as a DoS attack, or so that its own chain that is not so encumbered can finalize more quickly than the honest chain. This does not seem like a practical attack, because such a stall can only happen when the adversary can cause a σ‑block rollback or has subverted Πbft​. The definition of snapshot becomes more complicated, and there is a risk of this complexity introducing problems. It can be argued that, unless σ is chosen too small, an adversary that can cause a σ‑block rollback likely has 50% of hash power, and therefore can cause a Σ‑block rollback. That is, increasing confirmation depth does not help beyond a certain point, and therefore (it could be argued) this change will also not help. This argument may hold for private mining attacks, but does not necessarily hold for partitioning attacks, as discussed in the next section.","breadcrumbs":"Design » Crosslink » Potential Changes to Crosslink » Making exploitation of bc‑rollbacks more difficult","id":"108","title":"Making exploitation of bc‑rollbacks more difficult"},"109":{"body":"In the case of a private mining attack, the adversary will typically conceal the existence of the overtaking chain until it can be used to cause a rollback in \\LOG_{\\bda}}. So the approach used in the previous section seems to be all we can do against such an attack. In the case of a partitioning attack, on the other hand, the adversary relies on honest nodes to do mining work on each side of the partition. This relies on the successful miners on each side knowing about their chain, but not the chain on the other side. Subtly, it does not rely on a perfect network partition. An adversary could, for example, attempt to create partitions around the most successful mining pools. Occasional leaks of information across a partition also do not necessarily foil the attack unless that information gets to a successful miner. Therefore, measures that constrain the adversary’s ability to make use of an incomplete partition can be useful. This also has the benefit of making the protocol more robust against non‑malicious incomplete partitions. Given that in such an attack the competing chains may be visible to some proposers, there is the possibility of detecting a potential rollback even before it gets snapshotted, by using the fact that previous bft‑blocks created by honest bft‑proposers have been recording the bc‑best‑chain tip σ blocks ahead. Also, depending on what proportion of validators an adversary has, they may rely on honest validators on each side to ensure that a snapshot of each chain appears in a bft‑valid block; in that case, including information about competing chains in validators' votes (see the next subsection) may be useful. It is still possible that if an adversary has several consecutive proposal slots, it can get its chain snapshotted. However, if there is an intervening slot with an honest proposer, we can potentially compare its tip with the adversary’s tip and anticipate the need to go into bc_is_forked mode. In order to get this to work, we need to propose a definition to identify bc‑chains that are competing with the current best chain, such that there is some risk of a “long” rollback to a competing chain. Let δ be a measure of how close (in terms of bc‑blocks) a competing chain’s score needs to be to that of the bc‑best‑chain, and let μ>δ be a lower bound on the rollback depth we would consider significant if the competing chain were to immediately catch up. (The condition μ>δ is necessary to avoid false positives that might only be a single‑block fork.) A node i identifies (δ,μ)‑competing chains as follows based on its current view at time t: A bc‑block is a tip if it has no known descendants. Let chit​ be node i’s bc‑best‑chain. Identify all of the tips T such that score(T)≥score(chit​⌈bcδ​) and last-common-ancestor(T,chit​)⪯bc​chit​⌈bcμ​. TODO: Details, including how to modify the enter_forked and exit_forked conditions. TODO: For now we will assume that all of the competing chain information in a bft‑block has to be checked as bc‑block‑valid in order for that block to be bft‑block‑valid. This might introduce validation DoS attacks and needs to be considered more carefully.","breadcrumbs":"Design » Crosslink » Potential Changes to Crosslink » Using tip information to detect rollbacks and partitions","id":"109","title":"Using tip information to detect rollbacks and partitions"},"11":{"body":"Users who are interested in providing finality infrastructure, or users who want to delegate ZEC towards finality, will be able to earn rewards from the protocol for doing so, while also taking on some risk to their funds (to prevent malicious abuse of the protocol). This may be an important new category of ZEC users and use cases.","breadcrumbs":"Introduction » Trailing Finality Layer in a Nutshell » Proof-of-Stake Users","id":"11","title":"Proof-of-Stake Users"},"110":{"body":"This complements the above idea by letting a validator that has seen a competing chain signal it in its signed vote. Then, as long as the adversary is reliant on some votes from honest validators that are signalling the existence of competing chains, we would go into bc_is_forked mode without relying on honest proposers to have an intervening slot. The notarization proof that appears in a bft‑block would need to be modified to preserve these signals. More precisely, it is necessary for a bft‑block B to preserve at least: the best bc‑chain that credibly competes with tip(B), if any. the best bc‑chain that does not credibly compete with tip(B) (this necessarily exists because tip(B) does not credibly compete with itself). This is also motivated by the suggested change in the next section. Enforcing this is relatively straightforward if the evidence is a SNARK. It can also be enforced with aggregate signatures even for schemes that only allow aggregation of signatures over a common message: we just collect the distinct messages (corresponding to either “no competing chain” or each distinct competing chain) and aggregate them separately.","breadcrumbs":"Design » Crosslink » Potential Changes to Crosslink » Allowing validators to signal the existence of a competing chain in their votes","id":"110","title":"Allowing validators to signal the existence of a competing chain in their votes"},"111":{"body":"Assume that votes include competing chain information as discussed above. We can assume that an honest proposer has read all of this information from its parent bft‑block. Therefore, we can require the tip score of its proposal to have at least the score of the best tip implied by that information: Let best-tip(B) be the tip mentioned in bft‑block B with the highest score. A bft‑block B “mentions” the two best tips defined in the previous section. Strong Increasing Tip Score rule: Either score(best-tip(B⌈bft1​))<score(tip(B)) or best-tip(B⌈bft1​)=tip(B). Note that this rule is really quite constraining for a potential adversary, especially in partitioning attacks. It means that if the adversary does not want to acknowledge the existence of a given chain, it cannot use any votes or build on any previous bft‑block that signals the existence of that chain. Essentially, a partitioning adversary with control over only the minimum one‑third of the stake would have to have ensure a perfectly complete partition; it could not get away with any information leakage between honest validators.","breadcrumbs":"Design » Crosslink » Potential Changes to Crosslink » Strengthening the Increasing Tip Score rule","id":"111","title":"Strengthening the Increasing Tip Score rule"},"112":{"body":"","breadcrumbs":"Design » Crosslink » Potential Changes to Crosslink » Attempts to improve finalization latency","id":"112","title":"Attempts to improve finalization latency"},"113":{"body":"The Crosslink 1 design imposes a finalization latency of at least 2σ+1 block times. Intuitively, this is because in fin(H):=[snapshot(B) for B⪯bft​bft-last-final(H⌈bcσ​.context_bft)], snapshot(bft-last-final(H⌈bcσ​.context_bft)) is at least σ+1 blocks back from H⌈bcσ​ (as argued in Questions about Crosslink 1 ), and therefore 2σ+1 blocks back from H. So the total finalization latency is σ block times + BFT overhead + (σ+1) block times + snapshot overhead. However, the snapshot headers contain information about the proposer’s bc‑best‑chain. Define LF(H):=bft-last-final(H.context_bft). Although it is not guaranteed, normally snapshot(LF(H⌈bcσ​)) will be an ancestor of H⌈bcσ​. What if we were to optimistically allow the last snapshot to be taken as S(H):={last-common-ancestor(H⌈bcσ​,tip(LF(H⌈bcσ​)),snapshot(LF(H⌈bcσ​)),​if it extends snapshot(LF(H⌈bcσ​))otherwise​? After all, we know that last-common-ancestor(H⌈bcσ​,tip(LF(H⌈bcσ​)) is confirmed. Oh, this won’t work. The problem is that we want safety of LOGfin​ not to depend on safety of Πbc​. So we cannot assume (for this purpose) that nodes see the same H⌈bcσ​.","breadcrumbs":"Design » Crosslink » Potential Changes to Crosslink » [Broken] Adjusting the last snapshot definition","id":"113","title":"[Broken] Adjusting the last snapshot definition"},"114":{"body":"What if we instead take this to be the definition of \\LOG_{\\opt}}, replacing \\LOG_{\\bda}} (\"opt\" meaning optimistic)? As stated, a malicious proposer can try to maximize the latency of \\LOG_{\\opt}} (subject to the Increasing Score rule). For example, if there exists a fork of length μ, the malicious proposer can force the latency of \\LOG_{\\opt}} to be (σ+μ+1) block times + BFT overhead. However, this can be improved by applying the idea to each bft‑block in turn after the one pointed to by the best confirmed bc‑block. Then a malicious proposer cannot do anything that it could not do anyway (keeping the finalization point at its current position). Pros: This is always more conservative in terms of safety than the current design. The latency of \\LOG_{\\opt}} will typically be σ+1 bc‑block times, rather than 2σ+1 block times. Cons: \\LOG_{\\opt}} is not dynamically available in any sense. It just has lower latency and different security characteristics. Even under optimistic conditions, \\LOG_{\\opt}} will lag slightly behind where it would be for the Crosslink 1 design, because H⌈bcσ​ will necessarily be ahead of tip(LF(H⌈bcσ​)).","breadcrumbs":"Design » Crosslink » Potential Changes to Crosslink » Replacing \\LOG_{\\bda}} with \\LOG_{\\opt}}","id":"114","title":"Replacing \\LOG_{\\bda}} with \\LOG_{\\opt}}"},"115":{"body":"The following idea is broken for safety when Πbft​ has been subverted: Info We have two potential sources of information about blocks that could plausibly be considered finalized: H⌈bcσ​ the snapshots on the chain of the last seen final bft‑block, lsf. We cannot rely only on 1. because we want assured finalization even under partition. We cannot rely only on 2. because if Πbft​ has been subverted, then the chain of final bft‑blocks could fork. But intuitively, if we combine these sources of information, using them over the Crosslink 1 finalization only when they are consistent, the resulting protocol should still be as safe as the safer of Πbft​ and Πbc​. In particular, 2. will not roll back unless Πbft​ has been subverted. If this idea were to pan out, it could improve the latency of finalization by σ block times. This approach is essentially a hybrid of Snap‑and‑Chat and Crosslink 1: the Snap‑and‑Chat construction gives a finalized ledger under the assumption that Πbft​ has not been subverted; the main crosslink idea is used to make sure that outputs from all finalized transactions are eventually spendable; safety is still only dependent on the stronger of the safety of Πbft​ and Πbc​, because we use the additional information from snapshots in final bft‑blocks only up to the point at which they agree with the best confirmed bc‑block. To explain the safety problem with this idea: suppose that Πbft​ has been subverted. In that case it is possible for a snapshot to be finalized without having being confirmed as in any honest node’s bc‑best‑chain; that is, it is possible for LOGfin​ to include transactions T from a snapshot S in bft‑block A such that S is not on the consensus bc‑best‑chain. And, because Πbft​ has been subverted, it is also possible that a conflicting final bft‑block B omits S. And so a node that has seen B will think that it is consistent with the best bc‑chain (so that its LOGfin​ does not include T but does include later transactions on the consensus bc‑best‑chain), but a node that has seen A will compute a LOGfin​ that does include T. More detailed specification of the above broken idea. Define LF(H):=bft-last-final(H.context_bft) as before. For simplicity assume that lsf extends LF(H⌈bcσ​) by only one bft‑block. (This assumption could have been removed if the idea had panned out.) Then this proposal was to consider this bc‑block as contributing the last finalized snapshot: S(H):={last-common-ancestor(H⌈bcσ​,snapshot(lsf)),snapshot(LF(H⌈bcσ​)),​if it extends snapshot(LF(H⌈bcσ​))otherwise​ There is no need for a tie‑breaking rule for 2.: if we ever see two context bft‑blocks for which the last‑final blocks are conflicting, we know that Πbft​ has been subverted, so we should stall or crash. Caveat: for a given node, H⌈bcσ​ can in theory roll back past snapshot(lsf), therefore S(H) can also roll back. It is okay if we keep state here and refuse to roll back. We should set a “crisis flag”, and unset it if at any point LF(H⌈bcσ​) extends lsf_at_crisis. (If Πbft​ is safe and live, it will.) A similar rule that would give the same result in almost all circumstances is: S(H):=⎩⎨⎧​H⌈bcσ​,snapshot(lsf),snapshot(LF(H⌈bcσ​)),​if snapshot(LF(H⌈bcσ​))⪯bc​H⌈bcσ​⪯bc​snapshot(lsf)if snapshot(LF(H⌈bcσ​))⪯bc​snapshot(lsf)⪯bc​H⌈bcσ​otherwise​","breadcrumbs":"Design » Crosslink » Potential Changes to Crosslink » [Broken] Using snapshots from the last‑seen bft‑chain when it is consistent with the bc‑best‑chain","id":"115","title":"[Broken] Using snapshots from the last‑seen bft‑chain when it is consistent with the bc‑best‑chain"},"116":{"body":"The answer given for this question at The Crosslink 2 Construction is: If this were enforced, it could be an alternative way of ensuring that every bft‑proposal snapshots a new bc‑block with a higher score than previous snapshots, potentially making the Increasing Score rule redundant. However, it would require merging bc‑block‑producers and bft‑proposers, which could have concerning knock‑on effects (such as concentrating security into fewer participants). This may have been too hasty. It is not clear that merging bc‑block‑producers and bft‑proposers actually does “concentrate security into fewer participants” in a way that can have any harmful effect. Remember, the job of a bft‑proposer in Crosslink is primarily to snapshot the bc‑best‑chain (even more so if the Increasing Tip Score rule is adopted). An honest miner by definition is claiming to build on the best chain, and miners have a strong economic incentive to do so. Therefore, it is entirely reasonable for every newly produced block to be treated as a bft‑proposal. This arguably decentralizes the task of proposing bft‑blocks more effectively than using a leader election protocol would — especially given that in a hybrid protocol we necessarily still rely on there being sufficient honest miners. [DKT2021] , for example, argues for the importance of “the complete unpredictability of who will get to propose a block next, even by the winner itself.” The main basis of this argument is that it makes subversion of the proposer significantly more difficult. A PoW protocol has that property, and most PoS protocols do not. (It is not that PoS protocols are unable to provide this property; indeed, [DKT2021] constructs a PoS protocol, “PoSAT”, that provides it.) So let’s explore this in more detail. A newly produced bc‑block would implicitly be a bft‑proposal with itself as the tip. The bc_headers field is therefore not needed. The Tail Confirmation rule goes away since its intent is automatically satisfied. This is already a significant simplification. The inner proposer signature is also not needed (since the bc‑header is self-authenticating), but the block producer would have to include a public key H.pubkey that can be used to verify its outer signature. It would sign the notarized bft‑block with the corresponding private key. This change is a wash in terms of protocol complexity. Considered as a bft‑proposal, a bc‑block needs to refer to a parent bft‑block, which requires a parent_bft field in the bc‑header. With some caveats depending on the design of Πorigbft​, it might be possible to merge this with the context_bft field, but for now we will assume that it is not merged. What are the caveats? If we are in an execution where Final Agreement holds for Πbft​, then it is possible to show that merging the two fields has no negative effect, provided that Πorigbft​ has no additional rules that could disallow it in some cases. This is because, by Final Agreement , bft-last-final(H′.context_bft)⪯⪰∗bft​bft-last-final(C) for any potential bft‑block C that the bc‑block‑producer of a new block H could choose as H.context_bft. Suppose that the bc‑block‑producer freely chooses H.parent_bft according to the desired honest behaviour for a bft‑proposer in Πbft​, and then chooses C to be the same block (which is always reasonable as long as it is allowed). In the case bft-last-final(H′.context_bft)⪯∗bft​bft-last-final(C), we are done, because this choice of H.context_bft is allowed by the Extension rule . In the case bft-last-final(H′.context_bft)⪲∗bft​bft-last-final(C), we can argue that H′.context_bft would be a better choice than C for H.parent_bft as well as for H.context_bft, because it has a later final ancestor. This is where the argument might fall down if Πorigbft​ (and therefore Πbft​) has any additional rules that could disallow this choice. For now let’s suppose that situtation does not arise, but it is one of the caveats. Another potential problem is that in an execution where Final Agreement does not hold for Πbft​, we can no longer infer that either bft-last-final(H′.context_bft)⪯∗bft​bft-last-final(C) or bft-last-final(H′.context_bft)⪲∗bft​bft-last-final(C). In particular it could be the case that the producer of H′ was adversarial, and chose H′.context_bft in such a way as to favour its own bft‑block that is final in that context. However, in that situation it must be possible for the bc‑block‑producer to see (and prove) that the bft‑chain has a final fork. That is, it can produce a witness C to the violation of Final Agreement , showing that bft-last-final(H′.context_bft)⪯⪰∗bft​bft-last-final(C) does not hold, as discussed in the section Recording more info about the bft‑chain in bc‑blocks above. The second caveat is that in that situation, we still need to set H.parent_bft and H.context_bft in order to be able to recover, and they typically should not be the same in order to do so. The Increasing Tip Score rule is still needed, but it can be simplified. A newly produced bc‑block H is also a bft‑proposal such that snapshot(H)=H. This would yield the following bft‑proposal / bc‑block validity rule: [Candidate rule for discussion] Either score(tip(H.parent_bft))<score(H) or tip(H.parent_bft)=H. except that tip(H.parent_bft) cannot be H, because H is newly produced. It turns out we can just drop that part: Increasing Tip Score (producer = proposer) rule: score(tip(H.parent_bft))<score(H). This works because, if H does not have a higher score than the bc‑block tip(H.parent_bft), the bc‑block‑producer should instead have built on top of that bc‑block — which was necessarily known to the producer in order for it to set H.parent_bft in the header of the new block. The voting would be the same, performed by the same parties. Therefore, there is no concentration of voting into fewer parties. There is no change in the producer/proposer’s incentive to make the bft‑notarization‑proof or its soundness properties. Everything else is roughly the same, including the use of the context_bft field of a bc‑block and the validity rules related to it. As far as I can see, all of the security analysis goes through essentially unchanged. There may be some complication due to the fact that BFT protocols are typically designed to use epochs with a fixed period, whereas bc‑blocks are found at less predictable intervals. However, as long as BFT messages are labelled with the bc‑block they apply to, it seems like most BFT protocols would be tolerant to this change. In fact the adaptations of Snap‑and‑Chat to Hotstuff and PBFT in [NTT2020] already assume that BFT messages can be queued and processed at a later time, and rely on those BFT protocols' tolerance to this. Info In most PoS protocols, the requirement to have a minimum amount of stake in order to make a proposal acts as a gatekeeping filter on proposals , and potentially allows parties that make invalid proposals to be slashed. Strictly speaking, whether there is a stake requirement to make a proposal is independent of whether bc‑block‑producers (e.g. miners) are merged with bft‑proposers. It could be, for example, that a miner is still able to produce bc‑blocks, but is not able to make them into a proposal unless they satisfy a stake requirement. (This would have significant effects on the economics of mining that would need to be analyzed, and that might have governance consequences.) In a system that uses PoS, validators by definition need to have stake in order to control the ability to vote. This also allows validators to be slashed. On the other hand, there is no technical reason why the ability to make a bft‑proposal has to be gatekept by a stake requirement — given the situation of Zcash in which we already have a mining infrastructure, and that in a Snap‑and‑Chat or Crosslink‑style hybrid protocol we necessarily still rely on miners not to censor transactions. The potential to make proposals that are expensive to validate as a denial of service is made sufficiently difficult by proof‑of‑work. This option has probably been underexplored by previous PoS protocols because they cannot assume an existing mining infrastructure. It could be argued that this approach goes less far toward a pure PoS‑based block‑chain protocol, leaving more to be done in the second stage. However, there is a clear route to that second stage, by replacing PoW with a protocol like PoSAT that has full unpredictabilty and dynamic availability. PoSAT does this using a VDF, and as it happens, Halo 2 is a strong candidate to be used to construct such a VDF. If the arguments in [DKT2021] about the need for proposer unpredictability are persuasive, then this approach defers the complexity of requiring a VDF without losing any security, since Zcash’s PoW is already unpredictable.","breadcrumbs":"Design » Crosslink » Potential Changes to Crosslink » What about making the bc‑block‑producer the bft‑proposer?","id":"116","title":"What about making the bc‑block‑producer the bft‑proposer?"},"117":{"body":"Building on the previous idea, we can try to eliminate the explicit bft‑chain by piggybacking the information it would hold onto a bc‑block (in the header and/or the block contents). In the previous section we merged the concepts of a bft‑proposal and a bc‑block; the parentbft​(P) and P.context_bft fields of a bft‑proposal were moved into the H.parent_bft and H.context_bft fields of a bc‑header respectively. A field H.pubkey was also added to hold the producer’s public key, so that the producer can sign the bft‑block constructed from it using the corresponding private key. This left the concept of a bft‑block intact. Recall that in Crosslink 1, a bft‑block B consists of (P,proofP​) signed by the proposer. So in “Crosslink with proposer = producer”, a bft‑block consists of (H,proofH​) signed by the producer. What if a bc‑block H were to “inline” its parent and context bft‑blocks rather than referring to them? I.e. a bc‑block H with H.parent_bft referring to (H′,proofH′​) signed for H′.pubkey, would instead literally include (either in the header or the coinbase transaction) (H′,proofH′​) signed for H′.pubkey — and similarly for H.context_bft. It would still be necessary to have the message type that the proposer/producer previously used to submit a notarized bft‑block. (It cannot be merged with a bc‑block announcement: the producer of a new block is not in general the producer of its parent, and their incentives may differ; also we cannot wait until a new block is found before publishing the previous notarization.) It would also still be necessary for Crosslink nodes to keep track of notarizations that have not made it into any bc‑block. Nevertheless, this is a potential simplification. Note that unless notarization proofs are particularly short and constant-length, it would not be appropriate to include them in the bc‑block headers, and so they would have to go into the coinbase transaction or another similar variable-length data structure. In that case we would still have an indirection to obtain the bft‑block information; it would just be merged with the indirection to obtain a coinbase transaction (or similar) — which is already needed in order to check validity of the bc‑block. As discussed under Recording more info about the bft‑chain in bc‑blocks above, we might in any case want to record information about other proposed and notarized bft‑blocks, and the data structure needed for this would necessarily be variable-length. The complexity burden of doing so would be shared between these two changes. It would be possible to save some space in headers (while keeping them fixed-length), by inlining only one of H.parent_bft and H.context_bft in the header and keeping the other as a hash. As discussed under “What are the caveats?” above, the only reason for the two bft‑blocks referred to by these fields to be different, is that the bc‑block producer has observed a violation of Final Agreement in Πbft​. In that case, we can include an inlining of the other H.∗_bft block, and any other information needed to prove that a violation of Final Agreement has occurred, in a variable-length overflow structure. Pros: No additional mechanism or messages are needed to obtain bft‑blocks given their hashes. It could be a performance/latency improvement to not have to separately fetch bft‑blocks. Con: Additional complexity of the variable-length overflow mechanism suggested above, if it is used. Assumes that notarization proofs are not too large.","breadcrumbs":"Design » Crosslink » Potential Changes to Crosslink » Do we need an explicit bft‑chain at all?","id":"117","title":"Do we need an explicit bft‑chain at all?"},"118":{"body":"A potential simplification can be obtained by combining the following two ideas: Str4d suggested that the snapshot of each bft‑block should descend from the snapshot of its parent bft‑block. Nate suggested that each bc‑block H should descend from snapshot(LF(H)). Str4d’s suggestion can be written as: Linearity rule: snapshot(B⌈bft1​)⪯bc​snapshot(B). Notice that this implies the existing Increasing Score rule in Crosslink 1, because score necessarily increases within a bc‑valid‑chain. Therefore it would in practice be a replacement for the Increasing Score rule . It does not imply the Increasing Tip Score rule discussed above , and in fact it could make sense to enforce both the Linearity rule and the Increasing Tip Score rule . The Linearity rule implies that it is no longer possible for a bft‑valid‑chain to snapshot a bc‑chain that rolls back relative to the previous snapshot. This makes it unnecessary to sanitize LOGfin​: the sequence of snapshots considered by san-ctx is linear, and so the “sanitization” would just return the transactions in the last snapshot. To remove the need to sanitize LOGbda​ as well, we need a further modification to Πbc​. Recall that in Crosslink 1 we define: fin(H)bda-ctx(H,μ)​:=:=​[snapshot(B) for B⪯bft​LF(H⌈bcσ​)]san-ctx(fin(H)∣∣[H⌈bcμ​]).​ The Linearity rule ensures that fin(H) is a linear sequence of snapshots, but for fin(H)∣∣[H⌈bcμ​] to be linear, we also need last(fin(H))=snapshot(LF(H⌈bcσ​))⪯bc​H⌈bcμ​. In order for this to hold for any choice of μ with 0≤μ≤σ, we require the strongest version of this condition with μ=σ, i.e. snapshot(LF(H⌈bcσ​))⪯bc​H⌈bcσ​. Since we can only enforce that this holds for H⌈bcσ​ by enforcing that it holds for an arbitrary bc‑valid‑block H, the rule becomes: Last Final Snapshot rule: snapshot(LF(H))⪯bc​H. This is exactly Nate’s suggestion discussed in Questions about Crosslink . In that document we argued against this rule, but that argument was made in the context of a protocol without the Linearity rule (and originally, even without the Increasing Score rule ). Combining the Linearity rule and Last Final Snapshot rule , on the other hand, completely eliminates the need for sanitization. This could be a huge simplification — and potentially safer, since it would avoid breaking assumptions that may be made by existing Zcash node implementations and other consumers of the Zcash block chain. To spell out the resulting simplifications to the definitions of LOGfin,it​ and \\LOG^t_{\\bda,\\mu,i}, we would just have: \\begin{array}{rcl} \\LOG_{\\fin,i}^t &:=& \\snapshotlf{\\ch_i^t \\trunc_{\\bc}^\\sigma} \\ \\LOG_{\\bda,\\mu,i}^t &:=& \\ch_i^t \\trunc_{\\bc}^\\mu \\end{array} Here it is no longer necessary to define LOGfin,it​ and \\LOG^t_{\\bda,\\mu,i} as sequences of transactions, since the final and bounded-available chains are both just bc‑valid‑chains. The definition of finality-depth in the Finality depth rule becomes much simpler: finality-depth(H):=height(H)−height(snapshot(LF(H⌈bcσ​))) As before, either finality-depth(H)≤L or is-stalled-block(H). Avoiding sanitization also means that the bug we described in Snap‑and‑Chat, that could prevent spending outputs from a snapshotted chain after a σ-block rollback, cannot occur by construction. That is, the changes in Πbc​ contexual validity relative to Πorigbc​ are not needed any more. This almost seems too simple, and indeed we should be skeptical, because the security analysis essentially has to be redone. The reason why Snap‑and‑Chat didn’t take this approach is that it requires a more complicated argument to show that it is reasonable to believe in the safety assumptions of Πbc​ whenever it is reasonable to believe in the corresponding assumptions for Πorigbc​. This is because the ... We will need to do some work to show that the changes are benign.","breadcrumbs":"Design » Crosslink » Potential Changes to Crosslink » Linearity and Last Final Snapshot rules","id":"118","title":"Linearity and Last Final Snapshot rules"},"119":{"body":"The key observation needed for this analysis is that neither the Linearity rule nor the Last Final Snapshot rule affect the evolution of Πbc​ unless we are in a situation where its Prefix Consistency or Prefix Agreement properties would be violated. This implies that any safety property that we can prove given Prefix Consistency plus ****.","breadcrumbs":"Design » Crosslink » Potential Changes to Crosslink » Security Analysis","id":"119","title":"Security Analysis"},"12":{"body":"Miners who provide Proof‑of‑Work security will necessarily see some reduction in their block rewards, since this proposal maintains the same issuance schedule and supply cap of ZEC while also spending some rewards on finality. Important Note: The proportion and details of how much mining rewards will be impacted, and conversely how much finality/PoS providers will earn, are not yet specified in this proposal.","breadcrumbs":"Introduction » Trailing Finality Layer in a Nutshell » Miners","id":"12","title":"Miners"},"120":{"body":"","breadcrumbs":"References » References","id":"120","title":"References"},"121":{"body":"Ebb-and-Flow Protocols: A Resolution of the Availability-Finality Dilemma","breadcrumbs":"References » Ebb-and-Flow Protocols","id":"121","title":"Ebb-and-Flow Protocols"},"122":{"body":"","breadcrumbs":"Version History » Version History","id":"122","title":"Version History"},"123":{"body":"This version updated the Crosslink construction and its security analysis to version 2.","breadcrumbs":"Version History » 0.2.0 - Crosslink 2","id":"123","title":"0.2.0 - Crosslink 2"},"124":{"body":"Issue Tracking This version introduced the Crosslink construction by linking to an external write-up. It transitioned away from the implicit design presented at Zcon4 to a rigorously defined hybrid construction .","breadcrumbs":"Version History » 0.1.0 - Introducing Crosslink","id":"124","title":"0.1.0 - Introducing Crosslink"},"125":{"body":"Prior to the initial release of this book, a design sketch was presented in Interactive Design of a Zcash Trailing Finality Layer (Hybrid PoW+PoS) by Nate Wilcox: Zcon4 .","breadcrumbs":"Version History » Prehistory - Zcon4 Presentation","id":"125","title":"Prehistory - Zcon4 Presentation"},"13":{"body":"This design is appealing as a safer first step in transitioning the Zcash protocol for multiple reasons:","breadcrumbs":"Introduction » Trailing Finality Layer in a Nutshell » Why is this a Good Approach to a PoS Transition?","id":"13","title":"Why is this a Good Approach to a PoS Transition?"},"14":{"body":"This transition would enable PoS mechanisms, including the ability to operate PoS infratructure and delegate ZEC towards those providers to earn protocol rewards. While all PoS transitions would accomplish this, this approach does so in a conservative manner: it introduces these mechanics while striving to minimize the impact on existing use cases and protocol security. In a sense, we can think of this approach as enabling the Zcash community to “dip our toes in the PoS waters” rather than diving in. If the results pan out well, it gives us confidence for further transitions. If we discover challenges, flaws, or risks, we anticipate their impact will be more limited since this is a more cautious transition step.","breadcrumbs":"Introduction » Trailing Finality Layer in a Nutshell » It Enables Proof-of-Stake Mechanisms Conservatively","id":"14","title":"It Enables Proof-of-Stake Mechanisms Conservatively"},"15":{"body":"In many cases, existing products, services, and tools can continue using the mainnet chain with no changes to code assuming they rely on existing consensus nodes. We view this as a major benefit which allows Zcash’s existing user network effect to continue safely unperturbed. There will be certain narrow exceptional areas if those products, services, or tools need to be precise in areas where the protocol has changed, such as mining/staking reward calculations, transaction formats (in particular any new PoS-related fields or logic), or chain rollback logic.","breadcrumbs":"Introduction » Trailing Finality Layer in a Nutshell » Minimal Use-Case Disruption","id":"15","title":"Minimal Use-Case Disruption"},"16":{"body":"By conceptualizing the TFL as a distinct “layer” or subprotocol, the consensus rules can be described in terms of two consensus subprotocols , one embodying most of the current consensus logic of Zcash and another the TFL. These protocols interact through a hybrid construction . See Design at a Glance to learn more about these distinct subprotocols. Reasoning about the whole protocol can leverage analysis and understanding of each subprotocol and the hybrid construction somewhat independently due to this modular design. Note that although this design is modular, the hybrid construction may require modifications to the [PoW] and/or [PoS] subprotocols to protect safety and liveness properties. Nevertheless, the modularity still improves analysis and reasoning compared to a monolithic design. Finally, since one subprotocol is very similar to the existing Zcash NU5/NU6 consensus protocol, this lessens risk that the consensus properties within that subprotocol compromise current NU5/NU6 properties.","breadcrumbs":"Introduction » Trailing Finality Layer in a Nutshell » Modular Design","id":"16","title":"Modular Design"},"17":{"body":"In addition to the other benefits of protocol design modularity, we anticipate actual implementations can realize this modularity in code. This can help makes implementations more robust, easier to maintain, and more interoperable. For example, we can envision a standardized interface between PoW & TFL consensus components, enabling different development teams to provide these different components and for “full node” packagers to mix and match them. This is somewhat reminiscent of Ethereum’s execution/consensus layer separation which we believe has shown great success in implementation team and product diversity.","breadcrumbs":"Introduction » Trailing Finality Layer in a Nutshell » Modular Implementation","id":"17","title":"Modular Implementation"},"18":{"body":"In the rest of the introductory section of this book, we describe the status and next steps for the TFL proposal, provide a motivation for finality, and suggestions for getting involved.","breadcrumbs":"Introduction » Trailing Finality Layer in a Nutshell » Cracking the Nutshell","id":"18","title":"Cracking the Nutshell"},"19":{"body":"This is an early and incomplete protocol design proposal. It has not been well vetted for feasibility and safety. It has not had broad review from the Zcash community, so its status on any Zcash roadmap is undetermined.","breadcrumbs":"Introduction » Status and Next Steps » Status and Next Steps","id":"19","title":"Status and Next Steps"},"2":{"body":"There are multiple developer orgs working on different proposed features for Zcash. Some of these involve multiple large distinct upgrade steps, and several of these steps depend on other such steps. This could be represented as a directed acyclic graph. We have begun referring to this space of possible future improvements as the Zcash Tech-Tree , taking inspiration from an analogous concept in gaming. [1] We envision a proof-of-stake transition path as one of the potential paths within this tech-tree which is the primary protocol focus of this proposal. An example visualization of this Zcash Tech-Tree might look like this:","breadcrumbs":"Introduction » A Path to Proof-of-Stake Zcash » The Zcash Tech-Tree","id":"2","title":"The Zcash Tech-Tree"},"20":{"body":"","breadcrumbs":"Introduction » Status and Next Steps » Current Components","id":"20","title":"Current Components"},"21":{"body":"This book is intended to become both a high-level overview and introduction to TFL , and a full specification.","breadcrumbs":"Introduction » Status and Next Steps » This Book","id":"21","title":"This Book"},"22":{"body":"The current heart of the design work is an in-progress hybrid consensus protocol construction called Crosslink (definition) . This is defined in the Crosslink chapter .","breadcrumbs":"Introduction » Status and Next Steps » Crosslink","id":"22","title":"Crosslink"},"23":{"body":"We've begun creating a simulator called simtfl which we will use to model security and abstract performance concerns. Its development is tracked at https://github.com/zcash/simtfl .","breadcrumbs":"Introduction » Status and Next Steps » simtfl","id":"23","title":"simtfl"},"24":{"body":"PoS subprotocol selection, Issuance and supply mechanics, such as how much ZEC stakers may earn, Integrated Zcash transaction semantics, A transition plan from current Zcash mainnet to this protocol design, ZIP s specifying the above to the level of specificity required by ZIPs, Security and safety analyses, Economic analyses. This list may be incomplete, and as the design matures the need for major new components may be revealed.","breadcrumbs":"Introduction » Status and Next Steps » Major Missing Components","id":"24","title":"Major Missing Components"},"25":{"body":"This design proposal is being developed by Electric Coin Company and Shielded Labs as the first major milestone in our focus of deploying Proof-of-Stake to the Zcash protocol. Our rough near-term plan for this proposal is as follows: Complete the Crosslink description. Complete core security arguments for Crosslink . Define the Major Missing Components above, including considerations such as issuance mechanics and Proof-of-Stake mechanisms. Complete auxillary security arguments and analyses, such as specific attack scenarios, game-theoretic security, and so forth. Mature simtfl to analyze all cases of interest. Follow the general Zcash process for proposal/review/refinement, including proposing one or more ZIP s. Follow the general Zcash governance process for proposal review and refinement. If accepted, productionize the proposal in ECC products and collaborate with other implementors who implement the proposal. Celebrate when and if the proposal is activated on Mainnet. 🎉 The fine-grained day-to-day goals and tasks for this project are present in the Zcash Developers Hub in the TFL-focused DAG . Please also see Get Involved if you are interested in tracking this progress more closely, or in contributing.","breadcrumbs":"Introduction » Status and Next Steps » Next Steps","id":"25","title":"Next Steps"},"26":{"body":"In Zcash currently, consensus relies solely on PoW, which only provides probabilistic finality , rather than assured finality . [1] This style of consensus does not offer a guarantee that any given block may not be rolled back which may invalidate the transactions it contains. Instead, the probability that a block may be rolled back decreases as more blocks are mined subsquently on top of it. [2] Let’s walk through an example of how Zcash’s current PoW with probabilistic finality can impede important use cases. Consider a PoW node which sees this block sequence at time T=0: When should a user, wallet, or other system choose to act based on a transaction in a block? For this example, let’s assume a bridging system may have received a deposit for ZEC in block f and issued a corresponding number of proxy tokens on a different network. At a later time, T=1, this same node may see a longer PoW chain which invalidates some previously seen blocks: The node has observed a longer chain ending at block h', so PoW consensus calls for that new sequence to be treated as consensus. The previously seen blocks f and g are no longer part of the consensus history, and have been rolled back.","breadcrumbs":"Introduction » Motivating Finality » Motivating Finality","id":"26","title":"Motivating Finality"},"27":{"body":"In our example, the bridging system acted in response to a transaction in the original block f at T=0. If the new sequence ending at h' no longer contains the deposit to the bridging system, the integrity of the bridge has been violated [3] ; the associated proxy tokens may have already been used in a complex chain of Defi applications or deposited onto an exchange and sold, which would make any recovery impossible. The proxy tokens on the other network no longer correspond to the correct amount of ZEC on the Zcash network.","breadcrumbs":"Introduction » Motivating Finality » Impact of the Rollback","id":"27","title":"Impact of the Rollback"},"28":{"body":"This example demonstrates how a lack of assured finality can impede many useful real-world scenarios. In practice, systems and services which need greater assurances wait for more block confirmations. This has several drawbacks: It doesn’t remove the vulnerability , it only reduces the likelihood. Different applications/services may require different block depths, making it difficult to compose or chain together different applications/services. Different block depth policies potentially confuse users, i.e. “why do I have to wait one hour for my deposit in this exchange, but only 30 minutes on that exchange?”. It introduces a long delay which inhibits many useful applications. In addition to these user-facing and economic drawbacks, correctly handling rollbacks makes the code for nodes, wallets, and other infrastructure more complex. Worse still, many systems may not have correct behavior for rollbacks at different depths, and since large rollbacks are rarer, these implementation flaws may not surface until there is a large rollback. While a large rollback would be disruptive all by itself, it becomes even worse when previously undiscovered bugs exacerbate the situation.","breadcrumbs":"Introduction » Motivating Finality » Rollback Complications","id":"28","title":"Rollback Complications"},"29":{"body":"Trailing finality extends the existing PoW consensus so that older blocks become final, assuring they cannot be rolled back, and by extension neither can any of the transactions they contain. This directly addresses the first two flaws above: it completely removes the vulnerability, and it ensures all systems that need finality behave consistently with each other. As for delay, trailing finality also introduces delay since final blocks “trail behind” the most recent PoW blocks. This can be an improvement for some applications, depending on their latency requirements. For example, if the delay to finality averages around 10 minutes, then this would enable an improvement for an exchange that requires 60 minutes of PoW blocks for a deposit. On the other hand, it would not be an improvement for an application that needs finality faster than 10 minutes. Finally, implementations can be simplified by relying on the guarantee of finality. For example, a wallet can describe any transaction as pending or final, and does not need to provide difficult and potentially confusing UX (and the supporting database sophistication) for handling rollbacks.","breadcrumbs":"Introduction » Motivating Finality » Trailing Finality Benefits","id":"29","title":"Trailing Finality Benefits"},"3":{"body":"Given that context, we envision a path within the Zcash Tech-Tree for transitioning Zcash to PoS. At the top level we propose that this path should include at least two major milestones: Transitioning from Zcash NU5/NU6 PoW consensus to a PoW/PoS hybrid consensus protocol dubbed PoW+TFL . Transitioning from PoW+TFL to pure PoS protocol. After this transition to pure PoS, there are likely to be future improvements to the PoS protocol, or the consensus protocol more generally. This TFL book focuses almost exclusively on the first step in this sequence. Our primary motivation for proposing (at least) two steps is to minimize disruption to usability, safety, security, and the ecosystem during each step. This book primarily focuses this first step: the transition to PoW+TFL . To understand the specific goals for that, see Design Goals . With this approach, the Zcash Tech Tree with the TFL approach might look something like this:","breadcrumbs":"Introduction » A Path to Proof-of-Stake Zcash » A Proof-of-Stake Transition Path","id":"3","title":"A Proof-of-Stake Transition Path"},"30":{"body":"Throughout this book, when we say finality or final without other qualifiers, we specifically are referring to assured finality or an assured final block. Where we call out probabalistic finality we always use that qualifier. The estimated probability of a rollback relies on a variety of PoW security assumptions, and can be violated in various conditions, such as in mining efficiency breakthroughs, compromises of the PoW challenge algorithm (e.g. hash function collision resistance failure), difficulty-adjustment-algorithm failures, sudden/surprise mining capacity increases, and so on. So the estimated probability can be violated in potential “black swan” events. This discussion simplifies consideration of transaction rollback vs block rollback . When a block is rolled back, it is possible for some of the transactions contained in it to appear in new canonical blocks. The conditions when this can occur vs when it cannot are multifaceted and also subject to malicious influence, so for simplicity we assume all transactions within a rolled-back block are also rolled back.","breadcrumbs":"Introduction » Motivating Finality » Footnotes","id":"30","title":"Footnotes"},"31":{"body":"We welcome contributions! There are a variety of ways to contribute to this project:","breadcrumbs":"Introduction » Get Involved » Get Involved","id":"31","title":"Get Involved"},"32":{"body":"If you have a GitHub account, you can get hands-on via the GitHub repository for this book, including: Ask a Question - all questions welcome from basics to in-depth. Suggest an Improvement to the content: anything from typo fixes to major design change proposals. Report Rendering / Infrastructure Issues in case you're having trouble reading the content, viewing diagrams, rendering on your own computer, etc…","breadcrumbs":"Introduction » Get Involved » GitHub","id":"32","title":"GitHub"},"33":{"body":"The Zcash Forum is a hangout for many Zcash enthusiasts. This is a good spot for more open-ended discussion about this design proposal, alternatives, and other developments in Zcash.","breadcrumbs":"Introduction » Get Involved » Zcash Forum","id":"33","title":"Zcash Forum"},"34":{"body":"You can catch us on the Zcash R&D Discord in the #proof-of-stake channel.","breadcrumbs":"Introduction » Get Involved » Zcash R&D Discord","id":"34","title":"Zcash R&D Discord"},"35":{"body":"The Zcash Arborist Calls are bi-weekly Zcash protocol development calls, where proposals like this are discussed. Feel free to come lurk, ask questions, or provide feedback or suggestions.","breadcrumbs":"Introduction » Get Involved » Zcash Arborist Calls","id":"35","title":"Zcash Arborist Calls"},"36":{"body":"This book relies on the following terminology. We strive to keep these definitions as precise as possible and their definitions may deviate from uses elsewhere. Definitions are sorted alphabetically.","breadcrumbs":"Terminology » Terminology","id":"36","title":"Terminology"},"37":{"body":"Assured Finality : A protocol property that assures that transactions cannot be reverted by that protocol. As with all protocol guarantees, a protocol assumes certain conditions must be met. A transaction may either be final or not: transactions which are not final may not become final, whereas once transactions do achieve finality they retain that property indefinitely (so long as protocol requirements are met). Importantly, it is not feasible for any protocol to prevent reversing final transactions \"out of band\" from the protocol, such as if a sufficiently large and motivated group of users forks the network to include a specific new validity rule reverting transactions. In some cases this might be desirable, for example to mitigate exploitation of a security flaw. We are investigating the implications for governance and how to incorporate such situations into our security model. In any case, for this reason we eschew the term \"absolute finality\" sometimes used in technical discussions about consensus protocols. Consensus Subprotocols : The PoW and PoS subprotocols in PoW+TFL or other hybrid protocols . Crosslink : A hybrid construction consensus protocol striving to implement the TFL design goals. See Status and Next Steps: Current Components for current status. Final : A protocol property of transactions. In this book, this always implies assured finality , in contrast to concepts like \"probabilistic finality\" provided by PoW . Hybrid Consensus : A consensus protocol that integrates more than one consensus subprotocol. PoW+TFL is an instance of a hybrid protocol integrating PoW and PoS protocols. Hybrid Construction : The design component of a hybrid consensus which specifies how to integrate subprotocols and what modifications, if any, those subprotocols need to be safely integrated. Examples include Crosslink and Snap-and-Chat . Liveness : The property of a distributed protocol which ensures that the protocol may progress provided liveness requirements are met. TODO: Fix the definition of Liveness #120 NU5/NU6 : The Zcash consensus protocol as of NU5 or NU6 (which do not differ significantly in terms of the base block chain layer). [1] Objective Validity : A validity property of a protocol history (such as a ledger) which can be computed purely from that history with no other context. Objective validity is needed to define consensus rules that will lead to the same protocol state being eventually agreed on by all nodes. Proof-of-Stake : A PoS protocol achieves consensus on transaction status by taking into account the weighting of staking tokens. PoS protocols exist under a large umbrella and may or may not provide assured finality or other properties this design requires of TFL . Proof‑of‑Work : A PoW protocol uses Nakamoto consensus pioneered by Bitcoin. The PoW subprotocol within PoW+TFL is a different consensus protocol from NU5/NU6 and encompasses more than narrow Nakamoto PoW consensus, including transaction semantics such as for shielded transfers. PoW+TFL : the overall complete, integrated consensus protocol specified in this book. Safety : The property of a distributed protocol that guarantees a participant may safely rely on a consistent local state, provided safety requirements are met. TODO: Provide a rigorous definition of Safety #121 simtfl : a protocol simulator for analyzing TFL security and abstract performance. Development lives at https://github.com/zcash/simtfl . See Status and Next Steps: Current Components for current status. Snap-and-Chat : A hybrid construction consensus protocol introduced in Ebb-and-Flow Protocols . TFL : The Trailing Finality Layer subprotocol within PoW+TFL. This is a new PoS subprotocol which provides assured finality for Zcash. Trailing Finality : A protocol property wherein transactions become final some time after first appearing in PoW blocks. ZIP : a Zcash Improvement Proposal is the protocol development process the Zcash community uses to safely define potential protocol improvements. See https://zips.z.cash . TODO: Clarify the distinctions between pure PoW, the PoW subprotocol, NU6, and fork-choice vs all of transaction semantics. #119","breadcrumbs":"Terminology » Terms","id":"37","title":"Terms"},"38":{"body":"If new consensus changes are deployed to Zcash mainnet prior to PoW+TFL design finalization, this design must be updated to refer to the new delta (e.g. by reanalyzing all changes against NU6 or NU7, etc…)","breadcrumbs":"Terminology » Footnotes","id":"38","title":"Footnotes"},"39":{"body":"This design augments the existing Zcash Proof‑of‑Work (PoW) network with a new consensus layer which provides trailing finality , called the Trailing Finality Layer (TFL) . This layer enables blocks produced via PoW to become final which ensures they may never be rolled back. This enables safer and simpler wallets and other infrastructure, and aids trust-minimized cross-chain bridges. This consensus layer uses a finalizing Proof-of-Stake (PoS) consensus protocol, and enables ZEC holders to earn protocol rewards for contributing to the security of the Zcash network. By integrating a PoS layer with the current PoW Zcash protocol, this design specifies a hybrid consensus protocol . The integration of the current PoW consensus with the TFL produces a new top-level consensus protocol referred to as PoW+TFL . In the following subchapters we introduce the Design at a Glance , then provide an overview of the major components of the design. Following this overview chapter, we proceed into a detailed Protocol Specification (TODO) .","breadcrumbs":"Design » Design Overview","id":"39","title":"Design Overview"},"4":{"body":"One question we've gotten in proposing this approach is why take a two-step process with an intermediate hybrid consensus protocol, rather than a single transition directly to a PoS protocol? Here’s how we think about those trade-offs:","breadcrumbs":"Introduction » A Path to Proof-of-Stake Zcash » Why Two Steps?","id":"4","title":"Why Two Steps?"},"40":{"body":"The PoW+TFL consensus protocol is logically an extension of the Zcash consensus rules to introduce trailing finality . This is achieved by compartmentalizing the top-level PoW+TFL protocol into two consensus subprotocols , one embodying most of the current consensus logic of Zcash and another the TFL. These protocols interact through a hybrid construction , which specifies how the protocols interact, and what changes from \"off-the-shelf\" behavior, if any, need to be imposed on the subprotocols. Each of these components (the two subprotocols and the hybrid construction) are somewhat modular: different subprotocols or hybrid constructions may be combined (with some modification) to produce a candidate PoW+TFL protocol. TODO: Add a protocol component diagram to \"Design at a Glance\" #122","breadcrumbs":"Design » Design at a Glance » Design at a Glance","id":"40","title":"Design at a Glance"},"41":{"body":"The hybrid construction is a major design component of the full consensus protocol which specifies how the subprotocols integrate. So far we have considered three candidates: The implied/loosely defined hybrid construction presented at Zcon4 . The Snap-and-Chat from the Ebb-and-Flow paper . The Crosslink construction. We believe Crosslink is the best candidate, due to its more rigorous specification and security analysis , and due to the issues with Snap-and-Chat described in Notes on Snap-and-Chat .","breadcrumbs":"Design » Design at a Glance » Hybrid Construction","id":"41","title":"Hybrid Construction"},"42":{"body":"The PoW+TFL hybrid consensus consists of two interacting subprotocols: PoW Subprotocol : this subprotocol is very similar to NU5 or NU6 consensus. It is a design goal of the TFL design to minimize changes to this subprotocol. Note: the shorthand \"PoW\" is potentially misleading, because this subprotocol is also responsible for the bulk of all supply and transaction semantic consensus rules. PoS Subprotocol : this is a new subprotocol which provides trailing finality via a finalizing PoS protocol. TODO: Clarify the distinctions between pure PoW, the PoW subprotocol, NU6, and fork-choice vs all of transaction semantics. #119 Note that the hybrid construction may require modification to the \"off-the-shelf\" versions of these subprotocols. In particular Crosslink requires each protocol to refer to the state of the other to provide objective validity .","breadcrumbs":"Design » Design at a Glance » Subprotocols","id":"42","title":"Subprotocols"},"43":{"body":"Here we strive to lay out our high level TFL design goals.","breadcrumbs":"Design » Design Goals » Design Goals","id":"43","title":"Design Goals"},"44":{"body":"Here we lay out ideal goals. As we develop a complete design, we are likely to inevitably encounter trade-offs some of which may preclude achieving the full idealized goals. Wherever possible, we motivate design decisions by these goals, and when goals are impacted by trade-offs we describe that impact and the rationale for the trade-off decision. For example, one ideal user experience goal below is to avoid disruption to existing wallets. However, the Crosslink construction may require wallets to alter their context of valid transactions differently from the current NU5/NU6 consensus protocol.","breadcrumbs":"Design » Design Goals » Goals, Design, and Trade-offs","id":"44","title":"Goals, Design, and Trade-offs"},"45":{"body":"We strive to start our protocol design process from user experience (UX) and use case considerations foremost, since at the end of the day all that matters in a protocol is what user needs it meets and how well. All currently supported wallet user experience should continue to operate seamlessly without change during or after protocol transitions. This covers the use of addresses, payment flow, transfers, ZEC supply cap and issuance rate, backup/restore, and other features users currently rely on. There must be no security or safety degradation due to wallet user behavior introduced by PoS transitions, assuming users follow their current behaviors unchanged and continue to use the same cognitive model of the impacts of their behaviors. This goal encompasses all of security and safety, including privacy and transparency or more explicit disclosures. The protocol should enable users of shielded mobile wallets to delegate ZEC to PoS consensus providers and earn a return on that ZEC coming via ZEC issuance or fees. Doing this may expose users to a risk of loss of delegated ZEC (such as through “slashing fees”). The protocol must guarantee that PoS consensus providers have no discretionary control over such delegated funds (including that they cannot steal those funds). For any hybrid PoW/PoS protocol (including the PoW+TFL protocol we’re proposing), the process and UX of mining remains unchanged except that the return on investment may be altered. This is true both of consensus-level block miners (i.e. mining pools and solo miners) and mining pool participants. Any hybrid PoW/PoS protocol (including PoW+TFL) block explorers will continue to function with the same UX through transitions as far as displaying information about transactions, the mempool, and blocks. Block explorers and other network metrics sites may require UX changes with respect to mining rewards and issuance calculations. Network metrics sites may require UX changes with respect to the p2p protocol or other network-specific information. Users can rely on assured finality with an expected time‑to‑finality below 30 minutes. [1]","breadcrumbs":"Design » Design Goals » User Experience and Use Case Goals","id":"45","title":"User Experience and Use Case Goals"},"46":{"body":"For a full PoS transition, ecosystem developers for products such as consensus nodes, wallets, mining services, chain analytics, and more will certainly need to update their code to support transitions. However, we carve out a few goals as an exception to this for this category of users: Wallet developers should not be required to make any changes through protocol transitions as long as they rely solely on the lightwalletd protocol or a full node API (such as the zcashd RPC interface). For any hybrid PoW/PoS protocol (including PoW+TFL), mining pools and miners should not be required to make any software or protocol changes as long as they rely on zcashd-compatible GetBlockTemplate. One exception to this is software that bakes in assumptions about the block reward schedule, rather than relying on GetBlockTemplate solely.","breadcrumbs":"Design » Design Goals » Developer Experience Goals","id":"46","title":"Developer Experience Goals"},"47":{"body":"Zcash has always had exemplary safety, security, and privacy, and we aim to continue that tradition: For any hybrid PoW/PoS protocol (including PoW+TFL), the cost‑of‑attack for a 1-hour rollback should not be reduced, given a “reasonably rigorous” security argument. For any hybrid PoW/PoS protocol (including PoW+TFL), the cost‑of‑attack to halt the chain should be larger than the 24‑hour revenue of PoW mining rewards, given a “reasonably rigorous” security argument. TODO: Define privacy goals of TFL #118 TODO: Define PoS Subprotocol desiderata which are distinct from Crosslink integration #117","breadcrumbs":"Design » Design Goals » Safety, Security, and Privacy Goals","id":"47","title":"Safety, Security, and Privacy Goals"},"48":{"body":"We want to follow some conservative design heuristics to minimize risk and mistakes: Rely as much as possible on design components that are already proven in production environments. Rely as much as possible on design components with adequate theoretical underpinnings and security analyses. Minimize changes or variations on the above: strive to only alter existing work when necessary for overall design goals. For example, Zcash’s privacy or issuance constraints are likely less common among existing PoS designs.","breadcrumbs":"Design » Design Goals » Design Conservatism Goals","id":"48","title":"Design Conservatism Goals"},"49":{"body":"These are not goals of the TFL design, either to simplify the scope of the initial design (a.k.a. Out-of-Scope Goals ), or because we believe some potential goal should not be supported (a.k.a. Anti-goals ).","breadcrumbs":"Design » Design Goals » Non-goals","id":"49","title":"Non-goals"},"5":{"body":"Pros We already understand the current PoW protocol well, and if we transition to an existing proven PoS protocol, then we could skip the complexity of an intermediate hybrid stage. The node implementation might be simpler. Explaining to people what is happening might be simpler. Something like “Zcash has been PoW since it launched, but on DATE (e.g. at block height X) it will switch to PoS.” Given that the issuance in a given time period is bounded by the supply curve, the full amount that was previously allocated to mining rewards becomes immediately available for staking rewards at the switch-over, rather than having to share this amount between mining and staking during the hybrid stage. Cons If there is any unforeseen show‑stopping problem in the new protocol or the transition process, we’d have to react to a network‑wide issue. It may be more likely to cause ecosystem disruption; unforeseen differences between PoW and PoS might cause various kinds of snags or papercuts throughout the ecosystem, and these would all pop up around the same time, which may lead to a loss of confidence/retention/adoption or at the very least inconvenience many users for some time. Losing miners: since the transition would be all at once, we may lose some number of miners, who are participants and users in the ecosystem. Miners may leave prior to the transition in order to take care of their own needs. If there is some showstopper in the transition, one possible short‑term mitigation would be to fall back on PoW which is well known, but if we’ve lost most miners, that may no longer be viable.","breadcrumbs":"Introduction » A Path to Proof-of-Stake Zcash » Considering Single Transition (vs Hybrid Multi-Step)","id":"5","title":"Considering Single Transition (vs Hybrid Multi-Step)"},"50":{"body":"While these desiderata may be common across the block‑chain consensus design space, they are not specific goals for the initial TFL design. Note that these may be goals for future protocol improvements. Prioritizing minimal time-to-finality over other considerations (such as protocol simplicity, impact on existing use cases, or other goals above). In-protocol liquid staking derivatives. Maximizing the PoS staked-voter count ceiling. For example, Tendermint BFT has a relatively low ceiling of ~hundreds of staked voters, whereas Ethereum’s Gasper supports hundreds of thousands of staked voters. Reducing energy usage. While this would presumably be a goal of a pure PoS transition, it likely cannot be achieved for hybrid PoW/PoS without loss of security.","breadcrumbs":"Design » Design Goals » Out-of-Scope Goals","id":"50","title":"Out-of-Scope Goals"},"51":{"body":"Distinctly from Out-of-Scope Goals we track \"anti-goals\" which are potential goals that we explicitly reject , which are potential goals we aim to not support even in future protocol improvements. We currently have no defined anti-goals.","breadcrumbs":"Design » Design Goals » Anti-Goals","id":"51","title":"Anti-Goals"},"52":{"body":"This requirement comes from a request from a DEX developer . While we have not yet surveyed DEX and Bridge designs, we're relying on this as a good starting point.","breadcrumbs":"Design » Design Goals » Footnotes","id":"52","title":"Footnotes"},"53":{"body":"Crosslink is the proposed hybrid construction for the Trailing Finality Layer. The current version is Crosslink 2.","breadcrumbs":"Design » Crosslink » Crosslink","id":"53","title":"Crosslink"},"54":{"body":"The Arguments for Bounded Availability and Finality Overrides Notes On Snap‑and‑Chat The Crosslink 2 Construction Security Analysis of Crosslink 2 Questions About Crosslink Potential Changes to Crosslink","breadcrumbs":"Design » Crosslink » Contents","id":"54","title":"Contents"},"55":{"body":"This document considers disadvantages of allowing transactions to continue to be included at the chain tip while the gap from the last finalized block becomes unbounded, and our perspective on what should be done instead. This condition is allowed by Ebb‑and‑Flow protocols [NTT2020] . We also argue that it is necessary to allow for the possibility of overriding finalization in order to respond to certain attacks, and that this should be explicitly modelled and subject to a well-defined governance process. This is a rewritten version of this forum post , adapting the main argument to take into account the discussion of “tail‑thrashing attacks” and finalization availability from the Addendum. More details of how bounded availability could be implemented in the context of a Snap‑and‑Chat protocol are in Notes on Snap‑and‑Chat . The proposed changes end up being significant enough to give our construction a new name: “ Crosslink ”, referring to the cross-links between blocks of the BFT and best-chain protocols. Crosslink has evolved somewhat, and now includes other changes not covered in either this document or Notes on Snap‑and‑Chat . The current version is called Crosslink 2 .","breadcrumbs":"Design » Crosslink » The Arguments for Bounded Availability and Finality Overrides » The Arguments for Bounded Availability and Finality Overrides","id":"55","title":"The Arguments for Bounded Availability and Finality Overrides"},"56":{"body":"“Ebb‑and‑Flow”, as described in [NTT2020] ( arXiv version ), is a security model for consensus protocols that provide two transaction logs, one with dynamic availability, and a prefix of it with finality. The paper proposes an instantiation of this security model called a “Snap‑and‑Chat” construction. It composes two consensus subprotocols, a BFT subprotocol and a best-chain subprotocol (it calls this the “longest chain protocol”). The above logs are obtained from the output of these subprotocols in a non-trivial way. This is claimed by the paper to “resolve” the tension between finality and dynamic availability. However, a necessary consequence is that in a situation where the “final” log stalls and the “available” log does not, the “finalization gap” between the finalization point and the chain tip can grow without bound. In particular, this means that transactions that spend funds can remain unfinalized for an arbitrary length of time. In this document, we argue that this is unacceptable, and that it is preferable to sacrifice strict dynamic availability. The main idea behind Ebb‑and‑Flow protocols is a good one, and allowing the chain tip to run ahead of the finalization point does make sense and has practical advantages. However, we also argue that it should not be possible to include transactions that spend funds in blocks that are too far ahead of the finalization point. Note Naive ways of preventing an unbounded finalization gap, such as stopping the chain completely in the case of a finalization stall, turn out to run into serious security problems — at least when the best-chain protocol uses Proof‑of‑Work. We’ll discuss those in detail in the section on tail‑thrashing attacks . Our proposed solution will be to require coinbase-only blocks during a long finalization stall. This solution has the advantage of not complicating the security analysis. We argue that losing strict dynamic availability in favour of “bounded availability” is preferable to the consequences of the unbounded finality gap, if/when a “long finalization stall” occurs. We also argue that it is beneficial to explicitly allow “finality overrides” under the control of a well-documented governance process. Such overrides allow long rollbacks that may be necessary in the case of an exploited security flaw. This is complementary to the argument for bounded availability, because the latter limits the period of user transactions that could be affected. The governance process can impose a limit on the length of this long rollback if desired.","breadcrumbs":"Design » Crosslink » The Arguments for Bounded Availability and Finality Overrides » Background","id":"56","title":"Background"},"57":{"body":"Since partition between nodes sufficient for finalization cannot be prevented, loosely speaking the CAP theorem implies that any consistent protocol (and therefore any protocol with finality) may stall for at least as long as the partition takes to heal. Background That “loosely speaking” is made precise by [LR2020] . Dynamic availability implies that the chain tip will continue to advance, and so the finalization gap increases without bound. Partition is not necessarily the only condition that could cause a finalization stall, it is just the one that most easily proves that this conclusion is impossible to avoid.","breadcrumbs":"Design » Crosslink » The Arguments for Bounded Availability and Finality Overrides » Finality + Dynamic availability + Possibility of Partition implies an unbounded finalization gap","id":"57","title":"Finality + Dynamic availability + Possibility of Partition implies an unbounded finalization gap"},"58":{"body":"Both the available protocol, and the subprotocol that provides finality, will be used in practice — otherwise, one or both of them might as well not exist. There is always a risk that blocks may be rolled back to the finalization point, by definition. Suppose, then, that there is a long finalization stall. The final and available protocols are not separate: there is no duplication of tokens between protocols, but the rules about how to determine best-effort balance and guaranteed balance depend on both protocols, how they are composed, and how the history after the finalization point is interpreted. Discussion The guaranteed minimum balance of a given party is not just the minimum of their balance at the finalization point and their balance at the current tip. It is the minimum balance taken over all possible transaction histories that extend the finalized chain — taking into account that a party’s previously published transactions might be able to be reapplied in a different context without its explicit consent. The extent to which published transactions can be reapplied depends on technical choices that we must make, subject to some constraints (for example, we know that shielded transactions cannot be reapplied after their anchors have been invalidated). It may be desirable to further constrain re-use in order to make guaranteed minimum balances easier to compute. As the finalization gap increases, the negative consequences of rolling back user transactions that spend funds increase. (Coinbase transactions do not spend funds; they are a special case that we will discuss later.) There are several possible —not mutually exclusive— outcomes: Users of the currency start to consider the available protocol increasingly unreliable. Users start to consider a rollback to be untenable, and lobby to prevent it or cry foul if it occurs. Users start to consider finalization increasingly irrelevant. Services that depend on finality become unavailable. There is no free lunch that would allow us to avoid availability problems for services that also depend on finality. Service providers adopt temporary workarounds that may not have had adequate security analysis. Any of these might precipitate a crisis of confidence, and there are reasons to think this effect might be worse than if the chain had switched to a “Stalled Mode” designed to prevent loss of user funds. Any such crisis may have a negative effect on token prices and long-term adoption. Note that adding finalization using an Ebb‑and‑Flow protocol does not by itself increase the probability of a rollback in the available chain, provided the PoW remains as secure against rollbacks of a given length as before. But that is a big proviso. We have a design constraint (motivated by limiting token devaluation and by governance issues) to limit issuance to be no greater than that of the original Zcash protocol up to a given height. Since some of the issuance is likely needed to reward staking, the amount of money available for mining rewards is reduced, which may reduce overall hash rate and security of the PoW. Independently, there may be a temptation for design decisions to rely on finality in a way that reduces security of PoW (“risk compensation”). There is also pressure to reduce the energy usage of PoW, which necessarily reduces the global hash rate, and therefore the cost of performing an attack that depends on the adversary having any given proportion of global hash rate. It could be argued that the issue of availability of services that depend on finality is mainly one of avoiding over-claiming about what is possible. Nevertheless there are also real usability issues if balances as seen by those services can differ significantly and for long periods from balances at the chain tip. Regardless, incorrect assumptions about the extent to which the finalized and available states can differ are likely to be exposed if there is a finalization stall. And those who made the assumptions may (quite reasonably!) not accept “everything is fine, those assumptions were always wrong” as a satisfactory response.","breadcrumbs":"Design » Crosslink » The Arguments for Bounded Availability and Finality Overrides » Problems with allowing spends in an unbounded finalization gap","id":"58","title":"Problems with allowing spends in an unbounded finalization gap"},"59":{"body":"An intuitive notion of “availability” for block‑chain protocols includes the ability to use the protocol as normal to spend funds. So, just to be clear, in a situation where that cannot happen we have lost availability, even if the block chain is advancing. Bounded availability is a weakening of dynamic availability [DKT2020] . It means that we intentionally sacrifice availability when some potentially hazardous operation —a “hazard” for short— would occur too far after the current finalization point. For now, assume for simplicity that our only hazard is spending funds. More generally, the notion of bounded availability can be applied to a wider range of protocols by tailoring the definition of “hazard” to the protocol. Background This talk by Soubhik Deb accompanying [DKT2020] provides a good explanation of the advantages of dynamic availability. We do not define bounded availability formally in this document, but informally, we aim to preserve the ability to securely adapt to large changes in hash rate or total stake.","breadcrumbs":"Design » Crosslink » The Arguments for Bounded Availability and Finality Overrides » What is Bounded Availability?","id":"59","title":"What is Bounded Availability?"},"6":{"body":"Note: TFL is one instance of a multi‑step approach. Pros We can hopefully be less disruptive across the ecosystem so that there are fewer snags and disruptions with each step. If there is a show-stopping flaw in any step, the fall‑back possibility seems more plausible. For example, if there is a show-stopper when transitioning from PoW to PoW+TFL , falling back to pure PoW seems more feasible, since both protocols rely on mainnet PoW infrastructure, so those participants will be present in either case. Retaining miners during a hybrid phase: while it is true that a hybrid protocol will lower miner revenue (since we aim to maintain the issuance schedule constraints), there is also more possibility and likelihood of keeping some of these users engaged. For example, they may begin participating in staking services (either as delegators or as infrastructure operators). If that is successful, then they’re also more likely to remain engaged in the subsequent transition to pure PoS. This general approach was demonstrated successfully by Ethereum, which is the largest or second largest cryptocurrency network for several important metrics (e.g. market cap, fees paid, user and developer activity, …). So we know this can be done well without disruptions. Cons The intermediate hybrid step will be a more novel and less well understood protocol. (It will necessarily be fairly different from Ethereum’s Beacon chain era.) Consensus nodes will be more complex, involving logic for both sub‑protocols as well as their integration. (Ideally this complexity can be modularized so that the nodes are easier to maintain and improve.) This may be more complicated to explain to current and potential new users. Something like “Zcash launched as PoW, and on $DATE (block height X) it will transition to a hybrid system, then later to a pure PoS system.” The available issuance must be shared between mining and staking rewards during the hybrid stage. The security of the PoW layer and of the PoS layer during this stage is partially dependent on the funds allocated to issuance for each protocol, and it is not yet clear to what extent splitting rewards would affect overall security.","breadcrumbs":"Introduction » A Path to Proof-of-Stake Zcash » Considering Hybrid Multi-Step approach (vs Single-Step):","id":"6","title":"Considering Hybrid Multi-Step approach (vs Single-Step):"},"60":{"body":"[NTT2020] calls the dynamically available block‑chain protocol Πlc​ that provides input to the rest of the contruction, the “longest chain” protocol. There are two reasons to avoid this terminology: In Bitcoin, Zcash, and most other PoW-based protocols, what is actually used by each node is not its longest observed chain, but its observed consensus-valid chain with most accumulated work. In Zcash this is called the node’s “ best valid block chain ”, which we shorten to “best chain”. As footnote 2 on page 3 of [NTT2020] says, that paper does not require Πlc​ to be a “longest chain” protocol anyway. Historical note The error in conflating the “longest chain” with the observed consensus-valid chain with most accumulated work, originates in the Bitcoin whitepaper. [ Nakamoto2008 , page 3] We will use the term “best‑chain protocol” instead. Note that this corresponds roughly to Πlc​ in the Snap‑and‑Chat construction, although the Crosslink 2 protocol that we propose will end up having other significant differences from Snap-and-Chat.","breadcrumbs":"Design » Crosslink » The Arguments for Bounded Availability and Finality Overrides » Terminology note","id":"60","title":"Terminology note"},"61":{"body":"We have not yet decided how to block hazards during a long finalization stall. We could do so directly, or by stopping block production in the more-available protocol. For reasons explained in the section on tail‑thrashing attacks below, it’s desirable not to stop block production. And so it’s consistent to have bounded availability together with another liveness property —which can be defined similarly to dynamic availability— that says the more-available protocol’s chain is still advancing. This is what we will aim for. We will call this method of blocking hazards, without stopping block production, “going into Stalled Mode”. Historical note This concept of Stalled Mode is very similar to a feature that was discussed early in the development of Zcash , but never fully designed or implemented. (After originally being called “Stalled Mode”, it was at some point renamed to “Emergency Mode”, but then the latter term was used for something else .) For Zcash, we propose that the main restriction of Stalled Mode should be to require coinbase-only blocks. This achieves a similar effect, for our purposes, as actually stalling the more-available protocol’s chain. Since funds cannot be spent in coinbase-only blocks, the vast majority of attacks that we are worried about would not be exploitable in this state. Info It is possible that a security flaw could affect coinbase transactions. We might want to turn off shielded coinbase for Stalled Mode blocks in order to reduce the chance of that. Also, mining rewards cannot be spent in a coinbase-only block; in particular, mining pools cannot distribute rewards. So there is a risk that an unscrupulous mining pool might try to do a rug-pull after mining of non-coinbase-only blocks resumes, if there were a very long finalization stall. But this approach works at least in the short term, and probably for long enough to allow manual intervention into the finalization protocol, or governance processes if needed. An analogy for the effect of this on availability that may be familiar to many people, is that it works like video streaming. All video streaming services use a buffer to paper over short-term interruptions or slow-downs of network access. In most cases, this buffer is bounded. This allows the video to be watched uninterrupted and at a constant rate in most circumstances. But if there is a longer-term network failure or insufficient sustained bandwidth, the playback will unavoidably stall. In our case, block production does not literally stall, but it’s the same as far as users’ ability to perform “hazardous” operations is concerned.","breadcrumbs":"Design » Crosslink » The Arguments for Bounded Availability and Finality Overrides » How to block hazards","id":"61","title":"How to block hazards"},"62":{"body":"So, why do we advocate this over: A protocol that only provides dynamic availability; A protocol that only provides finality; An unmodified Ebb‑and‑Flow protocol? The reason to reject option 1 is straightforward: finality is a valuable security property that is necessary for some use cases. If a protocol only provides finality (option 2), then short-term availability is directly tied to finalization. It may be possible to make finalization stalls sufficiently rare or short-lived that this is tolerable. But that is more likely to be possible if and when there is a well-established staking ecosystem. Before that ecosystem is established, the protocol may be particularly vulnerable to stalls. Furthermore, it’s difficult to get to such a protocol from a pure PoW system like current Zcash. We argued in the previous section that allowing hazards in an unbounded finalization gap is bad. Option 3 entails an unbounded finalization gap that will allow hazards. However, that isn’t sufficient to argue that bounded availability is better. Perhaps there are no good solutions! What are we gaining from a bounded availability approach that would justify the complexity of a hybrid protocol without obtaining strict dynamic availability? The argument goes like this: It is likely that a high proportion of the situations in which a sustained finalization stall happens will require human intervention. If the finality protocol were going to recover without intervention, there is no reason to think that it wouldn’t do so in a relatively short time. When human intervention is required, the fact that the chain tip is still proceeding apace (in protocols with strict dynamic availability) makes restarting the finality protocol harder, for many potential causes of a finalization stall. This may be less difficult when only “non-hazardous” transactions are present— in particular, when only coinbase transactions (which are subject to fairly strict rules in Zcash and other Bitcoin-derived chains) are present. This argument carries even more force when the protocol also allows “finality overrides”, as discussed later in the Complementarity section. Nothing about bounded availability prevents us from working hard to design a system that makes finalization stalls as infrequent and short-lived as possible, just as we would for any other option that provides finality. We want to optimistically minimize the finalization gap under good conditions, because this improves the usability of services that depend on finality. This argues against protocols that try to maintain a fixed gap, and motivates letting the gap vary. In practice, the likelihood of short finalization stalls is high enough that heuristically retaining availability in those situations is useful. The argument that it is difficult to completely prevent finalization stalls is supported by experience on Ethereum in May 2023 , when there were two stalls within 24 hours, one for about 25 minutes and one for about 64 minutes . This experience is consistent with our argument: Neither stall required short-term human intervention, and the network did in fact recover from them quickly. The stalls were caused by a resource exhaustion problem in the Prysm consensus client when handling attestations. It’s plausible to think that if this bug had been more serious, or possibly if Prysm clients had made up more of the network, then it would have required a hotfix release (and/or a significant proportion of nodes switching to another client) in order to resolve the stall. So this lines up with the hypothesis that longer stalls are likely to require manual intervention. A bounded availability protocol would very likely have resulted in either a shorter or no interruption in availability. If, say, the availability bound were set to be roughly an hour, then the first finalization stall would have been “papered over” and the second would have resulted in only a short loss of availability. Retaining short-term availability does not result in a risk compensation hazard: A finalization stall is still very visible, and directly affects applications relying on finality. Precisely because of the availability bound, it is obvious that it could affect all applications if it lasted long enough. A potential philosophical objection to lack of strict dynamic availability is that it creates a centralization risk to availability. That is, it becomes more likely that a coalition of validators can deliberately cause a denial of service. This objection may be more prevalent among people who would object to adding a finality layer or PoS at all.","breadcrumbs":"Design » Crosslink » The Arguments for Bounded Availability and Finality Overrides » Why is this better?","id":"62","title":"Why is this better?"},"63":{"body":"Consensus protocols sometimes fail. Potential causes of failure include: A design problem with the finality layer that causes a stall, or allows a stall to be provoked. A balance violation or spend authorization flaw that is being exploited or is sufficiently likely to be exploited. An implementation bug in a widely used node implementation that causes many nodes to diverge from consensus. In these situations, overriding finality may be better than any other alternative. An example is a balance violation flaw due to a 64-bit integer overflow that was exploited on Bitcoin mainnet on 15th August 2010. The response was to roll back the chain to before the exploitation, which is widely considered to have been the right decision. The time between the exploit (at block height 74638) and the forked chain overtaking the exploited chain (at block height 74691) was 53 blocks, or around 9 hours. Of course, Bitcoin used and still uses a pure‑PoW consensus. But the applicability of the example does not depend on that: the flaw was independent of the consensus mechanism. Another example of a situation that prompted this kind of override was the DAO recursion exploit on the Ethereum main chain in June 2016 . The response to this was the forced balance adjustment hard fork on 20th July 2016 commonly known as the DAO fork . Although this adjustment was not implemented as a rollback, and although Ethereum was using PoW at the time and did not make any formal finality guarantees, it did override transfers that would heuristically have been considered final at the fork height. Again, this flaw was independent of the consensus mechanism. The DAO fork was of course much more controversial than the Bitcoin fork, and a substantial minority of mining nodes split off to form Ethereum Classic. In any case, the point of this example is that it’s always possible to override finality in response to an exceptional situation, and that a chain’s community may decide to do so. The fact that Ethereum 2.0 now does claim a finality guarantee, would not in practice prevent a similar response in future that would override that guarantee. The question then is whether the procedure to override finality should be formalized or ad hoc. We argue that it should be formalized, including specifying the governance process to be used. This makes security analysis — of the consensus protocol per se, of the governance process, and of their interaction — much more feasible. Arguably a complete security analysis is not possible at all without it. It also front‑loads arguing about what procedure should be followed, and so it is more likely that stakeholders will agree to follow the process in any time‑critical incident.","breadcrumbs":"Design » Crosslink » The Arguments for Bounded Availability and Finality Overrides » Finality Overrides","id":"63","title":"Finality Overrides"},"64":{"body":"There is another possible way to model a protocol that claims finality but can be overridden in practice. We could say that the protocol after the override is a brand‑new protocol and chain (inheriting balances from the previous one, possibly modulo adjustments such as those that happened in the DAO fork). Although that would allow saying that the finality property has technically not been violated, it does not match how users think about an override situation. They are more likely to think of it as a protocol with finality that can be violated in exceptional cases — and they would reasonably want to know what those cases are and how they will be handled. It also does nothing to help with security analysis of such cases.","breadcrumbs":"Design » Crosslink » The Arguments for Bounded Availability and Finality Overrides » A way of modelling overrides that is insufficient","id":"64","title":"A way of modelling overrides that is insufficient"},"65":{"body":"Finality overrides and bounded availability are complementary in the following way: if a problem is urgent enough, then validators can be asked to stop validating. For genuinely harmful problems, it is likely to be in the interests of enough validators to stop that this causes a finalization stall. If this lasts longer than the availability bound then the protocol will go into Stalled Mode, giving time for the defined governance process to occur and decide what to do. And because the unfinalized consensus chain will contain only a limited period of user transactions that spend funds, the option of a long rollback remains realistically open. If, on the other hand, there is time pressure to make a governance decision about a rollback in order to reduce its length, that may result in a less well-considered decision. A possible objection is that there might be a coalition of validators who ignore the request to stop (possibly including the attacker or validators that an attacker can bribe), in which case the finalization stall would not happen. But that just means that we don’t gain the advantage of more time to make a governance decision; it isn’t actively a disadvantage relative to alternative designs. This outcome can also be thought of as a feature rather than a bug: going into Stalled Mode should be a last resort, and if the argument given for the request to stop failed to convince a sufficient number of validators that it was reason enough to do so, then perhaps it wasn’t a good enough reason. Rationale This resolves one of the main objections to the original Stalled Mode idea that stopped us from implementing it in Zcash. The original proposal was to use a signature with a key held by ECC to trigger Stalled Mode, which would arguably have been too centralized. The Stalled Mode described in this document, on the other hand, can only be entered by consensus of a larger validator set, or if there is an availability failure of the finalization protocol. It is also possible to make the argument that the threshold of stake needed is imposed by technical properties of the finality protocol and by the resources of the attacker, which might not be ideal for the purpose described above. However, we would argue that it does not need to be ideal, and will be in the right ballpark in practice. There’s a caveat related to doing intentional rollbacks when using the Stalled Mode approach, where block production in the more-available protocol continues during a long finalization stall. What happens to incentives of block producers (miners in the case of Proof‑of‑Work), given that they know the consensus chain might be intentionally rolled back? They might reasonably conclude that it is less valuable to produce those blocks, leading to a reduction of hash rate or other violations of the security assumptions of Πlc​. This is actually fairly easy to solve. We have the governance procedures say that if we do an intentional rollback, the coinbase-only mining rewards will be preserved. I.e. we produce a block or blocks that include those rewards paid to the same addresses (adjusting the consensus to allow them to be created from thin air if necessary), have everyone check it thoroughly, and require the chain to restart from that block. So as long as block producers believe that this governance procedure will be followed and that the chain will eventually recover at a reasonable coin price, they will still have incentive to produce on Πlc​, at least for a time. Rationale Although the community operating the governance procedures has already obtained the security benefit of mining done on the rolled-back chain by the time it creates the new chain, there is a strong incentive not to renege on the agreement with miners, because the same situation may happen again.","breadcrumbs":"Design » Crosslink » The Arguments for Bounded Availability and Finality Overrides » Complementarity","id":"65","title":"Complementarity"},"66":{"body":"Earlier we said that there were two possible approaches to preventing hazards during a long finalization stall: a) go into a Stalled Mode that directly disallows hazardous transactions (for example, by requiring Πlc​ blocks to be coinbase-only in Zcash); b) temporarily cause the more-available chain to stall. This section describes an important class of potential attacks on approach b) that are difficult to resolve. They are based on the fact that when the unfinalized chain stalls, an adversary has more time to find blocks, and this might violate security assumptions of the more-available protocol. For instance, if the more-available protocol is PoW-based, then its security in the steady state is predicated on the fact that an adversary with a given proportion of hash power has only a limited time to use that power, before the rest of the network finds another block. Background For an analysis of the concrete security of Nakamoto-like protocols, see [DKT+2020] and [GKR2020] . These papers confirm the intuition that the “private attack” —in which an adversary races privately against the rest of the network to construct a forking chain— is optimal, obtaining the same tight security bound independently using different techniques. During a chain stall, the adversary no longer has a limited time to construct a forking chain. If, say, the adversary has 10% hash power, then it can on average find a block in 10 block times. And so in 100 block times it can create a 10-block fork. It may in fact be worse than this: once miners know that a finalization stall is happening, their incentive to continue mining is reduced, since they know that there is a greater chance that their blocks might be rolled back. So we would expect the global hash rate to fall —even before the finality gap bound is hit— and then the adversary would have a greater proportion of hash rate. Info Even in a pure Ebb‑and‑Flow protocol, a finalization stall could cause miners to infer that their blocks are more likely to be rolled back, but the fact that the chain is continuing would make that more difficult to exploit. This issue with the global hash rate is mostly specific to the more-available protocol being PoW: if it were PoS, then its validators might as well continue proposing blocks because it is cheap to do so. There might be other attacks when the more-available protocol is PoS; we haven’t spent much time analyzing that case. The problem is that the more-available chain does not necessarily just halt during a chain stall. In fact, for a finality gap bound of k blocks, an adversary could cause the k-block “tail” of the chain as seen by any given node to “thrash” between different chains. We will call this a tail‑thrashing attack . If a protocol allowed such attacks then it would be a regression relative to the security we would normally expect from an otherwise similar PoW-based protocol. It only occurs during a finalization stall, but note that we cannot exclude the possibility of an adversary being able to provoke a finalization stall. Note that in the Snap‑and‑Chat construction, snapshots of Πlc​ are used as input to the BFT protocol. That implies that the tail‑thrashing problem could also affect the input to that protocol, which would be bad (not least for security analysis of availability, which seems somewhat intractable in that case). Also, when restarting Πlc​, we would need to take account of the fact that the adversary has had an arbitrary length of time to build long chains from every block that we could potentially restart from. It could be possible to invalidate those chains by requiring blocks after the restart to be dependent on fresh randomness, but that sounds quite tricky (especially given that we want to restart without manual intervention if possible), and there may be other attacks we haven’t thought of. This motivates using approach a) instead. Note that we have still glossed over precisely how consensus rules would change to enforce a). This will be covered later in The Crosslink 2 Construction , but first we will discuss other issues with Snap-and-Chat.","breadcrumbs":"Design » Crosslink » The Arguments for Bounded Availability and Finality Overrides » Tail-thrashing attacks","id":"66","title":"Tail-thrashing attacks"},"67":{"body":"The discussion in The Argument for Bounded Availability and Finality Overrides is at an abstract level, applying to any Ebb‑and‑Flow-like protocol. This document considers specifics of the Snap‑and‑Chat construction proposed in [NTT2020] ( arXiv version ). Terminology We are trying to be precise in this document about use of the terms “Ebb‑and‑Flow”, which is the security model and goal introduced in [NTT2020] , vs “Snap‑and‑Chat”, which is the construction proposed in the same paper to achieve that goal. There are other ways to design an Ebb‑and‑Flow protocol that don’t run into the difficulties described in this section (or that run into different difficulties).","breadcrumbs":"Design » Crosslink » Notes On Snap‑and‑Chat » Notes on Snap‑and‑Chat","id":"67","title":"Notes on Snap‑and‑Chat"},"68":{"body":"A general problem with the Snap‑and‑Chat construction is that it does not follow, from enforcement of the original consensus rules on blocks produced in Πlc​, that the properties they are intended to enforce hold for the LOGfin,it​ or LOGda,it​ ledgers. Less obviously, the converse also does not follow: enforcing unmodified consensus rules on Πlc​ blocks is both too lax and too strict. Recall from the paper how LOGfin,it​ and LOGda,it​ are constructed (starting at the end of page 8 of [NTT2020] ): Ledger extraction: Finally, how honest nodes compute LOGfin,it​ and LOGda,it​ from Chit​ and chit​ is illustrated in Figure 6. Recall that Chit​ is an ordering of snapshots, i.e., a chain of chains of LC blocks. First, Chit​ is flattened, i.e. the chains of blocks are concatenated as ordered to arrive at a single sequence of LC blocks. Then, all but the first occurrence of each block are removed (sanitized) to arrive at the finalized ledger LOGfin,it​ of LC blocks. To form the available ledger LOGda,it​, chit​, which is a sequence of LC blocks, is appended to LOGfin,it​ and the result again sanitized. This says that LOGfin,it​ and LOGda,it​ are sequences of transactions, not sequences of blocks. Therefore, consensus rules defined at the block level are not applicable. Zcash-specific Most of these rules are Proof‑of‑Work‑related checks that can be safely ignored at this level. Some are related to the hashBlockCommitments field intended for use by the FlyClient protocol. It is not at all clear how to make FlyClient (or other uses of this commitment) work with the Snap‑and‑Chat construction. In particular, the hashEarliest{Sapling,Orchard}Root, hashLatest{Sapling,Orchard}Root, and n{Sapling,Orchard}TxCount fields don’t make sense in this context since they could only reflect the values in chit​, which have no relation in general to those for any subrange of transactions in LOGda,it​. This problem occurs as a result of sanitization and so will be avoided by Crosslink 2, which does not need sanitization. Since LOGda,it​ does not have blocks, it is not well-defined whether it has “coinbase‑only blocks” when in Stalled Mode. That by itself is not so much of a problem because it would be sufficient for it to have only coinbase transactions in that mode.","breadcrumbs":"Design » Crosslink » Notes On Snap‑and‑Chat » Effect on consensus","id":"68","title":"Effect on consensus"},"69":{"body":"The issuance schedule of Zcash was designed under the assumption that blocks only come from a single chain, and that the difficulty adjustment algorithm keeps the rate of block mining roughly constant over time. For Snap‑and‑Chat, if there is a rollback longer than σ blocks in Πlc​, additional coinbase transactions from the rolled-back chain will be included in LOGfin​. We can argue that this will happen rarely enough not to cause any significant problem for the overall issuance schedule. However, it does mean that issuance is less predictable, because the block subsidies will be computed according to their depth in the Πlc​ chain on which they were mined. So it would no longer be the case that coinbase transactions issue a deterministic, non-increasing sequence of block subsidies. (Again, this problem will be avoided by Crosslink 2.)","breadcrumbs":"Design » Crosslink » Notes On Snap‑and‑Chat » Effect on issuance","id":"69","title":"Effect on issuance"},"7":{"body":"See Wikipedia’s Technology Tree - History section for details.","breadcrumbs":"Introduction » A Path to Proof-of-Stake Zcash » Footnotes","id":"7","title":"Footnotes"},"70":{"body":"The order of transactions in any particular chit​ is not in general preserved in either LOGfin​ or LOGda​. This is considered in the paper (middle of the left column on page 10) but it is very easy to miss it: Thus, snapshots taken by different nodes or at different times can conflict. However, Πbft​ is still safe and thus orders these snapshots linearly. Any transactions invalidated by conflicts are sanitized during ledger extraction. That is, a transaction from one snapshot might double-spend an output already spent in a different transaction of a different snaphot earlier in the flattening order. If it is omitted, then later transactions could depend on the outputs of the omitted one. The paper is saying that each transaction is only included if (in Bitcoin and Zcash terminology) it satisfies contextual checks for double-spending and existence of inputs at the point in the ledger where it would be added. Since nullifiers for shielded spends are public, it is possible to do this even for shielded transactions. Each node i will construct commitment trees in the order given by LOGda,it​. Info This means that if LOGfin,i​ is extended by a block that is not the next block in LOGda,i​ after the finalization point (and that has different note commitments), then all shielded transactions from that point onward in the previous LOGda,i​ will be invalidated. It could be possible to do better at the expense of a more complicated note commitment tree structure. In any case, this situation is expected to be rare, because it can only occur if there is a rollback of more than σ blocks in the Πlc​ consensus chain or a failure of BFT safety.","breadcrumbs":"Design » Crosslink » Notes On Snap‑and‑Chat » Effect on transaction ordering","id":"70","title":"Effect on transaction ordering"},"71":{"body":"There are two possible ways to interpret how LOG{fin,da}​ are constructed in Snap‑and‑Chat: Concatenate the transactions from each final BFT block snapshot of an LC chain, and sanitize the resulting transaction sequence by including each transaction iff it is contextually valid. Concatenate the blocks from each final BFT block snapshot of an LC chain, remove duplicate blocks , and only then sanitize the resulting transaction sequence by including each transaction iff it is contextually valid. These are equivalent in the setting considered by [NTT2020] , but the argument for their equivalence is not obvious. We definitely want them to be equivalent: in practice there will be many duplicate blocks from chain prefixes in the input to sanitization, and so a literal implementation of the first variant would have to recheck all duplicate transactions for contextual validity. That would have at least O(n2) complexity (more likely O(n2logn)) in the length n of the block chain, because the length of each final snapshot grows with n. Suppose that, in a particular Πlc​, the only reasons for a transaction to be contextually invalid are double-spends and missing inputs. In that case the argument for equivalence is: If a transaction is omitted due to a double-spend, then any subsequent time it is checked, that input will still have been double-spent. If a transaction is omitted due to a missing input, this can only be because an earlier transaction in the input to sanitization was omitted. So the structure of omitted transactions forms a DAG in which parent links must be to earlier omitted transactions. The roots of the DAG are at double-spending transactions, which cannot be reinstated. A child cannot be reinstated until its parents have been reinstated. Therefore, no transactions are reinstated. Note that any other reason for transactions to be contextually invalid might interfere with this argument. Therefore, strictly speaking Snap‑and‑Chat should require of Πlc​ that there is no such other reason. This does not seem to be explicitly stated anywhere in [NTT2020] . Zcash-specific In Zcash a transaction can also be contextually invalid because it has expired, or because it has a missing anchor. Expiry can be handled by extending the above argument as follows: If a transaction is omitted due to having expired, then any subsequent time it is checked, it will still be expired. It is not obvious how to extend it to handle missing anchors, because it is technically possible for a duplicate transaction that was invalid because of a missing anchor to become valid in a subsequent block. That situation would require careful manipulation of the commitment trees, but there does not seem to be anything preventing it from being provoked intentionally. The argument that was used above for missing inputs does not work here, because there is no corresponding DAG formed by the transactions with missing anchors: the same commitment treestate can be produced by two unrelated transactions.","breadcrumbs":"Design » Crosslink » Notes On Snap‑and‑Chat » Subtlety in the definition of sanitization","id":"71","title":"Subtlety in the definition of sanitization"},"72":{"body":"Transactions in chit​ need to be able to spend outputs that are not necessarily from any previous transaction in chit​. This is because, from the point of view of a user of node i at time t, the block chain includes all transactions in LOGda,it​. All of the transactions after the finalization point are guaranteed to also be in chit​, but the ones before the finalization point (i.e. in LOGfin,it​) are not, because they could be from some other chju​ for u≤t and j=i (intuitively, from some long chain fork that was once considered confirmed by enough nodes). Info Honest nodes only ever vote for confirmed snapshots, that is, prefixes of their best Πlc​ chain truncated by the confirmation depth σ. Obviously the whole point of having the BFT protocol is that chain forks longer than σ can occur in Πlc​ — otherwise we'd just use Πlc​ and have done. So it is not that we expect this case to be common, but if it happens then it will never fix itself: the consensus chain in Πlc​ will continue on without ever including the transactions from LOGfint​ that were obtained from a snapshot of another fork. A user must be able to spend outputs for which they hold the spending key from any finalized transaction, otherwise there would be no point to the finalization. The authors of [NTT2020] probably just missed this: the paper only has evidence that they simulated their construction, rather than implementing it for Bitcoin or any other concrete block chain as Πlc​. Let’s try to repair it. Suppose that node j is trying to determine whether chit​ is a consensus-valid chain, which is necessary for deterministic consensus in Πlc​. It cannot decide whether to allow transactions in chit​ to spend outputs not in the history of chit​ on the basis of its own finalized view LOGfin,jt​, because LOGfin,it​ and LOGfin,jt​ are not in general the same. Of course, we hope that LOGfin,it​ and LOGfin,jt​ are consistent, i.e. one is a prefix of the other. But even if they are consistent, they are not necessarily the same length. In particular, if LOGfin,jt​ is shorter than LOGfin,it​, then node j does not have enough information to fill in the gap — and so it may incorrectly view a transaction in chit​ as spending an output that does not exist, when actually it does exist in LOGfin,it​∖LOGfin,jt​. Conversely if LOGfin,jt​ were longer and node j were to allow spending an output in LOGfin,jt​∖LOGfin,it​, that would be using information that is not necessarily available to other nodes, and so node j could diverge from consensus. Consensus validity of the block at the tip of chit​ can only be a deterministic function of the block itself and its ancestors in chit​. It is crucial to be able to eventually spend outputs from the finalized chain. We are forced to conclude that the chain chit​ must include the information needed to calculate LOGfin,is​ for some s not too far behind t. That is, Πlc​ must be modified to ensure that this is the case. This leads us to strengthen the required properties of an Ebb‑and‑Flow protocol to include another property, “finalization availability”.","breadcrumbs":"Design » Crosslink » Notes On Snap‑and‑Chat » Spending finalized outputs","id":"72","title":"Spending finalized outputs"},"73":{"body":"In the absence of security flaws and under the security assumptions required by the finality layer, the finalization point will not be seen by any honest node to roll back. However, that does not imply that all nodes will see the same finalized height — which is impossible given network delays and unreliable messaging. Both in order to optimize the availability of applications that require finality, and in order to solve the technical issue of spending finalized outputs described in the previous section, we need to consider availability of the information needed to finalize the chain up to a particular point. Note that in Bitcoin-like consensus protocols, we don’t generally consider it to be an availability flaw that a block header only commits to the previous block hash and to the Merkle tree of transactions in the block, rather than including them directly. These commitments allow nodes to check that they have the correct information, which can then be requested separately. Suppose, then, that each block header in Πlc​ commits to the L ast F inal BFT block known by the Πlc​ block producer. For an LC block chain with block H at its tip, we will refer to this commitment as LF(H). We refer to the parent block of H as H⌈bc1​ (this is a special case of a notation that will be defined in The Crosslink 2 Construction ). Consensus rule: Extension We require, as a consensus rule, that if H is not the genesis block header, then this BFT block either descends from or is the same as the final BFT block committed to by the Πlc​ block’s parent. That is, LF(H⌈bc1​)⪯bft​LF(H). This Extension rule will be preserved into Crosslink 2. The Extension rule does not prevent the BFT chain from rolling back, if the security assumptions of Πbft​ were violated. However, it means that if a node i does not observe a rollback in Πlc​ at confirmation depth σ, then it will also not observe any instability in LOGfin,i​, even if the security assumptions of Πbft​ are violated. This property holds by construction , and in fact regardless of Πbft​. Info In the Snap‑and‑Chat construction, we also have BFT block proposals committing to Πlc​ snapshots (top of right column of [ NTT2020 , page 7]): In addition, chit​ is used as side information in Πbft​ to boycott the finalization of invalid snapshots proposed by the adversary. This does not cause any circularity, because each protocol only commits to earlier blocks of the other. In fact, BFT validators have to listen to transmission of Πlc​ block headers anyway, so that could be also the protocol over which they get the information needed to make and broadcast their own signatures or proposals. (A possible reason not to broadcast individual signatures to all nodes is that with large numbers of validators, the proof that a sufficient proportion of validators/stake has signed can use an aggregate signature, which could be much smaller. Also, Πlc​ nodes only need to know about successful BFT block proposals.) Now suppose that, in a Snap‑and‑Chat protocol, the BFT consensus finalizes a Πlc​ snapshot that does not extend the snapshot in the previous block (which can happen if either Πbft​ is unsafe, or Πlc​ suffers a rollback longer than σ blocks). In that case we will initially not be able to spend outputs from the old snapshot in the new chain. But eventually for some node i that sees the header H at the tip of its best chain at time t, LF(H) will be such that from then on (i.e. at time u≥t), LOGfin,iu​ includes the output that we want to spend. This assumes liveness of Πlc​ and safety of Πbft​. That is, including a reference to a recent final BFT block in Πlc​ block headers both incentivizes nodes to propagate this information, and can be used to solve the “spending finalized outputs” problem. Optionally, we could incentivize the block producer to include the latest information it has, for example by burning part of the block reward or by giving the producer some limited mining advantage that depends on how many Πlc​ blocks back the finalization information is. This raises the question of how we measure how far ahead a given block is relative to the finalization information it provides. As we said before, LOGfin,it​ is a sequence of transactions, not blocks. The transactions will in general be in a different order, and also some transactions from chit​ may have been omitted from LOGfin,it​ (and even LOGda,it​) because they were not contextually valid. Info In Crosslink 2 , we will sidestep this problem by avoiding the need for sanitization — that is, LOGfin,it​ will correspond exactly to a chain of blocks that is a prefix of chit​. Actually we use the notation finit​ to reflect the fact that it is a bc‑chain, not a sequence of bc‑transactions. This invariant is maintained statefully on each node i: any rollback past finit​ will be ignored. If a new finit​ would conflict with the old one, the node will refuse to use it. This allows each node to straightforwardly measure how many blocks chit​ is ahead of finit​ as the difference in heights. Since this document is intended to explain the development of Crosslink from Snap‑and‑Chat, here we describe the more complicated approach that we originally came up with for Crosslink 1 — which also serves to motivate the simplification in Crosslink 2. Assume that a block unambiguously specifies its ancestor chain. For a block H, define:tailhead(H):=last-common-ancestor(snapshot(LF(H)),H)finality-depth(H):=height(H)−height(tailhead(H)) Here LF(H) is the BFT block we are providing information for, and snapshot(LF(H)) is the corresponding Πlc​ snapshot. For a node i that sees LF(H) as the most recent final BFT block at time t, LOGfin,it​ will definitely contain transactions from blocks up to tailhead(H), but usually will not contain subsequent transactions on H’s fork. Info Strictly speaking, it is possible that a previous BFT block took a snapshot H′ that is between tailhead(H) and H. This can only happen if there have been at least two rollbacks longer than σ blocks (i.e. we went more than σ blocks down H’s fork from tailhead(H), then reorged to more than σ blocks down snapshot(LF(H))’s fork, then reorged again to H’s fork). In that case, the finalized ledger would already have the non-conflicting transactions from blocks between tailhead(H) and H′ — and it could be argued that the correct definition of finality depth in such cases is the depth of H′ relative to H, not of tailhead(H) relative to H. However, The definition above is simpler and easier to compute. The effect of overestimating the finality depth in such corner cases would only cause us to enforce Stalled Mode slightly sooner, which seems fine (and even desirable) in any situation where there have been at least two rollbacks longer than σ blocks. By the way, the “tailhead” of a tailed animal is the area where the posterior of the tail joins the rump (also called the “dock” in some animals). We could alternatively just rely on the fact that some proportion of block producers are honest and will include the latest information they have. However, it turns out that having a definition of finality depth will also be useful to enforce going into Stalled Mode. Specifically, if we accept the above definition of finality depth, then the security property we want is Definition Bounded hazard-freeness for a finality gap bound of L blocks: There is never, for any node i at time t, observed to be a more-available ledger LOGda,it​ with a hazardous transaction that comes from block H of chit​ such that finality-depth(H)>L. Info This assumes that transactions in the non-finalized suffix LOGda,it​∖LOGfin,it​ come from blocks in chit​. In Snap‑and‑Chat they do by definition, but ideally we wouldn’t depend on that. The difficulty in finding a more general security definition is due to the ledgers in an Ebb‑and‑Flow protocol being specified as sequences of transactions, so that a depth in the ledger would have only a very indirect correspondence to time. We could instead base a definition on timestamps, but that could run into difficulties in ensuring timestamp accuracy. Another possibility would be to count the number of coinbase transactions in LOGda,it​∖LOGfin,it​ before the hazardous transaction. This would still be somewhat ad hoc (it depends on the fact that coinbase transactions happen once per block and cannot conflict with any other distinct transaction). In any case, if finality-depth sometimes overestimates the depth, that cannot weaken this security definition. Note that a node that is validating a chain chit​ must fetch all the chains referenced by BFT blocks reachable from it (back to an ancestor that it has seen before). In theory, there could be a partition that causes there to be multiple disjoint snapshots that get added to the BFT chain in quick succession. However, in practice we expect such long rollbacks to be rare if Πlc​ is meeting its security goals. Going into Stalled Mode if there is a long finalization stall helps to reduce the cost of validation when the stall resolves. That is, if there is a partition and nodes build on several long chains, then in unmodified Snap‑and‑Chat, it could be necessary to validate an arbitrary number of transactions on each chain when the stall resolves. Having only coinbase transactions after a certain point in each chain would significantly reduce the concrete validation costs in this situation. Nodes should not simply trust that the BFT blocks are correct; they should check validator signatures (or aggregate signatures) and finalization rules. Similarly, Πlc​ snapshots should not be trusted just because they are referenced by BFT blocks; they should be fully validated, including the proofs-of-work. It is also possible for a snapshot reference to include the subsequent σ block headers, which are guaranteed to be available for a confirmed snapshot. Having all nodes validate the proofs-of-work in these headers is likely to significantly increase the work that an attacker would need to perform to cause disruption under a partial failure of either Πbft​ or Πlc​’s security properties. Info Note that [NTT2020] (bottom of right column, page 9) makes a safety assumption about Πlc​ in order to prove the consistency of LOGfin​ with the output of Πlc​: As indicated by Algorithm 1, a snapshot of the output of Πlc​ becomes final as part of a BFT block only if that snapshot is seen as confirmed by at least one honest node. However, since Πlc​ is safe [i.e., does not roll back further than the confirmation depth σ], the fact that one honest node sees that snapshot as confirmed implies that every honest node sees the same snapshot as confirmed. We claim that, while this may be a reasonable assumption to make for parts of the security analysis, in practice we should always require any adversary to do the relevant amount of Proof‑of‑Work to construct block headers that are plausibly confirmed. This is useful even though we cannot require, for every possible attack, that it had those headers at the time they should originally have appeared.","breadcrumbs":"Design » Crosslink » Notes On Snap‑and‑Chat » Finalization Availability","id":"73","title":"Finalization Availability"},"74":{"body":"The following idea for enforcing finalization availability and a bound on the finality gap was originally conceived before we had switched to advocating the Stalled Mode approach. It’s simpler to explain first in that variant. Suppose that for an L-block availability bound, we required each block header to include the information necessary for a node to finalize to L blocks back. This would automatically enforce a chain stall after the availability bound without any further explicit check, because it would be impossible to produce a block after the bound. Note that if full nodes have access to the BFT chain, knowing LF(H) is sufficient to tell whether the correct version of any given BFT block in LF(H)’s ancestor chain has been obtained. Suppose that the finality gap bound is L blocks. Having already defined finality-depth, the necessary Πlc​ consensus rule is attractively simple: Consensus rule For every Πlc​ block H, finality-depth(H)≤L. To adapt this approach to enforce Stalled Mode instead of stalling the chain, we can allow the alternative of producing a block that follows the Stalled Mode restrictions: Consensus rule For every Πlc​ block H, either finality-depth(H)≤L, or H follows the Stalled Mode restrictions. Note that Stalled Mode will be exited automatically as soon as the finalization point catches up to within L blocks (if it does without an intentional rollback). Typically, after recovery from whatever was causing the finalization stall, the validators will be able to obtain consensus on the same chain as LOGda​, and so there will be no rollback (or at least not a long one) of LOGda​. Info An earlier iteration of this idea required the finalization information to be included in Πlc​ block headers. This is not necessary when we assume that full nodes have access to the BFT chain and can obtain arbitrary BFT blocks. This also sidesteps any need to relax the rule in order to bound the size of Πlc​ block headers. Πlc​ block producers are still incentivized to make the relevant BFT blocks available, because without them the above consensus rule cannot be checked, and so their Πlc​ blocks would not be accepted. There is, however, a potential denial-of-service attack by claiming the existence of a BFT block that is very far ahead of the actual BFT chain tip. This attack is not very serious as long as nodes limit the number of BFT blocks they will attempt to obtain in parallel before having checked validator signatures.","breadcrumbs":"Design » Crosslink » Notes On Snap‑and‑Chat » Enforcing Finalization Availability and Stalled Mode","id":"74","title":"Enforcing Finalization Availability and Stalled Mode"},"75":{"body":"Consider Lemma 5: Moreover, for a BFT block to become final in the view of an honest node i under (A2∗​,Z2​), at least one vote from an honest node is required, and honest nodes only vote for a BFT block if they view the referenced LC block as confirmed. The stated assumptions are: (A2​(β),Z2​) formalizes the model of P2 , a synchronous network under dynamic participation, with respect to a bound β on the fraction of awake nodes that are adversarial: At all times, A2​ is required to deliver all messages sent between honest nodes in at most Δ slots. At all times, A2​ determines which honest nodes are awake/asleep and when, subject to the constraint that at all times at most fraction β of awake nodes are adversarial and at least one honest node is awake. (A2∗​,Z2​) is defined as (A2​(21​,Z2​)). Now consider this statement and figure: Even if Πbft​ is unsafe (Figure 9c), finalization of a snapshot requires at least one honest vote, and thus only valid snapshots become finalized. Figure 9 of [NTT2020] This argument is technically correct but has to be interpreted with care. It only applies when the number of malicious nodes f is such that n/3<f<n/2. What we are trying to do with Crosslink is to ensure that a similar conclusion holds even if Πbft​ is completely subverted, i.e. the adversary has 100% of validators (but only < 50% of Πlc​ hash rate).","breadcrumbs":"Design » Crosslink » Notes On Snap‑and‑Chat » Comment on security assumptions","id":"75","title":"Comment on security assumptions"},"76":{"body":"We are now ready to give a description of a protocol that takes into account the issues described in Notes on Snap‑and‑Chat , and that implements bounded availability . We call this the “Crosslink” construction; more precisely the version described here is “Crosslink 2”. This description will attempt to be self-contained, but [NTT2020] ( arXiv version ) is useful background on the general model of Ebb-and-Flow protocols.","breadcrumbs":"Design » Crosslink » The Crosslink 2 Construction » The Crosslink 2 Construction","id":"76","title":"The Crosslink 2 Construction"},"77":{"body":"“∗” is a metavariable for the name of a protocol. We also use it as a wildcard in protocol names of a particular type, for example “∗bc” for the name of some best‑chain protocol. Protocols are referred to as Π∗​ for a name “∗”. Where it is useful to avoid ambiguity, when referring to a concept defined by Π∗​ we prefix it with “∗‑”. We do not take synchrony or partial synchrony as an implicit assumption of the communication model; that is, unless otherwise specified , messages between protocol participants can be arbitrarily delayed or dropped. A given message is received at most once, and messages are nonmalleably authenticated as originating from a given sender whenever needed by the applicable protocol. Particular subprotocols may require a stronger model. Background For an overview of communication models used to analyze distributed protocols, see this blog post by Ittai Abraham . Discussion of incorrect applications of the GST formalization of partial synchrony to continuously operating protocols. The original context for the definition of the partially synchronous model in [DLS1988] was for “one‑shot” Byzantine Agreement — called “the consensus problem” in that paper. The following argument is used to justify assuming that all messages from the Global Stabilization Time onward are delivered within the upper time bound Δ: Therefore, we impose an additional constraint: For each execution there is a global stabilization time (GST), unknown to the processors, such that the message system respects the upper bound Δ from time GST onward. This constraint might at first seem too strong: In realistic situations, the upper bound cannot reasonably be expected to hold forever after GST, but perhaps only for a limited time. However, any good solution to the consensus problem in this model would have an upper bound L on the amount of time after GST required for consensus to be reached; in this case it is not really necessary that the bound Δ hold forever after time GST, but only up to time GST +L. We find it technically convenient to avoid explicit mention of the interval length L in the model, but will instead present the appropriate upper bounds on time for each of our algorithms. Several subsequent authors applying the partially synchronous model to block chains appear to have forgotten or neglected this context. In particular, the argument depends on the protocol completing soon after GST. Obviously a block‑chain protocol does not satisfy this assumption; it is not a “one‑shot” consensus problem. This assumption could be removed, but some authors of papers about block‑chain protocols have taken it to be an essential aspect of modelling partial synchrony. I believe this is contrary to the intent of [DLS1988] : Instead of requiring that the consensus problem be solvable in the GST model, we might think of separating the correctness conditions into safety and termination properties. The safety conditions are that no two correct processors should ever reach disagreement, and that no correct processor should ever make a decision that is contrary to the specified validity conditions. The termination property is just that each correct processor should eventually make a decision. Then we might require an algorithm to satisfy the safety conditions no matter how asynchronously the message system behaves, that is, even if Δ does not hold eventually. On the other hand, we might only require termination in case Δ holds eventually. It is easy to see that these safety and termination conditions are [for the consensus problem] equivalent to our GST condition: If an algorithm solves the consensus problem when Δ holds from time GST onward, then that algorithm cannot possibly violate a safety property even if the message system is completely asynchronous. This is because safety violations must occur at some finite point in time, and there would be some continuation of the violating execution in which Δ eventually holds. This argument is correct as stated, i.e. for the one‑shot consensus problem. Subtly, essentially the same argument can be adapted to protocols with safety properties that need to be satisfied continuously. However, it cannot correctly be applied to liveness properties of non‑terminating protocols. The authors (Cynthia Dwork, Nancy Lynch, and Larry Stockmeyer) would certainly have known this: notice how they carefully distinguish “the GST model” from “partial synchrony”. They cannot plausibly have intended this GST formalization to be applied unmodified to analyze liveness in such protocols, which seems to be common in the block‑chain literature, including in the Ebb-and-Flow paper [NTT2020] and the Streamlet paper [CS2020] . The Ebb-and-Flow paper acknowledges the issue by saying “Although in reality, multiple such periods of (a‑)synchrony could alternate, we follow the long‑standing practice in the BFT literature and study only a single such transition.” This is not adequate: “long‑standing practice” notwithstanding, it is not valid in general to infer that properties holding for the first transition to synchrony also apply to subsequent transitions (where the protocol can be in states that would not occur initially), and it is plausible that this inference could fail for real protocols. The Streamlet paper also refers to “periods of synchrony” which indicates awareness of the issue, but then it uses the unmodified GST model in the proofs. Informally, to solve this issue it is necessary to also prove that existing progress is maintained during periods of asynchrony, and that during such periods the protocol remains in states where it will be able to take advantage of a future period of synchrony to make further progress. This provides further motivation to avoid taking the GST formalization of partial synchrony as a basic assumption. Note that the recent result [CGSW2024] does not contradict anything we say here. Although the GST and Unknown Latency models are “equally demanding” in the sense of existence of protocols that satisfy a given goal, this result does not show that the models are equivalent for any specific protocol. In particular the requirements of the “clock‑slowing” technique fail in practice for any protocol involving Proof‑of‑Work. A ∗‑execution is the complete set of events (message sends/receives and decisions by protocol participants) that occur in a particular run of Π∗​ from its initiation up to a given time. A prefix of a ∗‑execution is also a ∗‑execution. Since executions always start from protocol initiation, a strict suffix of a ∗‑execution is not a ∗‑execution. Times are modelled as values of a totally ordered type Time with minimum value 0. For convenience, we consider all protocol executions to start at time 0. Remark Although protocols may be nondeterministic, an execution fixes the events that occur and times at which they occur, for the purpose of modeling. For simplicity, we assume that all events occur at global times in a total ordering. This assumption is not realistic in an asynchronous communication model, but it is not essential to the design or analysis and could be removed: we could use a partial happens-before ordering on events in place of a total ordering on times. A “∗‑node” is a participant in Π∗​ (the protocol may be implicit). A ∗‑node is “honest at time t” in a given execution iff it has followed the protocol up to and including time t in that execution. A time series on type U is a function S⦂Time→U assigning a value of U to each time in an execution. By convention, we will write the time t as a superscript: St=S(t). A ∗‑chain is a nonempty sequence of ∗‑blocks, starting at the “genesis block” O∗​, in which each subsequent block refers to its preceding or “parent block” by a collision‑resistant hash. The “tip” of a ∗‑chain is its last element. For convenience, we conflate ∗‑blocks with ∗‑chains; that is, we identify a chain with the block at its tip. This is justified because, assuming that the hash function used for parent links is collision‑resistant, there is exactly one ∗‑chain corresponding to a ∗‑block; and conversely there is exactly one ∗‑block at the tip of a ∗‑chain. If C is a ∗‑chain, C⌈∗k​ means C with the last k blocks pruned, except that if len(C)≤k, the result is the genesis ∗‑chain consisting only of O∗​. The block at depth k∈N in a ∗‑chain C is defined to be the tip of C⌈∗k​. Thus the block at depth k in a chain is the last one that cannot be affected by a rollback of length k (this also applies when len(C)≤k because the genesis ∗‑chain cannot roll back). Terminology Our usage of “depth” is different from [NTT2020] , which uses “depth” to refer to what Bitcoin and Zcash call “height”. It also differs by 1 from the convention for confirmation depths in zcashd, where the tip is considered to be at depth 1, rather than 0. For ∗‑blocks B and C: The notation B⪯∗​C means that the ∗‑chain with tip B is a prefix of the one with tip C. This includes the case B=C. The notation B⪯⪰∗​C means that either B⪯∗​C or C⪯∗​B. That is, “one of B and C is a prefix of the other”. This also includes the case B=C. The notation B/⪯⪰∗​C means that both B⪯∗​C and C⪯∗​B. That is, “neither of B and C is a prefix of the other”. A function S⦂I→∗-block is ∗‑linear iff for every t,u⦂I where t≤u we have S(t)⪯∗​S(u). (This definition can be applied to time series where I=Time, or to sequences of ∗‑blocks where values of I are indices.) Lemma: Linear prefix If A⪯∗​C and B⪯∗​C then A⪯⪰∗​B. Proof: The chain of ancestors of C is ∗-linear, and A, B are both on that chain. The notation [f(X) for X⪯∗​Y] means the sequence of f(X) for each ∗‑block X in chain order from genesis up to and including Y. (X is a bound variable within this construct.) TODO: remove this if not used: We use log⪯log′ (without a subscript on ⪯) to mean that the transaction ledger log is a prefix of log′. Similarly to ⪯⪰∗​ above, log⪯⪰log′ means that either log⪯log′ or log′⪯log; that is, “one of log and log′ is a prefix of the other”.","breadcrumbs":"Design » Crosslink » The Crosslink 2 Construction » Conventions","id":"77","title":"Conventions"},"78":{"body":"In the simplest case, a block‑chain protocol Π∗​ provides a single “view” that, for a given ∗‑execution, provides each ∗‑node with a time series on ∗‑chains. More generally a protocol may define several “views” that provide each ∗‑node with time series on potentially different chain types. We model a ∗‑view as a function V⦂Node×Time→∗-chain. By convention, we will write the node index i as a subscript and the time t as a superscript: Vit​=V(i,t). Definition: Agreement on a view An execution of Π has Agreement on the view V⦂Node×Time→∗-chain iff for all times t, u and all Π nodes i, j (potentially the same) such that i is honest at time t and j is honest at time u, we have Vit​⪯⪰∗​Vju​.","breadcrumbs":"Design » Crosslink » The Crosslink 2 Construction » Views","id":"78","title":"Views"},"79":{"body":"As in Snap‑and‑Chat, we depend on a BFT protocol Πorigbft​, and a best‑chain protocol Πorigbc​. Info See this terminology note for why we do not call Πorigbc​ a “longest‑chain” protocol. We modify Πorigbft​ (resp. Πorigbc​) to give Πbft​ (resp. Πbc​) by adding structural elements, changing validity rules, and changing the specified behaviour of honest nodes. A Crosslink 2 node must participate in both Πbft​ and Πbc​; that is, it must maintain a view of the state of each protocol. Acting in more specific roles such as bft‑proposer, bft‑validator, or bc‑block‑producer is optional, but we assume that all such actors are Crosslink 2 nodes.","breadcrumbs":"Design » Crosslink » The Crosslink 2 Construction » Subprotocols","id":"79","title":"Subprotocols"},"8":{"body":"The hybrid PoW/PoS consensus protocol proposed in this book is similar to today’s Zcash NU5/NU6 consensus protocol with the addition of a Trailing Finality Layer : TODO: Add network topology / software subcomponent diagram #124 The Zcash Trailing Finality Layer refers to a new subprotocol of a new hybrid PoW/PoS protocol, which we refer to as PoW+TFL . This subprotocol introduces assured finality for the Zcash block chain, ensuring that final blocks (and the transactions within them) may never be rolled back. We use the term “layer” because we can understand this design as introducing a new layer to the Zcash network, making only minimal changes to the existing network and consensus protocol. This modular separation is present in the consensus rules, the network protocol, and the code architecture.","breadcrumbs":"Introduction » Trailing Finality Layer in a Nutshell » Trailing Finality Layer in a Nutshell","id":"8","title":"Trailing Finality Layer in a Nutshell"},"80":{"body":"A ∗bft‑node’s view includes a set of ∗bft‑block chains each rooted at a fixed genesis ∗bft‑block O∗bft​. There is a ∗bft‑block‑validity rule (specified below), which depends only on the content of the block and its ancestors. A non‑genesis block can only be ∗bft‑block‑valid if its parent is ∗bft‑block‑valid. A ∗bft‑valid‑chain is a chain of ∗bft‑block‑valid blocks. Execution proceeds in a sequence of epochs. In each epoch, an honest proposer for that epoch may make a ∗bft‑proposal. A ∗bft‑proposal refers to a parent ∗bft‑block, and specifies the proposal’s epoch. The content of a proposal is signed by the proposer using a strongly unforgeable signature scheme. We consider the proposal to include this signature. There is a ∗bft‑proposal‑validity rule, depending only on the content of the proposal and its parent block, and the validity of the proposer’s signature. We extend the ⌈k notation to ∗bft‑proposals in the obvious way: if k≥1, P is a ∗bft‑proposal and B its parent ∗bft‑block, then P⌈∗bftk​:=B⌈∗bftk−1​. Terminology We will shorten “∗bft‑block‑valid ∗bft‑block” to “∗bft‑valid‑block”, and “∗bft‑proposal‑valid ∗bft‑proposal” to “∗bft‑valid‑proposal”. For each epoch, there is a fixed number of voting units distributed between the ∗bft‑nodes, which they use to vote for a ∗bft‑proposal. We say that a voting unit has been cast for a ∗bft‑proposal P at a given time in a ∗bft‑execution, if and only if P is ∗bft‑proposal‑valid and a ballot for P authenticated by the holder of the voting unit exists at that time. Using knowledge of ballots cast for a ∗bft‑proposal P that collectively satisfy a notarization rule at a given time in a ∗bft‑execution, and only with such knowledge, it is possible to obtain a valid ∗bft‑notarization‑proof proofP​. The notarization rule must require at least a two‑thirds absolute supermajority of voting units in P’s epoch to have been cast for P. It may also require other conditions. A voting unit is cast non‑honestly for an epoch’s proposal iff: it is cast other than by the holder of the unit (due to key compromise or any flaw in the voting protocol, for example); or it is double‑cast (i.e. there are at least two ballots casting it for distinct proposals); or the holder of the unit following the conditions for honest voting in Π∗bft​, according to its view, should not have cast that vote. Note that a unit should be considered to be cast non-honestly in the case of key compromise, because it is then effectively under the control of an adversary. The key compromise may or may not be attributable to another flaw in the protocol, but such a flaw would not be a break of the consensus mechanism per se. Definition: One‑third bound on non‑honest voting An execution of Πbft​ has the one‑third bound on non‑honest voting property iff for every epoch, strictly fewer than one third of the total voting units for that epoch are ever cast non‑honestly. Info It may be the case that a ballot cast for P is not in honest view when it is used to create a notarization proof for P. Since we are not assuming synchrony, it may also be the case that such a ballot is in honest view but that any given node has not received it (and perhaps will never receive it). There may be multiple distinct ballots or distinct ballot messages attempting to cast a given voting unit for the same proposal; this is undesirable for bandwidth usage, but it is not necessary to consider it to be non‑honest behaviour for the purpose of security analysis, as long as such ballots are not double‑counted toward the two‑thirds threshold. Security caveat The one‑third bound on non‑honest voting property considers all ballots cast in the entire execution. In particular, it is possible that a validator’s key is compromised and then used to cast its voting units for a proposal of an epoch long finished. If the number of voting units cast non-honestly for any epoch ever reaches one third of the total voting units for that epoch during an execution, then the one‑third bound on non‑honest voting property is violated for that execution. Therefore, validator keys of honest nodes must remain secret indefinitely. Whenever a key is rotated, the old key must be securely deleted. For further discussion and potential improvements, see tfl-book issue #140 . A ∗bft‑block consists of (P,proofP​) re‑signed by the same proposer using a strongly unforgeable signature scheme. It is ∗bft‑block‑valid iff: P is ∗bft‑proposal‑valid; and proofP​ is a valid proof that some subset of ballots cast for P are sufficient to satisfy the notarization rule; and the proposer’s outer signature on (P,proofP​) is valid. A ∗bft‑proposal’s parent reference hashes the entire parent ∗bft‑block, i.e. proposal, proof, and outer signature. Info Neither proofP​ nor the proposer’s outer signature is unique for a given P. The proposer’s outer signature is however third‑party nonmalleable, by definition of a strongly unforgeable signature scheme. An “honest ∗bft‑proposal” is a ∗bft‑proposal made for a given epoch by a proposer who is honest in that epoch. Such a proposer will only create one proposal and only sign at most once for each epoch, and so there will be at most one “honestly submitted” ∗bft‑block for each epoch. It is possible for there to be multiple ∗bft‑valid‑blocks for the same proposal, with different notarization proofs and/or outer signatures, if the proposer is not honest. However, the property that there will be at most one “honestly submitted” ∗bft‑block for each epoch is important for liveness, even though we cannot guarantee that any particular proposer for an epoch is honest. TODO: check that we are correctly using this in the liveness analysis. There is an efficiently computable function ∗bft-last-final⦂∗bft-block→∗bft-block∪{⊥}. For a ∗bft‑block‑valid input block C, this function outputs the last ancestor of C that is final in the context of C. Info The chain of ancestors is unambiguously determined because a ∗bft‑proposal’s parent reference hashes the entire parent ∗bft‑block; each ∗bft‑block commits to a proposal; and the parent hashes are collision‑resistant. This holds despite the caveat mentioned above that there may be multiple ∗bft‑valid‑blocks for the same proposal. ∗bft-last-final must satisfy all of the following: ∗bft-last-final(C)=⊥⟺C is not ∗bft‑block‑valid. If C is ∗bft‑block‑valid, then: ∗bft-last-final(C)⪯∗bft​C (and therefore it must also be ∗bft‑block‑valid); for all ∗bft‑valid‑blocks D such that C⪯∗bft​D, ∗bft-last-final(C)⪯∗bft​∗bft-last-final(D). ∗bft-last-final(O∗bft​)=O∗bft​. Info It is correct to talk about the “last final block” of a given chain (that is, each ∗bft‑valid-block C unambiguously determines a ∗bft‑valid-block ∗bft-last-final(C)), but it is not correct to refer to a given ∗bft‑block as objectively “∗bft‑final”. A particular BFT protocol might need adaptations to fit it into this model for Πorigbft​, before we apply the Crosslink 2 modifications to obtain Πbft​. Any such adaptions are necessarily protocol-specific. In particular: origbft‑proposal‑validity should correspond to the strongest property of an origbft‑proposal that is objectively and feasibly verifiable from the content of the proposal and its parent origbft‑block at the time the proposal is made. It must include verification of the proposer’s signature. origbft‑block‑validity should correspond to the strongest property of an origbft‑block that is objectively and feasibly verifiable from the content of the block and its ancestors at the time the block is added to an origbft‑chain. It should typically include all of the relevant checks from origbft‑proposal‑validity that apply to the created block (or equivalent checks). It must also include verification of the notarization proof and the proposer’s outer signature. If a node observes an origbft‑valid block C, then it should be infeasible for an adversary to cause a rollback in that node’s view past origbft-last-final(C), and the view of the chain up to origbft-last-final(C) should agree with that of all other honest nodes. This is formalized in the next section. Safety of Π∗bft​ The intuition behind the following safety property is that: For Π∗bft​ to be safe, it should never be the case that two honest nodes observe (at any time) ∗bft‑blocks B and B′ respectively that they each consider final in some context, but B⪯⪰∗bft​B′ does not hold. By definition, an honest node observes a ∗bft‑block to be final in the context of another ∗bft‑block C, iff B⪯∗bft​∗bft-last-final(C). We say that a ∗bft‑block is “in honest view” if a party observes it at some time at which that party is honest. Definition: Final Agreement An execution of Π∗bft​ has Final Agreement iff for all ∗bft‑valid blocks C in honest view at time t and C′ in honest view at time t′, we have ∗bft-last-final(C)⪯⪰∗bft​∗bft-last-final(C′). Note that it is possible for this property to hold for an execution of a BFT protocol in an asynchronous communication model. As previously mentioned , if the one‑third bound on non‑honest voting property is ever broken at any time in an execution, then it may not be possible to maintain Final Agreement from that point on. Adapting the Streamlet BFT protocol. Streamlet as described in [CS2020] has three possible states of a block in a player’s view: “valid” (but not notarized or final); “notarized” (but not final); “final”. By “valid” the Streamlet paper means just that it satisfies the structural property of being part of a block chain with parent hashes. The role of ∗bft‑block‑validity in our model corresponds roughly to Streamlet’s “notarized”. It turns out that with some straightforward changes relative to Streamlet, we can identify “origbft‑block‑valid” with “notarized” and consider an origbft‑valid‑chain to only consist of notarized blocks. This is not obvious, but is a useful simplification. Here is how the paper defines “notarized”: When a block gains votes from at least 2n/3 distinct players, it becomes notarized. A chain is notarized if its constituent blocks are all notarized. This implies that blocks can be added to chains independently of notarization. However, the paper also says that an honest leader always proposes a block extending from a notarized chain. Therefore, only notarized chains really matter in the protocol. In unmodified Streamlet, the order in which a player sees signatures might cause it to view blocks as notarized out of order. Streamlet’s security analysis is in a synchronous model, and assumes for liveness that any vote will have been received by all players (Streamlet nodes) within two epochs. In Crosslink 2, however, we need origbft‑block‑validity to be an objectively and feasibly verifiable property. We also would prefer reliable message delivery within bounded time not to be a basic assumption of our communication model. (This does not dictate what assumptions about message delivery are made for particular security analyses.) If we did not make a modification to the protocol to take this into account, then some Crosslink 2 nodes might receive a two‑thirds absolute supermajority of voting messages and consider a BFT block to be notarized, while others might never receive enough of those messages. Obviously a proposal cannot include signatures on itself — but the block formed from it can include proofs about the proposal and signatures. We can therefore say that when a proposal gains a two‑thirds absolute supermajority of signatures, a block is created from it that contains a proof (such as an aggregate signature) that it had such a supermajority. For example, we can have the proposer itself make this proof once it has enough votes, sign the resulting (P,proofP​) to create a block, then submit that block in a separate message. (The proposer has most incentive to do this in order to gain whatever reward attaches to a successful proposal; it can outsource the proving task if needed.) Then the origbft‑block‑validity rule can require a valid supermajority proof, which is objectively and feasibly verifiable. Players that see an origbft‑valid‑block can immediately consider it notarized. Note that for the liveness analysis to be unaffected, we need to assume that the combined latency of messages, of collecting and aggregating signatures, and of block submission is such that all adapted‑Streamlet nodes will receive a notarized block corresponding to a given proposal (rather than just all of the votes for the proposal) within two epochs. Alternatively we could re‑do the timing analysis. With this change, “origbft‑block‑valid” and “notarized” do not need to be distinguished. Streamlet’s finality rule is: If in any notarized chain, there are three adjacent blocks with consecutive epoch numbers, the prefix of the chain up to the second of the three blocks is considered final. When a block becomes final, all of its prefix must be final too. We can straightforwardly express this as an origbft-last-final function of a context block C, as required by the model: For an origbft‑valid‑block C, origbft-last-final(C) is the last origbft‑valid‑block B⪯origbft​C such that either B=Oorigbft​ or B is the second block of a group of three adjacent blocks with consecutive epoch numbers. Note that “When a block becomes final, all of its prefix must be final too.” is implicit in the model.","breadcrumbs":"Design » Crosslink » The Crosslink 2 Construction » Model for BFT protocols (Π{origbft,bft})","id":"80","title":"Model for BFT protocols (Π{origbft,bft})"},"81":{"body":"A node’s view in Π∗bc​ includes a set of ∗bc‑block chains each rooted at a fixed genesis ∗bc‑block O∗bc​. There is a ∗bc‑block‑validity rule (often described as a collection of “consensus rules”), depending only on the content of the block and its ancestors. A non‑genesis block can only be ∗bc‑block‑valid if its parent is ∗bc‑block‑valid. By “∗bc‑valid‑chain” we mean a chain of ∗bc‑block‑valid blocks. Terminology The terminology commonly used in the block‑chain community does not distinguish between rules that are part of the consensus protocol proper, and rules required for validity of the economic computation supported by the block chain. Where it is necessary to distinguish, the former can be called “L0” consensus rules, and the latter “L1” consensus rules. The definition of ∗bc‑block‑validity is such that it is hard for a block producer to extend a ∗bc‑valid‑chain unless they are selected by a random process that chooses a block producer in proportion to their resources with an approximately known and consistent time distribution, subject to some assumption about the total proportion of resources held by honest nodes. There is a function score⦂∗bc-valid-chain→Score, with < a strict total ordering on Score. An honest node will choose one of the ∗bc‑valid‑chains with highest score as the ∗bc‑best‑chain in its view. Any rule can be specified for breaking ties. The score function is required to satisfy score(H⌈∗bc1​)<score(H) for any non‑genesis ∗bc‑valid‑chain H. Info For a Proof‑of‑Work protocol, the score of a ∗bc‑chain should be its accumulated work. Unless an adversary is able to censor knowledge of other chains from a node’s view, it should be difficult to cause the node to switch to a chain with a last common ancestor more than σ blocks back from the tip of its previous ∗bc‑best‑chain. Let ch be a view such that chit​ is node i’s ∗bc‑best‑chain at time t. (This matches the notation used in [NTT2020] .) We define chi0​ to be Obc​. A ∗bc‑valid‑block is assumed to commit to a collection (usually, a sequence) of ∗bc‑transactions. Unlike in Crosslink 1 or Snap-and-Chat, we do not need to explicitly model ∗bc‑transaction validity or impose any additional constraints on it. The consensus rules applying to ∗bc‑transactions are entirely unchanged, including any rules that depend on ∗bc‑block height or previous ∗bc‑blocks. This is because Crosslink 2 never reorders or selectively “sanitizes” transactions as Snap-and-Chat does. If a bc‑block is included in a Crosslink 2 block chain then its entire parent bc‑block chain is included just as it would have been in Πorigbc​ (only modified by the structural additions described later), so block heights are also preserved. A “coinbase transaction” is a ∗bc‑transaction that only distributes newly issued funds and has no inputs. Define is-coinbase-only-block⦂∗bc-block→boolean so that is-coinbase-only-block(B)=true iff B has exactly one transaction that is a coinbase transaction. Each ∗bc‑block is summarized by a ∗bc‑header that commits to the block. There is a notion of ∗bc‑header‑validity that is necessary, but not sufficient, for validity of the block. We will only make the distinction between ∗bc‑headers and ∗bc‑blocks when it is necessary to avoid ambiguity. Header validity for Proof‑of‑Work protocols. In a Proof‑of‑Work protocol, it is normally possible to check the Proof‑of‑Work of a block using only the header. There is a difficulty adjustment function that determines the target difficulty for a block based on its parent chain. So, checking that the correct difficulty target has been used relies on knowing that the header’s parent chain is valid. Checking header validity before expending further resources on a purported block can be relevant to mitigating denial‑of‑service attacks that attempt to inflate validation cost. Typically, Bitcoin‑derived best chain protocols do not need much adaptation to fit into this model. The model still omits some details that would be important to implementing Crosslink 2, but distracting for this level of abstraction. Safety of Π∗bc​ We make an assumption on executions of Πorigbc​ that we will call Prefix Consistency (introduced in [ PSS2016 , section 3.3] as just “consistency”): Definition: Prefix Consistency An execution of Π∗bc​ has Prefix Consistency at confirmation depth σ, iff for all times t≤u and all nodes i, j (potentially the same) such that i is honest at time t and j is honest at time u, we have that chit​⌈∗bcσ​⪯∗bc​chju​. Explain the confusion in the literature about what variants of this property are called. The literature uses the same name, “common‑prefix property”, for two different properties of very different strength. [ PSS2016 , section 3.3] introduced the stronger variant. That paper first describes the weaker variant, calling it the “common‑prefix property by Garay et al [GKL2015] .” Then it explains what is essentially a bug in that variant, and describes the stronger variant which it just calls “consistency”: The common‑prefix property by Garay et al [GKL2015] , which was already considered and studied by Nakamoto [Nakamoto2008] , requires that in any round r, the record chains of any two honest players i, j agree on all, but potentially the last T, records. We note that this property (even in combination with the other two desiderata [of Chain Growth and Chain Quality]) provides quite weak guarantees: even if any two honest parties perfectly agree on the chains, the chain could be completely different on, say, even rounds and odd rounds. We here consider a stronger notion of consistency which additionally stipulates players should be consistent with their “future selves”. Let consistentT(view)=1 iff for all rounds r≤r′, and all players i, j (potentially the same) such that i is honest at viewr and j is honest at viewr′, we have that the prefixes of Cir​(view) and Cjr′​(view) consisting of the first ℓ=∣Cir​(view)∣−T records are identical. Unfortunately, [GKL2020] , which is a revised version of [GKL2015] , switches to the stronger variant without changing the name . (The eprint version history may be useful; the change was made in version 20181013:200033 , page 17.) Note that [GKL2020] uses an adaptive‑corruption model, “meaning that the adversary is allowed to take control of parties on the fly”, and so their wording in Definition 3: ... for any pair of honest players P1​, P2​ adopting the chains C1​, C2​ at rounds r1​≤r2​ in viewΠ,A,Zt,n​ respectively, it holds that C1⌈k​⪯C2​. is intended to mean the same as our ... for all times t≤u and all nodes i, j (potentially the same) such that i is honest at time t and j is honest at time u, we have that chit​⌈∗bcσ​⪯∗bc​chju​. The latter is closer to [PSS2016] . Incidentally, this property does not seem to be mentioned in [Nakamoto2008] , contrary to the [PSS2016] authors’ assertion. Maybe implicitly, but it’s a stretch. Discussion of [GKL2020]’s communication model and network partition. When Prefix Consistency is taken to hold of typical PoW-based block‑chain protocols like Bitcoin (as it often is), this implies that, in the relevant executions, the network of honest nodes is never partitioned — unless any partition lasts only for a short length of time relative to σ block times. If node i is on one side of a full partition and node j on the other, then after node i’s best chain has been extended by more than σ blocks, chit​⌈∗bcσ​ will contain information that has no way to get to node j. And even if the partition is incomplete, we cannot guarantee that the Prefix Consistency property will hold for any given pair of nodes. It might be possible to maintain Prefix Consistency if the honest nodes on one side of the partition knew that they should not continue building on their chain until the partition has healed, but it is unclear how that would be done in general without resorting to a BFT protocol (as opposed to in specific cases like a single node being unable to connect to the rest of the network). Certainly there is no mechanism to explicitly detect and respond to partitions in protocols derived from Bitcoin. And yet, [GKL2020] claims to prove Prefix Consistency from other assumptions. So we know that those assumptions must also rule out a long partition between honest nodes. In fact the required assumption is implicit in the communication model: A synchronous network cannot be partitioned. A partially synchronous network —that is, providing reliable delivery with bounded but unknown delay— cannot be partitioned for longer than the delay. We might be concerned that these implicit assumptions are stronger than we would like. In practice, the peer‑to‑peer network protocol of Bitcoin and Zcash attempts to flood blocks to all nodes. This protocol might have weaknesses, but it is not intended to (and plausibly does not) depend on all messages being received. (Incidentally, Streamlet also implicitly floods messages to all nodes.) Also, Streamlet and many other BFT protocols do not assume for safety that the network is not partitioned. That is, BFT protocols can be safe in a fully asynchronous communication model with unreliable messaging. That is why we avoid taking synchrony or partial synchrony as an implicit assumption of the communication model, or else we could end up with a protocol with weaker safety properties than Πorigbft​ alone. This leaves the question of whether the Prefix Consistency property is still too strong, even if we do not rely on it for the analysis of safety when Πbft​ has not been subverted. In particular, if a particular node h is not well-connected to the rest of the network, then that will inevitably affect node h’s security, but should not affect other honest nodes’ security. Fortunately, it is not the case that disconnecting a single node h from the network causes the security assumption to be voided. The solution is to view h as not honest in that case (even though it would follow the protocol if it could). This achieves the desired effect within the model, because other nodes can no longer rely on h’s honest input. Although viewing h as potentially adversarial might seem conservative from the point of view of other nodes, bear in mind that an adversary could censor an arbitrary subset of incoming and outgoing messages from the node, and this may be best modelled by considering it to be effectively controlled by the adversary. Prefix Consistency compares the σ-truncated chain of some node i with the untruncated chain of node j. For our analysis of safety of the derived ledgers, we will also need to make an assumption on executions of Πorigbc​ that at any given time t, any two honest nodes i and j agree on their confirmed prefixes — with only the caveat that one may have observed more of the chain than the other. That is: Definition: Prefix Agreement An execution of Π∗bc​ has Prefix Agreement at confirmation depth σ iff it has Agreement on the view (i,t)↦chit​⌈∗bcσ​. Why are this property, and Prefix Consistency above, stated as unconditional properties of protocol executions , rather than as probabilistic assumptions? Our security arguments that depend on these properties will all be of the form “in an execution where ⟨safety properties⟩ are not violated, ⟨undesirable thing⟩ cannot happen”. It is not necessary to involve probability in arguments of this form. Any probabilistic reasoning can be done separately. In particular, if a statement of this form holds, and ⟨safety properties⟩ are violated with probability at most p under certain conditions, then it immediately follows that under those conditions ⟨undesirable thing⟩ happens with probability at most p. Furthermore, ⟨undesirable thing⟩ can only happen after ⟨safety properties⟩ have been violated, because the execution up to that point has been an execution in which ⟨safety properties⟩ are not violated. With few exceptions, involving probability in a security argument is best done only to account for nondeterministic choices in the protocol itself. This is opinionated advice, but a lot of security proofs would likely be simpler if inherently probabilistic arguments were more distinctly separated from unconditional ones. In the case of the Prefix Agreement property, an alternative approach would be to prove that Prefix Agreement holds with some probability given Prefix Consistency and some other chain properties. This is what [NTT2020] does in its Theorem 2, which essentially says that under certain conditions Prefix Agreement holds except with probability e−Ω(σ​). The conclusions that can be obtained from this approach are necessarily probabilistic, and depending on the techniques used, the proof may not be tight; that is, the proof may obtain a bound on the probability of failure that is (either asymptotically or concretely) higher than needed. This is the case for [ NTT2020 , Theorem 2]; footnote 10 in that paper points out that the expression for the probability can be asymptotically improved: Using the recursive bootstrapping argument developed in [ DKT+2020 , Section 4.2], it is possible to bring the error probability e−Ω(σ​) as close to an exponential decay as possible. In this context, for any ϵ>0, it is possible to find constants Aϵ​, aϵ​ such that Πlc​(p) is secure after C (max(GST,GAT)) with confirmation time Tlc​=σ except with probability Aϵ​e−aϵ​σ1−ϵ. (Here p is the probability that any given node gets to produce a block in any given time slot.) In fact none of the proofs of security properties for Snap‑and‑Chat depend on the particular expression e−Ω(σ​); for example in the proofs of Lemma 5 and Theorem 1, this probability just “passes through” the proof from the premisses to the conclusion, because the argument is not probabilistic. The same will be true of our safety arguments. Talking about what is possible in particular executions has further advantages: It sidesteps the issue of how to interpret results in the GST model of partial synchrony, when we do not know what C (max(GST,GAT)) is. See also the critique of applying the GST model to block‑chain protocols under “Discussion of [GKL2020]’s communication model and network partition” above. (This is not an inherent problem with analyzing the protocol in the partially synchronous setting, but only with inappropriate use of the GST model of that setting.) We do not require Πbc​ to be a Nakamoto‑style Proof‑of‑Work block chain protocol. Some other kind of protocol could potentially satisfy Prefix Consistency and Prefix Agreement. It is not clear whether a e−Ω(σ​) probability of failure would be concretely adequate. That would depend on the value of σ and the constant hidden by the Ω notation. The asymptotic property using Ω tells us whether a sufficiently large σ could be chosen, but we are more interested in what needs to be assumed for a given concrete choice of σ. If a violation of a required safety property occurs in a given execution , then the safety argument for Crosslink that depended on the property fails for that execution, regardless of what the probability of that occurrence was. This approach therefore more precisely models the consequences of such violations. Why, intuitively, should we believe that Prefix Agreement and Prefix Consistency for a large enough confirmation depth hold with high probability for executions of a PoW‑based best‑chain protocol? Roughly speaking, the intuition behind both properties is as follows: Honest nodes are collectively able to find blocks faster than an adversary, and communication between honest nodes is sufficiently reliable that they act as a combined network racing against that adversary. Then by the argument in [Nakamoto2008] , modified by [GP2020] to correct an error in the concrete analysis, a private mining attack that attempts to cause a σ‑block rollback will, with high probability, fail for large enough σ. A private mining attack is optimal by the argument in [DKT+2020] . Any further analysis of the conditions under which these properties hold should be done in the context of a particular Π∗bc​. Why is the quantification in Prefix Agreement over two different times t and t′ ? This strengthens the security property, relative to quantifying over a single time. The question can then be split into several parts: What does the strengthened property mean, intuitively? Consider the full tree of ∗bc‑valid-blocks that honest nodes have considered to be part of their ∗bc‑best-chain at any times during the execution. This property holds iff, when we strip off all branches of length up to and including σ blocks, the resulting tree is ∗bc‑linear. Why is the strengthening needed? Suppose that time were split into periods such that honest nodes agreed on one chain in odd periods, and a completely different chain in even periods. This would obviously not satisfy the intent, but it would satisfy a version of the property that did not quantify over different times t and u. Why should we expect the strengthened property to hold? If node j were far ahead, i.e. u≫t, then it is obvious that chit​⌈∗bcσ​⪯∗bc​chju​⌈∗bcσ​ should hold. Conversely, if node i were far ahead then it is obvious that chju​⌈∗bcσ​⪯∗bc​chit​⌈∗bcσ​ should hold. The case where t=u is the same as quantifying over a single time. By considering intermediate cases where t and u converge from the extremes or where they diverge from being equal, you should be able to convince yourself that the property holds for any relative values of t and u, in executions of a reasonable best‑chain protocol.","breadcrumbs":"Design » Crosslink » The Crosslink 2 Construction » Model for best-chain protocols (Π{origbc,bc})","id":"81","title":"Model for best-chain protocols (Π{origbc,bc})"},"82":{"body":"","breadcrumbs":"Design » Crosslink » The Crosslink 2 Construction » Definition of Crosslink 2","id":"82","title":"Definition of Crosslink 2"},"83":{"body":"Crosslink 2 is parameterized by a bc‑confirmation‑depth σ∈N+ (as in Snap‑and‑Chat), and also a finalization gap bound L∈N+ with L significantly greater than σ. Each node i always uses the fixed confirmation depth σ to obtain its view of the finalized chain finit​⦂bc-valid-chain. Unlike in Snap‑and‑Chat or Crosslink 1, this is just a block chain; because we do not need sanitization, there is no need to express it as a log of transactions rather than blocks. Each node i chooses a potentially different bc‑confirmation‑depth μ∈N where 0<μ≤σ to obtain its view of the bounded‑available ledger at time t, (baμ​)it​⦂bc-valid-chain. (We make the restriction μ≤σ because there is no reason to choose a larger μ.) Security caveat Choosing μ<σ is at the node’s own risk and may increase the risk of rollback attacks against (baμ​)it​ (it does not affect finit​). Using small values of μ is not recommended. The default should be μ=σ.","breadcrumbs":"Design » Crosslink » The Crosslink 2 Construction » Parameters","id":"83","title":"Parameters"},"84":{"body":"Consider, roughly speaking, the number of bc‑blocks that are not yet finalized at time t (a more precise definition will be given in the section on Πbc​ changes from Πorigbc​ ). We call this the “finality gap” at time t. Under an assumption about the distribution of bc‑block intervals, if this gap stays roughly constant then it corresponds to the approximate time that transactions take to be finalized after being included in a bc‑block (if they are finalized at all) just prior to time t. As explained in detail by The Arguments for Bounded Availability and Finality Overrides , if this bound exceeds a threshold L, then it likely signals an exceptional or emergency condition, in which it is undesirable to keep accepting user transactions that spend funds into new bc‑blocks. In practice, L should be at least 2σ. The condition that the network enters in such cases will be called “Stalled Mode”. For a given higher‑level transaction protocol, we can define a policy for which bc‑blocks will be accepted in Stalled Mode. This will be modelled by a predicate is-stalled-block⦂bc-block→boolean. A bc‑block for which is-stalled-block returns true is called a “stalled block”. Caution A bc‑block producer is only constrained to produce stalled blocks while, roughly speaking, its view of the finalization point is not advancing. In particular an adversary that has subverted the BFT protocol in a way that does not keep the finalization point from advancing, can always avoid being constrained by Stalled Mode. The desired properties of stalled blocks and a possible Stalled Mode policy for Zcash are discussed in the How to block hazards section of The Arguments for Bounded Availability and Finality Overrides . In practice a node's view of the finalized chain, finit​, is likely to lag only a few blocks behind bai,σt​ (depending on the latency overhead imposed by Πbft​), unless the chain has entered Stalled Mode. So when μ=σ, the main factor influencing the choice of a given application to use finit​ or (baμ​)it​ is not the average latency, but rather the desired behaviour in the case of a finalization stall: i.e. stall immediately, or keep processing user transactions until L blocks have passed.","breadcrumbs":"Design » Crosslink » The Crosslink 2 Construction » Stalled Mode","id":"84","title":"Stalled Mode"},"85":{"body":"Each bc‑header has, in addition to origbc‑header fields, a context_bft field that commits to a bft‑block. Each bft‑proposal has, in addition to origbft‑proposal fields, a headers_bc field containing a sequence of exactly σ bc‑headers (zero‑indexed, deepest first). Each non‑genesis bft‑block has, in addition to origbft‑block fields, a headers_bc field containing a sequence of exactly σ bc‑headers (zero-indexed, deepest first). The genesis bft‑block has headers_bc=∅. For a bft‑block or bft‑proposal B, define LF(H)snapshot(B)​:={Obc​,B.headers_bc[0]⌈bc1​,​if B.headers_bc=∅otherwise.​​ For a bc‑block H, define snapshot(B)LF(H)candidate(H)​:=bft-last-final(H.context_bft):=last-common-ancestor(snapshot(LF(H)),H⌈bcσ​)​ When H is the tip of a node’s bc‑best‑chain, candidate(H) will give the candidate finalization point, subject to a condition described below that prevents local rollbacks. Use of the headers_bc field, and its relation to the ch field in Snap‑and‑Chat. For a bft‑proposal or bft‑block B, the role of the bc‑chain snapshot referenced by B.headers_bc[0]⌈bc1​ is comparable to the Πlc​ snapshot referenced by B.ch in the Snap‑and‑Chat construction from [NTT2020] . The motivation for the additional headers is to demonstrate, to any party that sees a bft‑proposal (resp. bft‑block), that the snapshot had been confirmed when the proposal (resp. the block’s proposal) was made. Typically, a node that is validating an honest bft‑proposal or bft‑block will have seen at least the snapshotted bc‑block (and possibly some of the subsequent bc‑blocks in the headers_bc chain) before. For this not to be the case, the validator’s bc‑best‑chain would have to be more than σ bc‑blocks behind the honest proposer’s bc‑best‑chain at a given time, which would violate the Prefix Consistency property of Πbc​. If the headers do not connect to any bc‑valid‑chain known to the validator, then the validator should be suspicious that the proposer might not be honest. It can assign a lower priority to validating the proposal in this case, or simply drop it. The latter option could drop a valid proposal, but this does not in practice cause a problem as long as a sufficient number of validators are properly synced (so that Prefix Consistency holds for them). If the headers do connect to a known bc‑valid‑chain, it could still be the case that the whole header chain up to and including B.headers_bc[σ−1] is not a bc‑valid‑chain. Therefore, to limit denial‑of‑service attacks the validator should first check the Proofs‑of‑Work and difficulty adjustment —which it can do locally using only the headers— before attempting to download and validate any bc‑blocks that it has not already seen. This is why we include the full headers rather than just the block hashes. Nodes may “trim” (i.e. not explicitly store) headers in a bft‑block that overlap with those referred to by its ancestor bft‑block(s). Why is a distinguished value needed for the headers_bc field in the genesis bft‑block? It would be conceptually nice for Obft​.headers[0]⌈bc1​ to refer to Obc​, as well as Obc​.context_bft being Obft​ so that bft-last-final(Obc​.context_bft)=Obft​. That reflects the fact that we know “from the start” that neither genesis block can be rolled back. This is not literally implementable using block hashes because it would involve a hash cycle, but we achieve the same effect by defining a snapshot function that allows us to “patch” snapshot(Obft​) to be Obc​. We do it this way around rather than “patching” the link from a bc‑block to a bft‑block, because the genesis bft‑block already needs a special case since there are not σ bc‑headers available. Why is the context_bft field needed? Why not use a final_bft field to refer directly to the last final bft‑block before the context block? The finality of some bft‑block is only defined in the context of another bft‑block. One possible design would be for a bc‑block to have both final_bft and context_bft fields, so that the finality of final_bft could be checked objectively in the context of context_bft. However, specifying just the context block is sufficient information to determine its last final ancestor. There would never be any need to give a context block and a final ancestor that is not the last one. The bft-last-final function can be computed efficiently for typical BFT protocols. Therefore, having just the context_bft field is sufficient.","breadcrumbs":"Design » Crosslink » The Crosslink 2 Construction » Structural additions","id":"85","title":"Structural additions"},"86":{"body":"Each node i keeps track of a “locally finalized” bc‑chain finit​ at time t. Each node’s locally finalized bc‑chain starts at fini0​=Obc​. However, this chain state should not be exposed to clients of the node until it has synced . Definition: Local finalization linearity Node i has Local finalization linearity up to time t iff the time series of bc‑blocks finir≤t​ is bc‑linear . When node i’s bc‑best‑chain view is updated from chis​ to chit​, the node’s finit​ will become candidate(chit​) if and only if this is a descendant of finis​. Otherwise finit​ will stay at finis​. This guarantees Local finalization linearity by construction. If when making this update, candidate(chit​)/⪯⪰finis​ (i.e. candidate(chit​) and finis​ are on different forks), then the node should record a finalization safety hazard. This can only happen if global safety assumptions are violated. Note that Local finalization linearity on each node is not sufficient for Assured Finality , but it is necessary. This can be expressed by the following state update algorithm, where s is the time of the last update and t is the time of the current update: update-fin(mut fini​,chit​,s):let N=candidate(chit​)if N⪰finis​:finit​←Nelse:finit​←finis​if N⪯finis​:record finalization safety hazard A safety hazard record should include chit​ and the history of fini​ updates including and since the last one that was an ancestor of N. Lemma: Local fin‑depth In any execution of Crosslink 2, for any node i that is honest at time t, there exists a time r≤t such that finit​⪯chir​⌈bcσ​. Proof: By the definition of candidate we have candidate(chiu​)⪯chiu​⌈bcσ​ for all times u. Let r≤t be the last time at which finit​ changed, or the genesis time 0 if it has never changed. Then for r>0 we have finit​=finir​=candidate(chir​)⪯chir​⌈bcσ​, and for r=0 we have finit​=finir​=Obc​⪯chir​⌈bcσ​ (because chi0​=Obc​, and truncating Obc​ always yields Obc​). Why does fini need to be maintained using local state? When a node’s view of the bc‑best‑chain reorgs to a different fork (even if the reorg is shorter than σ blocks), it may be the case that candidate(chit​) rolls back. If Final Agreement of Πbft​ holds up to time t, the new snapshot should in that case be an ancestor of the old one. If all is well then this snapshot will subsequently roll forward along the same path. However, we do not want applications using the node to see the temporary rollback. Assured Finality is our main safety goal for Crosslink 2. It is essentially the same goal as Final Agreement but applied to nodes’ locally finalized bc‑chains; intuitively it means that honest nodes never see conflicting locally finalized chains. We intend to prove that this goal holds under reasonable assumptions about either Πorigbft​ or Πorigbc​. Definition: Assured Finality An execution of Crosslink 2 has Assured Finality iff for all times t, u and all nodes i, j (potentially the same) such that i is honest at time t and j is honest at time u, we have finit​⪯⪰bc​finju​. Note that if an execution of Crosslink 2 has Assured Finality , then all nodes that are honest for that execution have Local finalization linearity . That is because the restriction of Assured Finality to the case i=j is equivalent to Local finalization linearity for node i up to any time at which node i is honest. Why do we need to use candidate(H) rather than snapshot(LF(H))? This ensures that the candidate is at least σ‑confirmed. In practice candidate(H) will rarely differ from snapshot(LF(H)), but using the former patches over a potential gap in the safety proof. The Last Final Snapshot rule specified later will guarantee that snapshot(LF(H))⪯bc​H, and this ensures that snapshot(LF(H))⪯⪰bc​H⌈bcσ​. However, the depth of snapshot(LF(H)) relative to H is not guaranteed to be ≥σ. For the proof we will need candidate(H)⪯bc​H⌈bcσ​, so that we can use the Local fin‑depth lemma together with Prefix Agreement of Πbc​ at confirmation depth σ to prove Assured Finality . An alternative would be to change the Last Final Snapshot rule to directly require snapshot(LF(H))⪯bc​H⌈bcσ​. TODO: Choose between these options based on what works well for the security proofs and finalization latency.","breadcrumbs":"Design » Crosslink » The Crosslink 2 Construction » Locally finalized chain","id":"86","title":"Locally finalized chain"},"87":{"body":"Define the locally bounded‑available chain on node i for bc‑confirmation‑depth μ, as (baμ​)it​={chit​⌈bcμ​,finit​,​if finit​⪯chit​⌈bcμ​otherwise.​ Like the locally finalized bc‑chain, this chain state should not be exposed to clients of the node until it has synced . Theorem: Ledger prefix property For any node i that is honest at time t, and any confirmation depth μ, finit​⪯(baμ​)it​. Proof: By construction of (baμ​)it​. Lemma: Local ba‑depth In any execution of Crosslink 2, for any confirmation depth μ≤σ and any node i that is honest at time t, there exists a time r≤t such that (baμ​)it​⪯bc​chir​⌈bcμ​. Proof: Either (baμ​)it​=finit​, in which case the result follows by the Local fin‑depth lemma since μ≤σ, or (baμ​)it​=chit​⌈bcμ​ in which case it follows trivially with r=t. Our security goal for baμ​ will be Agreement on baμ​ as already defined. Why is the ‘otherwise’ case in the definition of (baμ)it necessary? Assume for this discussion that Πbc​ uses PoW. Depending on the value of σ, the timestamps of bc‑blocks, and the difficulty adjustment rule, it can be the case that if fini​ switches to a different fork, the difficulty on that fork is greater than on the chain of the previous snapshot. Then, the new bc‑chain could reach a higher score than the previous chain in fewer than σ blocks from the fork point, and so chit​⌈bcμ​ might not be a descendant of finit​ (which is more likely if μ=σ). This can occur even when all safety assumptions are satisfied. For Zcash’s difficulty adjustment algorithm , the difficulty of each block is adjusted based on the median timestamps and difficulty target thresholds over a range of PoWAveragingWindow=17 blocks, where each median is taken over PoWMedianBlockSpan=11 blocks. Other damping factors and clamps are applied in order to prevent instability and to reduce the influence that adversarially chosen timestamps can have on difficulty adjustment. This makes it unlikely that an adversary could gain a significant advantage by manipulating the difficulty adjustment. So it is safe to use finit​ in this case: even though it does not have μ confirmations relative to chit​, it does have at least the required amount of work “on top” of it. Defining (baμ​)it​ this way also has the advantage of making the proof of the Ledger prefix property trivial.","breadcrumbs":"Design » Crosslink » The Crosslink 2 Construction » Locally bounded‑available chain","id":"87","title":"Locally bounded‑available chain"},"88":{"body":"It is recommended that node implementations “bake in” a checkpointed bft‑block Bcheckpoint​ to each released version, and that node i should only expose finit​ and (baμ​)it​ to its clients once it is “synced”, that is: Bcheckpoint​⪯bft​LF(chit​); and snapshot(Bcheckpoint​)⪯bc​finit​; and the timestamp of finit​ is within some threshold of the current time.","breadcrumbs":"Design » Crosslink » The Crosslink 2 Construction » Syncing and checkpoints","id":"88","title":"Syncing and checkpoints"},"89":{"body":"Πbft proposal and block validity Genesis bft‑block rule: Obft​ is bft‑block‑valid. A bft‑proposal (resp. non‑genesis bft‑block) B is bft‑proposal‑valid (resp. bft‑block‑valid) iff all of the following hold: Inherited origbft rules: The corresponding origbft‑proposal‑validity (resp. origbft‑block‑validity) rules hold for B. Linearity rule: snapshot(B⌈bft1​)⪯bc​snapshot(B). Tail Confirmation rule: B.headers_bc form the σ‑block tail of a bc‑valid‑chain. The “corresponding validity rules” are assumed to include the Parent rule that B’s parent is bft‑valid. Note: origbft‑block‑validity rules may be different to origbft‑proposal‑validity rules. For example, in adapted Streamlet, a origbft‑block needs evidence that it was voted for by a supermajority, and an origbft‑proposal doesn’t. Such differences also apply to bft‑block‑validity vs bft‑proposal‑validity. Why have validity rules been separated from the honest voting condition below? The reason to separate the validity rules from the honest voting condition, is that the validity rules are objective: they don’t depend on an observer’s view of the bc‑best‑chain. Therefore, they can be checked independently of validator signatures. Even a proposal voted for by 100% of validators will not be considered bft‑proposal‑valid by other nodes unless it satisfies the above rules. If more than two thirds of voting units are cast for an invalid proposal, something is seriously and visibly wrong; in any case, the block will not be accepted as a bft‑valid‑block. Importantly, a purportedly valid bft‑block will not be recognized as such by any honest Crosslink 2 node even if it includes a valid notarization proof, if it does not meet other bft‑block‑validity rules. This is essential to making the finalized chain safe against a flaw in Πbft​ or its security assumptions (even, say, a complete break of the validator signature algorithm), as long as Πbc​ remains safe. What does the Linearity rule do? This rule is key to combining simplicity with strong security properties in Crosslink 2. It essentially says that, in a given bft‑valid‑chain, the snapshots pointed to by blocks in that chain cannot roll back. This allows the informal safety argument for Crosslink 2 to be rather intuitive. Informally, if Πbft​ has Final Agreement , then all nodes see only one consistent bft‑linear chain (restricting to bft‑blocks that are final in the context of some bft‑block in honest view ). Within such a bft‑chain, the Linearity rule ensures by construction that the sequence of referenced bc‑chain snapshots is bc‑linear . This implies Assured Finality , without needing to assume any safety property of Πbc​. We will also be able to prove safety of the finalized snapshots based only on safety of Πbc​ (for a confirmation depth of σ), without needing to assume any safety property of Πbft​. Informally, that is because each node sees each candidate final snapshot at a given time as a σ-confirmed prefix of its bc‑best‑chain at that time (this can be proven based on the Last Final Snapshot rule and the fact that a snapshot includes σ subsequent headers), and Prefix Agreement implies that honest nodes agree on this prefix. We will leave a more detailed argument until after we have presented the Πbc​ changes from Πorigbc​ . The Linearity rule replaces the “Increasing Score rule” used in Crosslink 1. The Increasing Score rule required that each snapshot in a bft‑valid‑chain either be the same snapshot, or a higher-scoring snapshot to that of its parent block. Since scores strictly increase within a bc‑valid‑chain, the Linearity rule implies the Increasing Score rule. It retains the same or stronger positive effects: It prevents potential attacks that rely on proposing a bc‑valid‑chain that forks from a much earlier block. This is necessary because the difficulty (or stake threshold) at that point could have been much lower. It limits the extent of disruption an adversary can feasibly cause to the bounded‑available chain chi​, even if it has subverted Πbft​. Informally, because the finalized chain is a Πbc​ chain, its safety is no worse than Πbc​ alone for a rollback of any depth. It ensures that either progress is made (the snapshot advances relative to that of the parent bft‑block), or there is no further validation that needs to be done for the snapshot because it was already validated. Note that the adversary could take advantage of an “accidental” fork and start its attack from the base of that fork, so that not all of this work is done by it alone. This is also possible in the case of a standard “private mining” attack, and is not so much of a problem in practice because accidental forks are expected to be short. In any case, σ should be chosen to take it into account. The Linearity rule is also critical to removing the need for one of the most complex elements of Snap‑and‑Chat and Crosslink 1, “sanitization”. In those protocols, because bc‑chain snapshots could be unrelated to each other, it was necessary to sanitize the chain formed from these snapshots to remove transactions that were contextually invalid (e.g. because they double‑spend). The negative consequences of this are described in Notes on Snap‑and‑Chat ; avoiding it is much simpler. The linearity property is intentionally always relative to the snapshot of the parent bft‑block, even if it is not final in the context of the current bft‑block. This is because the rule needs to hold if and when it becomes final in the context of some descendant bft‑block. TODO: PoS Desideratum: we want leader selection with good security / performance properties that will be relevant to this rule. (Suggested: PoSAT .) Why does the Linearity rule allow keeping the same snapshot as the parent? This is necessary in order to preserve liveness of Πbft​ relative to Πorigbft​. Liveness of Πorigbft​ might require honest proposers to make proposals at a minimum rate. That requirement could be consistently violated if it were not always possible to make a valid proposal. But given that it is allowed to repeat the same snapshot as in the parent bft‑block, neither the Linearity rule nor the Tail Confirmation rule can prevent making a valid proposal — and all other rules of Πbft​ affecting the ability to make valid proposals are the same as in Πorigbft​. (In principle, changes to voting in Πbft​ could also affect its liveness; we’ll discuss that in the liveness proof later.) For example, Streamlet requires three notarized blocks in consecutive epochs in order to finalize a block [ CS2020 , section 1.1]. Its proof of liveness depends on the assumption that in each epoch for which the leader is honest, that leader will make a proposal, and that during a “period of synchrony” this proposal will be received by every node [ CS2020 , section 3.6]. This argument can also be extended to adapted‑Streamlet. We could alternatively have allowed to always make a “null” proposal, rather than to always make a proposal with the same snapshot as the parent. We prefer the latter because the former would require specifying the rules for null proposals in Πorigbft​. As a clarification, no BFT protocol that uses leader election can require a proposal in each epoch, because the leader might be dishonest. The above issue concerns liveness of the protocol when assumptions about the attacker’s share of bft‑validators or stake are met, so that it can be assumed that sufficiently long periods with enough honest leaders to make progress (5 consecutive epochs in the case of Streamlet), will occur with significant probability. Πbft block finality in context The finality rule for bft‑blocks in a given context is unchanged from origbft‑finality. That is, bft-last-final is defined in the same way as origbft-last-final (modulo referring to bft‑block‑validity and Obft​). Πbft honest proposal An honest proposer of a bft‑proposal P chooses P.headers_bc as the σ‑block tail of its bc‑best‑chain, provided that it is consistent with the Linearity rule . If it would not be consistent with that rule, it sets P.headers_bc to the same headers_bc field as P’s parent bft‑block. It does not make proposals until its bc‑best‑chain is at least σ+1 blocks long. Why σ + 1? If the length were less than σ+1 blocks, it would be impossible to construct the headers_bc field of the proposal. Note that when the length of the proposer’s bc‑best‑chain is exactly σ+1 blocks, the snapshot must be of Obc​. But this does not violate the Linearity rule , because Obc​ matches the previous snapshot by Obft​. How is it possible that the Linearity rule would not be satisfied by choosing headers from an honest proposer’s bc‑best‑chain? As in the answer to Why is the ‘otherwise’ case in the definition of (baμ​)it​ necessary? above, after a reorg on the bc‑chain, the σ-confirmed block on the new chain might not be a descendant of the σ-confirmed block on the old chain, which could break the Linearity rule . Πbft honest voting An honest validator considering a proposal P, first updates its view of both subprotocols with the bc‑headers given in P.headers_bc, downloading bc‑blocks for these headers and checking their bc‑block‑validity. For each downloaded bc‑block, the bft‑chain referenced by its context_bft field might need to be validated if it has not been seen before. Wait what, how much validation is that? In general the entire referenced bft‑chain needs to be validated, not just the referenced block — and for each bft‑block, the bc‑chain in headers_bc needs to be validated, and so on recursively. If this sounds overwhelming, note that: We should check the requirement that a bft‑valid‑block must have been voted for by a two‑thirds absolute supermajority of validators, and any other non‑recursive bft‑validity rules, first . Before validating a bc‑chain referenced by a headers_bc field, we check that it connects to an already-validated bc‑chain and that the Proofs‑of‑Work are valid. This implies that the amount of bc‑block validation is constrained by how fast the network can find valid Proofs‑of‑Work. The Linearity rule reduces the worst‑case validation effort, by ensuring that only one bc‑chain needs to be validated for any bft‑chain. Assuming safety of Πbft​ and that the adversary does not have an overwhelming advantage in computing the Proof‑of‑Work, this is effectively only one bc‑chain overall with, at most, short side branches. In summary, the order of validation is important to avoid denial‑of‑service — but it already is in Bitcoin and Zcash. After updating its view, the validator will vote for a proposal P only if : Valid proposal criterion: it is bft‑proposal‑valid, and Confirmed best‑chain criterion: snapshot(P) is part of the validator’s bc‑best‑chain at a bc‑confirmation‑depth of at least σ. Blocks in a bc‑best‑chain are by definition bc‑block‑valid. If we’re checking the Confirmed best‑chain criterion , why do we need to have separately checked that the blocks referenced by the headers are bc‑block‑valid? The Confirmed best‑chain criterion is quite subtle. It ensures that snapshot(P)=P.headers_bc[0]⌈bc1​ is bc‑block‑valid and has σ bc‑block‑valid blocks after it in the validator’s bc‑best‑chain. However, it need not be the case that P.headers_bc[σ−1] is part of the validator’s bc‑best‑chain after it updates its view. That is, the chain could fork after snapshot(P). The bft‑proposal‑validity rule must be objective; it can’t depend on what the validator’s bc‑best‑chain is. The validator’s bc‑best‑chain may have been updated to P.headers_bc[σ−1] (if it has the highest score), but it also may not. However, if the validator’s bc‑best‑chain was updated, that makes it more likely that it will be able to vote for the proposal. In any case, if the validator does not check that all of the blocks referenced by the headers are bc‑block‑valid, then its vote may be invalid. How does this compare to Snap‑and‑Chat? Snap‑and‑Chat already had the voting condition: An honest node only votes for a proposed BFT block B if it views B.ch as confirmed. but it did not give the headers potentially needed to update the validator’s view, and it did not require a proposal to be for an objectively confirmed snapshot as a matter of validity. If a Crosslink‑like protocol were to require an objectively confirmed snapshot but without including the bc‑headers in the proposal, then validators would not immediately know which bc‑blocks to download to check its validity. This would increase latency, and would be likely to lead proposers to be more conservative and only propose blocks that they think will already be in at least a two‑thirds absolute supermajority of validators’ best chains. That is, showing P.headers_bc to all of the validators is advantageous to the proposer, because the proposer does not have to guess what blocks the validators might have already seen. It is also advantageous for the protocol goals in general, because it improves the trade‑off between finalization latency and security.","breadcrumbs":"Design » Crosslink » The Crosslink 2 Construction » Πbft changes from Πorigbft","id":"89","title":"Πbft changes from Πorigbft"},"9":{"body":"There are three categories of users this proposed TFL protocol would impact:","breadcrumbs":"Introduction » Trailing Finality Layer in a Nutshell » Why Should Users Care?","id":"9","title":"Why Should Users Care?"},"90":{"body":"Πbc block validity Genesis bc‑block rule: For the genesis bc‑block we must have Obc​.context_bft=Obft​, and therefore bft-last-final(Obc​.context_bft)=Obft​. A bc‑block H is bc‑block‑valid iff all of the following hold: Inherited origbc rules: H satisfies the corresponding origbc‑block‑validity rules. Valid context rule: H.context_bft is bft‑block‑valid. Extension rule: LF(H⌈bc1​)⪯bft​LF(H). Last Final Snapshot rule: snapshot(LF(H))⪯bc​H. Finality depth rule: Define: finality-depth(H):=height(H)−height(snapshot(LF(H))) Then either finality-depth(H)≤L or is-stalled-block(H). Explain the definition of finality‑depth. The finality depth must be objectively defined, since it is used in a consensus rule. Therefore it should measure the height of H relative to snapshot(LF(H)), which is an objectively defined function of H, rather than relative to finit​. (These will only differ for H=chit​ when node i has just reorged, and only then in corner cases.) Note that the Last Final Snapshot rule ensures that it is meaningful to simply use the difference in heights, since snapshot(LF(H))⪯bc​H. Πbc contextual validity The consensus rule changes above are all non-contextual. Modulo these changes, contextual validity in Πbc​ is the same as in Πorigbc​. Πbc honest block production An honest producer of a bc‑block H must follow the consensus rules under Πbc​ block validity above. In particular, it must produce a stalled block if required to do so by the Finality depth rule . To choose H.context_bft, the producer considers a subset of the tips of bft‑valid‑chains in its view: {T:T is bft‑block‑valid and LF(H⌈bc1​)⪯bft​bft-last-final(T)} It chooses one of the longest of these chains, C, breaking ties by maximizing score(snapshot(bft-last-final(C))), and if there is still a tie then by taking C with the smallest hash. The honest block producer then sets H.context_bft to C. Attention An honest bc‑block‑producer must not use information from the BFT protocol, other than the specified consensus rules, to decide which bc‑valid‑chain to follow. The specified consensus rules that depend on Πbft​ have been carefully constructed to preserve safety of Πbc​ relative to Πorigbc​. Imposing any additional constraints could potentially allow an adversary that is able to subvert Πbft​, to influence the evolution of the bc‑best‑chain in ways that are not considered in the safety argument. Why not choose T such that H ⌈1bc . context_bft ⪯bft bft‑last‑final( T )? The effect of this would be to tend to more often follow the last bft‑block seen by the producer of the parent bc‑block, if there is a choice. It is not always possible to do so, though: the resulting set of candidates for C might be empty. Also, it is not clear that giving the parent bc‑block‑producer the chance to “guide” what bft‑block should be chosen next is beneficial, since that producer might be adversarial and the resulting incentives are difficult to reason about. Why choose the longest C , rather than the longest bft‑last‑final( C )? We could have instead chosen C to maximize the length of bft-last-final(C). The rule we chose follows Streamlet, which builds on the longest notarized chain, not the longest finalized chain. This may call for more analysis specific to the chosen BFT protocol. Why this tie‑breaking rule? Choosing the bft‑chain that has the last final snapshot with the highest score, tends to inhibit an adversary’s ability to finalize its own chain if it has a lesser score. (If it has a greater score, then it has already won a hash race and we cannot stop the adversary chain from being finalized.) For discussion of potentially unifying the roles of bc‑block producer and bft‑proposer, see What about making the bc‑block‑producer the bft‑proposer? in Potential changes to Crosslink . At this point we have completed the definition of Crosslink 2. In Security Analysis of Crosslink 2 , we will prove it secure.","breadcrumbs":"Design » Crosslink » The Crosslink 2 Construction » Πbc changes from Πorigbc","id":"90","title":"Πbc changes from Πorigbc"},"91":{"body":"This document analyzes the security of Crosslink 2 in terms of liveness and safety. It assumes that you have read The Crosslink 2 Construction which defines the protocol.","breadcrumbs":"Design » Crosslink » Security Analysis of Crosslink 2 » Security Analysis of Crosslink 2","id":"91","title":"Security Analysis of Crosslink 2"},"92":{"body":"First note that Crosslink 2 intentionally sacrifices availability if there is a long finalization stall. Info This is technically independent of the other changes; you can omit the Finality depth rule and the protocol would still have security advantages over Snap‑and‑Chat, as well as being much simpler and solving its “spending from finalized outputs” issue. In that case the incentives to “pull” the finalization point forward to include new final bft‑blocks would be weaker, but honest bc‑block‑producers would still do it. It would still be a bug if there were any situation in which Πbc​ failed to be live , though, because that would allow tail‑thrashing attacks . Crosslink 2 involves a bidirectional dependence between Πbft​ and Πbc​. The Ebb‑and‑Flow paper [NTT2020] argues in Appendix E (\"Bouncing Attack on Casper FFG\") that it can be more difficult to reason about liveness given a bidirectional dependence between protocols: To ensure consistency among the two tiers [of Casper FFG], the fork choice rule of the blockchain is modified to always respect ‘the justified checkpoint of the greatest height [*]’ [22] . There is thus a bidirectional interaction between the block proposal and the finalization layer: blocks proposed by the blockchain are input to finalization, while justified checkpoints constrain future block proposals. This bidirectional interaction is intricate to reason about and a gateway for liveness attacks. Info [*] The quotation changes this to “[depth]”, but our terminology is consistent with Ethereum here and not with [NTT2020]’s idiosyncratic use of “depth” to mean “block height”. The argument is correct as far as it goes. The main reason why this does not present any great difficulty to proving liveness of Crosslink, is due to a fundamental difference from Casper FFG: in Crosslink 2 the fork‑choice rule of Πbc​ is not modified. Let S be the subset of bc‑blocks {B:is-coinbase-only-block(B) and is-stalled-block(B)}. Assume that S is such that a bc‑block‑producer can always produce a block in S. In that case it is straightforward to convince ourselves that the additional bc‑block‑validity rules are never an obstacle to producing a new bc‑block in S: The changes to the definition of contextual validity do not interfere with liveness, since they do not affect coinbase transactions, and therefore do not affect blocks in S. That is, the bc‑block‑producer can always just omit non‑coinbase transactions that are contextually invalid. The Genesis bc‑block rule doesn’t apply to new bc‑blocks. The Valid context rule , Extension rule , and Last Final Snapshot rule are always satisfiable by referencing the same context bft‑block as the parent bc‑block. The Finality depth rule always allows the option of producing a stalled block, and therefore does not affect blocks in S. The instructions to an honest bc‑block‑producer allow it to produce a block in S. Therefore, Πbc​ remains live under the same conditions as Πorigbc​. The additional bft‑proposal‑validity, bft‑block‑validity, bft‑finality, and honest voting rules are also not an obstacle to making, voting for, or finalizing bft‑proposals: Because Πbc​ is live, there will always be some point in time at which a fresh valid bc‑header chain that satisfies both the Linearity rule and the Tail confirmation rule exists for use in the P.headers_bc field. If no fresh valid bc‑header chain is available, the Linearity rule and Tail confirmation rule allow an honest bft‑proposer to choose headers_bc to be the same as in the previous bft‑block. So, if liveness of Πorigbft​ depends on an honest proposer always being able to make a proposal (as it does in adapted‑Streamlet for example), then this requirement will not be violated. The changes to voting are only requiring a vote to be for a proposal that could have been honestly proposed. The bft‑finality rules are unchanged from origbft‑finality. Therefore, Πbft​ remains live under the same conditions as Πorigbft​. The only other possibility for a liveness issue in Crosslink 2 would be if the change to the constructions of fini​ or (baμ​)i​ could cause either of them to stall, even when Πbft​ and Πbc​ are both still live. However, liveness of Πbft​ and the Linearity rule together imply that at each point in time, provided there are sufficient honest bft‑proposers/validators, eventually a new bft‑block with a higher-scoring snapshot will become final in the context of the longest bft‑valid‑chain. TODO: make that more precise. Because of the Extension rule , this new bft‑block must be a descendent of the previous final bft‑block in the context visible to bc‑block‑producers. Therefore, the new finalized chain will extend the old finalized chain. Finally, we need to show that Stalled Mode is only triggered when it should be; that is, when the assumptions needed for liveness of Πbft​ are violated. Informally, that is the case because, as long as there are sufficient honest bc‑block‑producers and sufficient honest bft‑proposers/validators, the finalization point implied by the context_bft field at the tip of the bc‑best chain in any node’s view will advance fast enough for the finalization gap bound L not to be hit. This depends on the value of L relative to σ, the network delay, the hash rate of honest bc‑block‑producers, the number of honest bft‑proposers and the proportion of voting units they hold, and other details of the BFT protocol. TODO: more detailed argument needed, especially for the dependence on L.","breadcrumbs":"Design » Crosslink » Security Analysis of Crosslink 2 » Liveness argument","id":"92","title":"Liveness argument"},"93":{"body":"TODO: Not updated for Crosslink 2 below this point.","breadcrumbs":"Design » Crosslink » Security Analysis of Crosslink 2 » Safety argument","id":"93","title":"Safety argument"},"94":{"body":"Recall the definition of Assured Finality . Definition: Assured Finality An execution of Crosslink 2 has Assured Finality iff for all times t, u and all nodes i, j (potentially the same) such that i is honest at time t and j is honest at time u, we have finit​⪯⪰bc​finju​. First we prove that Assured Finality is implied by Prefix Agreement of Πbc​. Definition: Prefix Agreement An execution of Π∗bc​ has Prefix Agreement at confirmation depth σ, iff for all times t, u and all nodes i, j (potentially the same) such that i is honest at time t and j is honest at time u, we have chit​⌈∗bcσ​⪯⪰∗bc​chju​⌈∗bcσ​. Safety Theorem: Prefix Agreement of Πbc implies Assured Finality In an execution of Crosslink 2 for which the Πbc​ subprotocol has Prefix Agreement at confirmation depth σ, that execution has Assured Finality . Proof: Suppose that we have times t, u and nodes i, j (potentially the same) such that i is honest at time t and j is honest at time u. Then, by the Local fin-depth lemma applied to each of node i and node j, there exist times r at which node i is honest and r′ at which node j is honest, such that finit​⪯bc​chir​⌈bcσ​ and finju​⪯bc​chjr′​⌈bcσ​. By Prefix Agreement at confirmation depth σ , we have chir​⌈bcσ​⪯⪰bc​chjr′​⌈bcσ​. Wlog due to symmetry, suppose chir​⌈bcσ​⪯bc​chjr′​⌈bcσ​. Then finit​⪯bc​chjr′​⌈bcσ​ (by transitivity of ⪯bc​) and finju​⪯bc​chjr′​⌈bcσ​ (as above), so finit​⪯⪰bc​finju​ by the Linear prefix lemma . Then we prove that Assured Finality is also implied by Final Agreement of Πbft​. Definition: Final Agreement An execution of Π∗bft​ has Final Agreement iff for all ∗bft‑valid blocks C in honest view at time t and C′ in honest view at time t′, we have ∗bft-last-final(C)⪯⪰∗bft​∗bft-last-final(C′). Safety Theorem: Final Agreement of Πbft implies Assured Finality In an execution of Crosslink 2 for which the Πbft​ subprotocol has Final Agreement , that execution has Assured Finality . Proof: Suppose that we have times t, u and nodes i, j (potentially the same) such that i is honest at time t and j is honest at time u. Then, by the Local fin-depth lemma applied to each of node i and node j, there exist times r at which node i is honest and r′ at which node j is honest, such that finit​⪯bc​chir​⌈bcσ​ and finju​⪯bc​chjr′​⌈bcσ​. By Prefix Agreement at confirmation depth σ , we have chir​⌈bcσ​⪯⪰bc​chjr′​⌈bcσ​. Wlog due to symmetry, suppose chir​⌈bcσ​⪯bc​chjr′​⌈bcσ​. Then finit​⪯bc​chjr′​⌈bcσ​ (by transitivity of ⪯bc​) and finju​⪯bc​chjr′​⌈bcσ​ (as above), so finit​⪯⪰bc​finju​ by the Linear prefix lemma . Safety Theorem: Prefix Consistency of Πbc implies Prefix Consistency of ba By the Local ba-depth lemma , we have: In any execution of Crosslink 2, for any confirmation depth μ≤σ and any node i that is honest at time t, there exists a time r≤t such that (baμ​)it​⪯bc​chir​⌈bcμ​. Renaming t to r and σ to μ in the definition of Prefix Consistency gives: An execution of Πbc​ has Prefix Consistency at confirmation depth μ, iff for all times r≤u and all nodes i, j (potentially the same) such that i is honest at time r and j is honest at time u, we have that chir​⌈bcμ​⪯bc​chju​. Since any node i that is honest at time t is also honest at time r≤t, and by transitivity of ⪯bc​, we therefore have: In any execution of Crosslink 2 that has Prefix Consistency at confirmation depth μ, for all times t≤u and all nodes i, j (potentially the same) such that i is honest at time t and j is honest at time u, we have that (baμ​)it​⪯bc​chju​. ---- The Extension rule ensures that, informally, if a given node i’s view of its bc‑best‑chain at a depth of σ blocks does not roll back, then neither does its view of the bft‑final block referenced by its bc‑best‑chain, and therefore neither does its view of LOGfin,it​. This does not by itself imply that all nodes are seeing the “same” confirmed bc‑best‑chain (up to propagation timing), or the same LOGfin,it​. If the network is partitioned and Πbft​ is subverted, it could be that the nodes on each side of the partition follow a different fork, and the adversary arranges for each node’s view of the last final bft‑block to be consistent with the fork it is on. It can potentially do this if it has more than one third of validators, because if the validators are partitioned in the same way as other nodes, it can vote with an additional one third of them on each side of the fork. This is, if you think about it, unavoidable. Πbc​ doesn’t include the mechanisms needed to maintain finality under partition; it takes a different position on the CAP trilemma. In order to maintain finality under partition, we need Πbft​ not to be subverted (and to actually work!) So what is the strongest security property we can realistically get? It is stronger than what Snap‑and‑Chat provides. Snap‑and‑Chat is unsafe even without a partition if Πbft​ is subverted. Ideally we would have a protocol with safety that is only limited by attacks “like” the unavoidable attack described above — which also applies to Πbc​ used on its own.","breadcrumbs":"Design » Crosslink » Security Analysis of Crosslink 2 » Discussion","id":"94","title":"Discussion"},"95":{"body":"In order to capture the intuition that it is hard in practice to cause a consistent partition of the kind described in the previous section, we will need to assume that the Prefix Agreement safety property holds for the relevant executions of Πbc​. The structural and consensus modifications to Πbc​ relative to Πorigbc​ seem unlikely to have any significant effect on this property, given that we proved above that they do not affect liveness. ==TODO: that is a handwave; we should be able to do better, as we do for Πbft​ below.== So, to the extent that it is reasonable to assume that Prefix Agreement holds for executions of Πorigbc​ under some conditions, it should also be reasonable to assume it holds for executions of Πbc​ under the same conditions. Recall that LF(H):=bft-last-final(H.context_bft). Prefix Lemma If H1​, H2​ are bc‑valid blocks with H1​⪯bc​H2​, then LF(H1​)⪯bft​LF(H2​). Proof: Using the Extension rule , by induction on the distance between H1​ and H2​. Using the Prefix Lemma once for each direction, we can transfer the Prefix Agreement property to the referenced bft‑blocks: Prefix Agreement Lemma In an execution of Πbc​ that has Prefix Agreement at confirmation depth σ, for all times t, u and all nodes i, j (potentially the same) such that i is honest at time t and j is honest at time u, we have LF(chit​⌈bcσ​)⪯⪰bft​LF(chju​⌈bcσ​). Let chain-txns(ch) be the sequence of transactions in the given chain ch, starting from genesis. Recall that san-ctxsan-ctx(S)finfin(H)fin-ctx(H)ba-ctx(H,μ)LOGfin,it​LOGba,μ,it​​⦂:=⦂:=:=:=:=:=​[bc-chain]→bc-contextsanitize(concat([chain-txns(C) for C in S]))bc-block→[bc-chain][snapshot(B) for B⪯bft​LF(H⌈bcσ​)]san-ctx(fin(H))san-ctx(fin(H)∣∣[H⌈bcμ​])context-txns(fin-ctx(chit​))context-txns(ba-ctx(chit​,μ))​ Because fin takes the form fin(H):=[f(X) for X⪯bft​g(H)], we have that g(H)⪯bft​g(H′)⟹fin(H)⪯fin(H′). (This would be true for any well‑typed f and g.) By this observation and the Prefix Agreement Lemma , we also have that, in an execution of Crosslink 2 where Πbc​ has the Prefix Agreement safety property at confirmation depth σ, for all times t, u and all nodes i, j (potentially the same) such that i is honest at time t and j is honest at time u, fin(chit​)⪯⪰fin(chju​). Because sanitize only considers previous state, context-txns ∘ san-ctx must be a prefix-preserving map; that is, if S1​⪯S2​ then context-txns(san-ctx(S1​))⪯context-txns(san-ctx(S2​)). Therefore: Theorem: LOGfin Safety (from Prefix Agreement of Πbc) In an execution of Crosslink 2 where Πbc​ has Prefix Agreement at confirmation depth σ, for all times t, u and all nodes i, j (potentially the same) such that i is honest at time t and j is honest at time u, LOGfin,it​⪯⪰LOGfin,ju​. Notice that this does not depend on any safety property of Πbft​, and is an elementary proof. ([ NTT2020 , Theorem 2] is a much more complicated proof that takes nearly 3 pages, not counting the reliance on results from [PS2017] .) In addition, just as in Snap‑and‑Chat, safety of LOGfin​ can be inferred from safety of Πbft​, which follows from safety of Πorigbft​. We prove this based on the Final Agreement property for executions of Πorigbft​: Definition: Final Agreement An execution of Πorigbft​ has the Final Agreement safety property iff for all origbft‑valid blocks C in honest view at time t and C′ in honest view at time t′, we have origbft-last-final(C)⪯⪰origbft​origbft-last-final(C′). The changes in Πbft​ relative to Πorigbft​ only add structural components and tighten bft‑block‑validity and bft‑proposal‑validity rules. So for any legal execution of Πbft​ there is a corresponding legal execution of Πorigbft​, with the structural additions erased and with the same nodes honest at any given time. A safety property, by definition, only asserts that executions not satisfying the property do not occur. Safety properties of Πorigbft​ necessarily do not refer to the added structural components in Πbft​. Therefore, for any safety property of Πorigbft​, including Final Agreement , the corresponding safety property holds for Πbft​. By the definition of fin as above, in an execution of Crosslink 2 where Πbft​ has Final Agreement , for all times t, u and all nodes i, j (potentially the same) such that i is honest at time t and j is honest at time u, fin(chit​)⪯⪰fin(chju​). Therefore, by an argument similar to the one above using the fact that context-txns ∘ san-ctx is a prefix-preserving map: Theorem: LOGfin Safety (from Final Agreement of Πbft or Πorigbft) In an execution of Crosslink 2 where Πbft​ has Final Agreement , for all times t, u and all nodes i, j (potentially the same) such that i is honest at time t and j is honest at time u, LOGfin,it​⪯⪰LOGfin,ju​.","breadcrumbs":"Design » Crosslink » Security Analysis of Crosslink 2 » Proof of safety for LOGfin","id":"95","title":"Proof of safety for LOGfin"},"96":{"body":"From the two LOGfin​ Safety theorems and the Ledger prefix property , we immediately have: Theorem: LOGba does not roll back past the agreed LOGfin Let μi​ be an arbitrary choice of LOGba​ confirmation depth for each node i. Consider an execution of Crosslink 2 where either Πbc​ has Prefix Agreement at confirmation depth σ or Πbft​ has Final Agreement . In such an execution, for all times t, u and all nodes i, j (potentially the same) such that i is honest at time t and j is honest at time u, either LOGfin,it​⪯bc​LOGfin,ju​⪯bc​LOGba,μj​,ju​ or LOGfin,ju​⪯bc​LOGfin,it​⪯bc​LOGba,μi​,it​. Corollary: Under the same conditions, if wlog LOGfin,it​⪯bc​LOGfin,ju​, then LOGfin,it​⪯bc​{LOGba,μi​,it​,LOGba,μj​,ju​}. The above property is not as strong as we would like for practical uses of LOGba​, because it does not say anything about rollbacks up to the finalization point, and such rollbacks may be of unbounded length. (Loosely speaking, the number of non‑Stalled Mode bc‑blocks after the consensus finalization point is bounded by L, but we have also not proven that so far.) As documented in the Model for BFT protocols section of The Crosslink 2 Construction ): For each epoch, there is a fixed number of voting units distributed between the players, which they use to vote for a ∗bft‑proposal. We say that a voting unit has been cast for a ∗bft‑proposal P at a given time in a ∗bft‑execution, if and only if P is ∗bft‑proposal‑valid and a ballot for P authenticated by the holder of the voting unit exists at that time. Using knowledge of ballots cast for a ∗bft‑proposal P that collectively satisfy a notarization rule at a given time in a ∗bft‑execution, and only with such knowledge, it is possible to obtain a valid ∗bft‑notarization‑proof proofP​. The notarization rule must require at least a two‑thirds absolute supermajority of voting units in P’s epoch to have been cast for P. It may also require other conditions. A voting unit is cast non‑honestly for an epoch’s proposal iff: it is cast other than by the holder of the unit (due to key compromise or any flaw in the voting protocol, for example); or it is double‑cast (i.e. there are two ballots casting it for distinct proposals); or the holder of the unit following the conditions for honest voting in Π∗bft​, according to its view, should not have cast that vote. Definition: One‑third bound on non‑honest voting An execution of Πbft​ has the one‑third bound on non‑honest voting property iff for every epoch, strictly fewer than one third of the total voting units for that epoch are ever cast non‑honestly. Theorem: On bft‑valid blocks for a given epoch in honest view By a well known argument often used to prove safety of BFT protocols, in an execution of Crosslink 2 where Πbft​ has the one‑third bound on non‑honest voting property (and assuming soundness of notarization proofs), any bft‑valid block for a given epoch in honest view must commit to the same proposal. Proof (adapted from [ CS2020 , Lemma 1]): Suppose there are two bft‑proposals P and P′, both for epoch e, such that P is committed to by some bft‑block‑valid block B, and P′ is committed to by some bft‑block‑valid block B′. This implies that B and B′ have valid notarization proofs. Let the number of voting units for epoch e be ne​. Assuming soundness of the notarization proofs, it must be that at least 2ne​/3 voting units for epoch e, denoted as the set S, were cast for P, and at least 2ne​/3 voting units for epoch e, denoted as the set S′, were cast for P′. Since there are ne​ voting units for epoch e, S∩S′ must have size at least ne​/3. In an execution of Crosslink 2 where Πbft​ has the one‑third bound on non‑honest voting property, S∩S′ must therefore include at least one voting unit that was cast honestly. Since a voting unit for epoch e that is cast honestly is not double-cast, it must be that P=P′. Info In the case of a network partition, votes may not be seen on both/all sides of the partition. Therefore, it is not necessarily the case that honest nodes can directly see double‑voting. The above argument does not depend on being able to do so. Therefore, in an execution of Crosslink 2 for which Πbft​ has the one‑third bound on non‑honest voting property, for each epoch e there will be at most one bft‑proposal‑valid proposal Pe​, and at least one third of honestly cast voting units must have been cast for it. Let Ie​ be the (necessarily nonempty) set of nodes that cast these honest votes; then, snapshot(Pe​)⪯bc​chite,i​​⌈bcσ​ for all i∈Ie​ at the times te,i​ of their votes in epoch e. (For simplicity, we assume that for each honest node i there is only one time te,i​ at which it obtains a successful check for the voting condition in epoch e, which it uses for any votes that it casts in that epoch.) Let B be any bft‑block for epoch e such that B⪯bft​bft-last-final(C), where C is some bft‑block‑valid block. Since B⪯bft​C, B is bft‑block‑valid. So by the argument above, B commits to the only bft‑proposal‑valid proposal Pe​ for epoch e, and snapshot(B)=snapshot(Pe​) was voted for in that epoch by a nonempty subset of honest nodes Ie​. Let H be any bc‑valid block. We have by definition: fin(H)​==​[snapshot(B) for B⪯bft​LF(H⌈bcσ​)][snapshot(B) for B⪯bft​bft-last-final(H⌈bcσ​.context_bft)]​ So, taking C=H⌈bcσ​.context_bft, each snapshot(B) for B of epoch e in the result of fin(H) satisfies snapshot(B)⪯bc​chite,i​​⌈bcσ​ for all i in some nonempty honest set of nodes Ie​. For an execution of Crosslink 2 in which Πbc​ has the Prefix Consistency property at confirmation depth σ, for every epoch e, for every such (i,te,i​), for every node j that is honest at any time u≥te,i​, we have chite,i​​⌈bcσ​⪯bc​chju​. Let te​=min{te,i​:i∈Ie​}. Then, by transitivity of ⪯bc​: Theorem: On snapshots in LOGfin In an execution of Crosslink 2 where Πbft​ has the one‑third bound on non‑honest voting property and Πbc​ has the Prefix Consistency property at confirmation depth σ, every bc‑chain snapshot(B) in fin(chit​) (and therefore every snapshot that contributes to LOGfin,it​) is, at any time u≥te​, in the bc‑best‑chain of every node j that is honest at time u (where B commits to Pe​ at epoch e and te​ is the time of the first honest vote for Pe​). A similar (weaker) statement holds if we replace u≥te​ with u≥t, since the time of the first honest vote for P necessarily precedes the time at which the signed (P,proofP​) is submitted as a bft‑block, which necessarily precedes t. (Whether or not the notarization proof depends on the first honest vote for B’s proposal Pe​, it must depend on some honest vote for that proposal that was not made earlier than te​.) Furthermore, we have ba-ctx(H,μ)LOGba,μ,it​​==​san-ctx(fin(H)∣∣[H⌈bcμ​])context-txns(ba-ctx(chit​,μ))​ So in an execution of Crosslink 2 where Πbc​ has the Prefix Consistency property at confirmation depth μ, if node i is honest at time t then H⌈bcμ​ is also, at any time u≥t, in the bc‑best‑chain of every node j that is honest at time u. If an execution of Πbc​ has the Prefix Consistency property at confirmation depth μ≤σ, then it necessarily also has it at confirmation depth σ. Therefore we have: Theorem: On snapshots in LOGba In an execution of Crosslink 2 where Πbft​ has the one‑third bound on non‑honest voting property and Πbc​ has the Prefix Consistency property at confirmation depth μ≤σ, every bc‑chain snapshot in fin(chit​)∣∣[chit​⌈bcμ​] (and therefore every snapshot that contributes to LOGba,μ,it​) is, at any time u≥t, in the bc‑best‑chain of every node j that is honest at time u. Sketch: we also need the sequence of snapshots output from fin to only be extended in the view of any node. In that case we can infer that the node does not observe a rollback in LOG_ba. Recall that in the proof of safety for LOGfin​, we showed that in an execution of Crosslink 2 where Πbft​ (or Πorigbft​) has Final Agreement , for all times t, u and all nodes i, j (potentially the same) such that i is honest at time t and j is honest at time u, fin(chit​)⪯⪰fin(chju​). What we want to show is that, under some conditions on executions, ...","breadcrumbs":"Design » Crosslink » Security Analysis of Crosslink 2 » Proof of safety for LOGba","id":"96","title":"Proof of safety for LOGba"},"97":{"body":"","breadcrumbs":"Design » Crosslink » Security Analysis of Crosslink 2 » Disadvantages of Crosslink","id":"97","title":"Disadvantages of Crosslink"},"98":{"body":"Unlike Snap‑and‑Chat, Crosslink 2 requires structural and consensus rule changes to both Πbc​ and Πbft​. On the other hand, several of those changes are arguably necessary to fix a show‑stopping bug in Snap‑and‑Chat (not being able to spend some finalized outputs).","breadcrumbs":"Design » Crosslink » Security Analysis of Crosslink 2 » More invasive changes","id":"98","title":"More invasive changes"},"99":{"body":"For a given choice of σ, the finalization latency is higher. The snapshot of the BFT chain used to obtain LOGfin,μ,it​ is obtained from the block at depth μ on node i’s best Πbc​ chain, which will on average lead to a finalized view that is about μ+1+σ blocks back (in Πbc​), rather than \\sigma_{\\sac}} blocks in Snap‑and‑Chat. This is essentially the cost of ensuring that safety is given by the stronger of the safety of Πbc​ (at μ confirmations) and the safety of Πbft​. 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+                <ol class="chapter"><li class="chapter-item expanded "><a href="introduction.html"><strong aria-hidden="true">1.</strong> Introduction</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="introduction/a-path-to-pos-zcash.html"><strong aria-hidden="true">1.1.</strong> A Path to Proof-of-Stake Zcash</a></li><li class="chapter-item expanded "><a href="introduction/trailing-finality-layer-in-a-nutshell.html"><strong aria-hidden="true">1.2.</strong> Trailing Finality Layer in a Nutshell</a></li><li class="chapter-item expanded "><a href="introduction/status-and-next-steps.html"><strong aria-hidden="true">1.3.</strong> Status and Next Steps</a></li><li class="chapter-item expanded "><a href="introduction/motivating-finality.html"><strong aria-hidden="true">1.4.</strong> Motivating Finality</a></li><li class="chapter-item expanded "><a href="introduction/get-involved.html"><strong aria-hidden="true">1.5.</strong> Get Involved</a></li></ol></li><li class="chapter-item expanded "><a href="terminology.html" class="active"><strong aria-hidden="true">2.</strong> Terminology</a></li><li class="chapter-item expanded "><a href="design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/design-at-a-glance.html"><strong aria-hidden="true">3.1.</strong> Design at a Glance</a></li><li class="chapter-item expanded "><a href="design/goals.html"><strong aria-hidden="true">3.2.</strong> Design Goals</a></li><li class="chapter-item expanded "><a href="design/crosslink.html"><strong aria-hidden="true">3.3.</strong> Crosslink</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/crosslink/the-arguments-for-bounded-availability-and-finality-overrides.html"><strong aria-hidden="true">3.3.1.</strong> The Arguments for Bounded Availability and Finality Overrides</a></li><li class="chapter-item expanded "><a href="design/crosslink/notes-on-snap-and-chat.html"><strong aria-hidden="true">3.3.2.</strong> Notes On Snap‑and‑Chat</a></li><li class="chapter-item expanded "><a href="design/crosslink/construction.html"><strong aria-hidden="true">3.3.3.</strong> The Crosslink 2 Construction</a></li><li class="chapter-item expanded "><a href="design/crosslink/security-analysis.html"><strong aria-hidden="true">3.3.4.</strong> Security Analysis of Crosslink 2</a></li><li class="chapter-item expanded "><a href="design/crosslink/questions.html"><strong aria-hidden="true">3.3.5.</strong> Questions About Crosslink</a></li><li class="chapter-item expanded "><a href="design/crosslink/potential-changes.html"><strong aria-hidden="true">3.3.6.</strong> Potential Changes to Crosslink</a></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">3.4.</strong> Model Code Architecture</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">3.5.</strong> Network Architecture</div></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">4.</strong> Protocol Specification</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">4.1.</strong> Abstract Protocol</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">4.1.1.</strong> Token Dynamics</div></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">4.2.</strong> Concrete Protocol</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">4.2.1.</strong> Token Dynamics</div></li></ol></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.</strong> Security</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.</strong> Abstract Analysis</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.1.</strong> Ebb-and-Flow analysis</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.2.</strong> Subprotocol Compromise Analysis</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.3.</strong> Model Code Architecture Analysis</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.4.</strong> Network Architecture Analysis</div></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.2.</strong> Concrete Analysis</div></li></ol></li><li class="chapter-item expanded "><a href="references.html"><strong aria-hidden="true">6.</strong> References</a></li><li class="chapter-item expanded "><a href="version-history.html"><strong aria-hidden="true">7.</strong> Version History</a></li></ol>
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+<h1 id="terminology"><a class="header" href="#terminology">Terminology</a></h1>
+<p>This book relies on the following terminology. We strive to keep these definitions as precise as possible and their definitions may deviate from uses elsewhere.</p>
+<p>Definitions are sorted alphabetically.</p>
+<h2 id="terms"><a class="header" href="#terms">Terms</a></h2>
+<!-- Unlike true html, it seems `mdbook` requires the span tags to be empty and immediately pre-fix the intended anchor target. -->
+<p><span id="definition-assured-finality"></span>
+<strong>Assured Finality</strong>: A protocol property that assures that transactions cannot be reverted by that protocol. As with all protocol guarantees, a protocol assumes certain conditions must be met. A transaction may either be final or not: transactions which are not final may not become final, whereas once transactions do achieve finality they retain that property indefinitely (so long as protocol requirements are met).</p>
+<p>Importantly, it is not feasible for any protocol to prevent reversing final transactions &quot;out of band&quot; from the protocol, such as if a sufficiently large and motivated group of users forks the network to include a specific new validity rule reverting transactions. In some cases this might be desirable, for example to mitigate exploitation of a security flaw. We are investigating the implications for governance and how to incorporate such situations into our security model. In any case, for this reason we eschew the term &quot;absolute finality&quot; sometimes used in technical discussions about consensus protocols.</p>
+<p><span id="definition-consensus-subprotocols"></span>
+<strong>Consensus Subprotocols</strong>: The <a href="#definition-pow">PoW</a> and <a href="#defintion-pos">PoS</a> subprotocols in <a href="#definition-pow-tfl">PoW+TFL</a> or other <a href="#definition-hybrid-consensus">hybrid protocols</a>.</p>
+<p><span id="definition-crosslink"></span>
+<strong>Crosslink</strong>: A <a href="#definition-hybrid-construction">hybrid construction</a> consensus protocol striving to implement the <a href="#definition-tfl">TFL</a> design goals. See <a href="./introduction/status-and-next-steps.html#current-components">Status and Next Steps: Current Components</a> for current status.</p>
+<p><span id="definition-final"></span>
+<strong>Final</strong>: A protocol property of transactions. In this book, this always implies <a href="#definition-assured-finality">assured finality</a>, in contrast to concepts like &quot;probabilistic finality&quot; provided by <a href="#definition-pow">PoW</a>.</p>
+<p><span id="definition-hybrid-consensus"></span>
+<strong>Hybrid Consensus</strong>: A consensus protocol that integrates more than one consensus subprotocol. <a href="#definition-pow-tfl">PoW+TFL</a> is an instance of a hybrid protocol integrating <a href="#definition-pow">PoW</a> and <a href="#definition-pos">PoS</a> protocols.</p>
+<p><span id="definition-hybrid-construction"></span>
+<strong>Hybrid Construction</strong>: The design component of a <a href="#defintion-hybrid-consensus">hybrid consensus</a> which specifies how to integrate <a href="#definition-consensus-subprotocols">subprotocols</a> and what modifications, if any, those subprotocols need to be safely integrated. Examples include <a href="#definition-crosslink">Crosslink</a> and <a href="#definition-snap-and-chat">Snap-and-Chat</a>.</p>
+<p><span id="definition-liveness"></span>
+<strong>Liveness</strong>: The property of a distributed protocol which ensures that the protocol may progress provided liveness requirements are met. <strong>TODO:</strong> <a href="https://github.com/Electric-Coin-Company/tfl-book/issues/120">Fix the definition of Liveness #120</a></p>
+<p><span id="definition-nu5nu6"></span>
+<strong>NU5/NU6</strong>: The Zcash consensus protocol as of NU5 or NU6 (which do not differ significantly in terms of the base block chain layer).<sup class="footnote-reference"><a href="#new-mainnet-precursors">1</a></sup></p>
+<p><span id="definition-objective-validity"></span>
+<strong>Objective Validity</strong>: A validity property of a protocol history (such as a ledger) which can be computed purely from that history with no other context. Objective validity is needed to define consensus rules that will lead to the same protocol state being eventually agreed on by all nodes.</p>
+<p><span id="definition-pos"></span>
+<strong>Proof-of-Stake</strong>: A PoS protocol achieves consensus on transaction status by taking into account the weighting of staking tokens. PoS protocols exist under a large umbrella and may or may not provide <a href="#definition-assured-finality">assured finality</a> or other properties this design requires of <a href="#definition-tfl">TFL</a>.</p>
+<p><span id="definition-pow"></span>
+<strong>Proof‑of‑Work</strong>: A PoW protocol uses Nakamoto consensus pioneered by Bitcoin. The PoW subprotocol within PoW+TFL is a different consensus protocol from <a href="#definition-nu5nu6">NU5/NU6</a> and encompasses more than narrow Nakamoto PoW consensus, including transaction semantics such as for shielded transfers.</p>
+<p><span id="definition-pow-tfl"></span>
+<strong>PoW+TFL</strong>: the overall complete, integrated consensus protocol specified in this book.</p>
+<p><span id="definition-safety"></span>
+<strong>Safety</strong>: The property of a distributed protocol that guarantees a participant may safely rely on a consistent local state, provided safety requirements are met. <strong>TODO:</strong> <a href="https://github.com/Electric-Coin-Company/tfl-book/issues/121">Provide a rigorous definition of Safety #121</a></p>
+<p><span id="definition-simtfl"></span>
+<strong><code>simtfl</code></strong>: a protocol simulator for analyzing <a href="#definition-tfl">TFL</a> security and abstract performance. Development lives at <a href="https://github.com/zcash/simtfl">https://github.com/zcash/simtfl</a></span>. See <a href="./introduction/status-and-next-steps.html#current-components">Status and Next Steps: Current Components</a> for current status.</p>
+<p><span id="definition-snap-and-chat"></span>
+<strong>Snap-and-Chat</strong>: A <a href="#definition-hybrid-construction">hybrid construction</a> consensus protocol introduced in <a href="./references.html#ebb-and-flow-protocols">Ebb-and-Flow Protocols</a>.</p>
+<p><span id="definition-tfl"></span>
+<strong>TFL</strong>: The <em>Trailing Finality Layer</em> subprotocol within PoW+TFL. This is a new <a href="#definition-pos">PoS</a> subprotocol which provides <a href="#definition-assured-finality">assured finality</a> for Zcash.</p>
+<p><span id="definition-trailing-finality"></span>
+<strong>Trailing Finality</strong>: A protocol property wherein transactions become final some time after first appearing in <a href="#definition-pow">PoW</a> blocks.</p>
+<p><span id="definition-zip"></span>
+<strong>ZIP</strong>: a Zcash Improvement Proposal is the protocol development process the Zcash community uses to safely define potential protocol improvements. See <a href="https://zips.z.cash">https://zips.z.cash</a></span>.</p>
+<p><strong>TODO:</strong> <a href="https://github.com/Electric-Coin-Company/tfl-book/issues/119">Clarify the distinctions between pure PoW, the PoW subprotocol, NU6, and fork-choice vs all of transaction semantics. #119</a></p>
+<h1 id="footnotes"><a class="header" href="#footnotes">Footnotes</a></h1>
+<div class="footnote-definition" id="new-mainnet-precursors"><sup class="footnote-definition-label">1</sup>
+<p>If new consensus changes are deployed to Zcash mainnet prior to PoW+TFL design finalization, this design must be updated to refer to the new delta (e.g. by reanalyzing all changes against NU6 or NU7, etc…)</p>
+</div>
+<!-- This trailing whitespace ensures that readers who follow a link to a definition will always see that term at the top of their view. -->
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+                <ol class="chapter"><li class="chapter-item expanded "><a href="introduction.html"><strong aria-hidden="true">1.</strong> Introduction</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="introduction/a-path-to-pos-zcash.html"><strong aria-hidden="true">1.1.</strong> A Path to Proof-of-Stake Zcash</a></li><li class="chapter-item expanded "><a href="introduction/trailing-finality-layer-in-a-nutshell.html"><strong aria-hidden="true">1.2.</strong> Trailing Finality Layer in a Nutshell</a></li><li class="chapter-item expanded "><a href="introduction/status-and-next-steps.html"><strong aria-hidden="true">1.3.</strong> Status and Next Steps</a></li><li class="chapter-item expanded "><a href="introduction/motivating-finality.html"><strong aria-hidden="true">1.4.</strong> Motivating Finality</a></li><li class="chapter-item expanded "><a href="introduction/get-involved.html"><strong aria-hidden="true">1.5.</strong> Get Involved</a></li></ol></li><li class="chapter-item expanded "><a href="terminology.html"><strong aria-hidden="true">2.</strong> Terminology</a></li><li class="chapter-item expanded "><a href="design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/design-at-a-glance.html"><strong aria-hidden="true">3.1.</strong> Design at a Glance</a></li><li class="chapter-item expanded "><a href="design/goals.html"><strong aria-hidden="true">3.2.</strong> Design Goals</a></li><li class="chapter-item expanded "><a href="design/crosslink.html"><strong aria-hidden="true">3.3.</strong> Crosslink</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/crosslink/the-arguments-for-bounded-availability-and-finality-overrides.html"><strong aria-hidden="true">3.3.1.</strong> The Arguments for Bounded Availability and Finality Overrides</a></li><li class="chapter-item expanded "><a href="design/crosslink/notes-on-snap-and-chat.html"><strong aria-hidden="true">3.3.2.</strong> Notes On Snap‑and‑Chat</a></li><li class="chapter-item expanded "><a href="design/crosslink/construction.html"><strong aria-hidden="true">3.3.3.</strong> The Crosslink 2 Construction</a></li><li class="chapter-item expanded "><a href="design/crosslink/security-analysis.html"><strong aria-hidden="true">3.3.4.</strong> Security Analysis of Crosslink 2</a></li><li class="chapter-item expanded "><a href="design/crosslink/questions.html"><strong aria-hidden="true">3.3.5.</strong> Questions About Crosslink</a></li><li class="chapter-item expanded "><a href="design/crosslink/potential-changes.html"><strong aria-hidden="true">3.3.6.</strong> Potential Changes to Crosslink</a></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">3.4.</strong> Model Code Architecture</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">3.5.</strong> Network Architecture</div></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">4.</strong> Protocol Specification</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">4.1.</strong> Abstract Protocol</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">4.1.1.</strong> Token Dynamics</div></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">4.2.</strong> Concrete Protocol</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">4.2.1.</strong> Token Dynamics</div></li></ol></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.</strong> Security</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.</strong> Abstract Analysis</div></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.1.</strong> Ebb-and-Flow analysis</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.2.</strong> Subprotocol Compromise Analysis</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.3.</strong> Model Code Architecture Analysis</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.1.4.</strong> Network Architecture Analysis</div></li></ol></li><li class="chapter-item expanded "><div><strong aria-hidden="true">5.2.</strong> Concrete Analysis</div></li></ol></li><li class="chapter-item expanded "><a href="references.html"><strong aria-hidden="true">6.</strong> References</a></li><li class="chapter-item expanded "><a href="version-history.html" class="active"><strong aria-hidden="true">7.</strong> Version History</a></li></ol>
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+<h1 id="version-history"><a class="header" href="#version-history">Version History</a></h1>
+<h2 id="020---crosslink-2"><a class="header" href="#020---crosslink-2">0.2.0 - Crosslink 2</a></h2>
+<p>This version updated the <a href="./design/crosslink/construction.html">Crosslink construction</a> and its <a href="./design/crosslink/security-analysis.html">security analysis</a> to version 2.</p>
+<h2 id="010---introducing-crosslink"><a class="header" href="#010---introducing-crosslink">0.1.0 - Introducing Crosslink</a></h2>
+<p><a href="https://github.com/Electric-Coin-Company/tfl-book/milestone/1?closed=1">Issue Tracking</a></p>
+<p>This version introduced the <a href="./terminology.html#crosslink">Crosslink construction</a> by linking to an external write-up. It transitioned away from the implicit design presented at Zcon4 to a rigorously defined <a href="./terminology.html#definition-hybrid-construction">hybrid construction</a>.</p>
+<h2 id="prehistory---zcon4-presentation"><a class="header" href="#prehistory---zcon4-presentation">Prehistory - Zcon4 Presentation</a></h2>
+<p>Prior to the initial release of this book, a design sketch was presented in <a href="https://www.youtube.com/watch?v=qhMzMYeEPMM&amp;list=PL40dyJ0UYTLII7oQRQmNOFf0d2iKT35tL&amp;index=19">Interactive Design of a Zcash Trailing Finality Layer (Hybrid PoW+PoS) by Nate Wilcox: Zcon4</a>.</p>
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