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<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="UTF-8">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<title>Ruliad Lambda — Machine-Verified λ-Calculus Proofs</title>
<style>
* { margin: 0; padding: 0; box-sizing: border-box; }
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font-family: 'Courier New', monospace;
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</style>
</head>
<body>
<div class="header">
<h1>RULIAD LAMBDA</h1>
<div class="subtitle">Machine-Verified Proofs for λ-Calculus Ruliology</div>
<div class="badges">
<span class="badge">Lean 4.14.0</span>
<span class="badge badge-proven">Church-Rosser ✓</span>
<span class="badge badge-proven">sorry: 0</span>
<span class="badge">10,000+ lines</span>
</div>
</div>
<div class="container">
<div class="section">
<h2>Interactive Visualizations</h2>
<div class="viz-grid">
<div class="viz-card">
<img src="visualizations/church_rosser_diamond.svg" alt="Church-Rosser Diamond">
<div class="viz-card-content">
<h3>Church-Rosser Diamond</h3>
<p>The fundamental confluence property: all paths converge.</p>
<a href="visualizations/church_rosser_diamond.svg">View Full</a>
</div>
</div>
<div class="viz-card">
<img src="visualizations/multiway_confluence.svg" alt="Multiway Confluence">
<div class="viz-card-content">
<h3>Multiway β-Reduction</h3>
<p>Branching and merging in the multiway system.</p>
<a href="visualizations/multiway_confluence.svg">View Full</a>
</div>
</div>
<div class="viz-card">
<img src="visualizations/multiway_size6.svg" alt="Size-6 Graph">
<div class="viz-card-content">
<h3>Size-6 Term Graph</h3>
<p>Complete multiway graph for a size-6 λ-term.</p>
<a href="visualizations/multiway_size6.svg">View Full</a>
</div>
</div>
<div class="viz-card">
<img src="visualizations/ruliad_branchial.svg" alt="Branchial Graph">
<div class="viz-card-content">
<h3>Branchial Structure</h3>
<p>Simultaneity surfaces in the Ruliad slice.</p>
<a href="visualizations/ruliad_branchial.svg">View Full</a>
</div>
</div>
</div>
</div>
<div class="section">
<h2>3D Interactive Explorations</h2>
<div class="viz-grid">
<div class="viz-card">
<div style="height:200px;background:#1a0033;display:flex;align-items:center;justify-content:center;border-bottom:1px solid #3c096c;">
<span style="font-size:3em;">🔗</span>
</div>
<div class="viz-card-content">
<h3>Proof Dependency Graph</h3>
<p>Interactive 3D visualization of theorem dependencies leading to Church-Rosser.</p>
<a href="visualizations/proof_dependencies.html">Launch 3D View</a>
</div>
</div>
<div class="viz-card">
<div style="height:200px;background:#1a0033;display:flex;align-items:center;justify-content:center;border-bottom:1px solid #3c096c;">
<span style="font-size:3em;">λ</span>
</div>
<div class="viz-card-content">
<h3>Lambda Multiway Explorer</h3>
<p>Interactive 3D graph showing term reductions and confluence.</p>
<a href="visualizations/ruliad_lambda_3d.html">Launch 3D View</a>
</div>
</div>
</div>
</div>
<div class="section">
<h2>Key Theorems</h2>
<ul class="theorem-list">
<li>
<span class="theorem-name">Steps.churchRosser</span>
<span class="theorem-status"> ✓ PROVEN</span>
<div class="theorem-desc">Church-Rosser theorem: β-reduction is confluent</div>
</li>
<li>
<span class="theorem-name">Bridge.ofComb_simulates_step_joinable</span>
<span class="theorem-status"> ✓ PROVEN</span>
<div class="theorem-desc">SK combinator steps are simulated by joinable β-reductions</div>
</li>
<li>
<span class="theorem-name">Par.diamond</span>
<span class="theorem-status"> ✓ PROVEN</span>
<div class="theorem-desc">Parallel reduction has the diamond property</div>
</li>
<li>
<span class="theorem-name">stepEdgesList_sound</span>
<span class="theorem-status"> ✓ PROVEN</span>
<div class="theorem-desc">Multiway enumeration is sound</div>
</li>
<li>
<span class="theorem-name">stepEdgesList_complete</span>
<span class="theorem-status"> ✓ PROVEN</span>
<div class="theorem-desc">Multiway enumeration is complete</div>
</li>
</ul>
</div>
<div class="section">
<h2>Documentation</h2>
<div class="links-grid">
<a href="https://github.com/Abraxas1010/ruliad-lambda" class="link-card">
<h4>GitHub Repository</h4>
<p>Full source code, build instructions, and README</p>
</a>
<a href="docs/CLAIMS_AND_PROOFS.md" class="link-card">
<h4>Claims & Proofs</h4>
<p>Mapping Wolfram's claims to Lean theorems</p>
</a>
<a href="docs/NOTEBOOK_STYLE_README.md" class="link-card">
<h4>Notebook Narrative</h4>
<p>Computational essay with embedded visualizations</p>
</a>
<a href="https://writings.stephenwolfram.com/2025/09/the-ruliology-of-lambdas/" class="link-card">
<h4>Wolfram's Article</h4>
<p>"The Ruliology of Lambdas" (2025)</p>
</a>
</div>
</div>
<div class="section">
<h2>The Ruliad Aesthetic</h2>
<p style="text-align:center;margin-bottom:20px;">Neon purple palette inspired by Wolfram's Ruliad visualizations</p>
<div class="color-palette">
<div class="color-swatch" style="background:#9D4EDD;" title="#9D4EDD"></div>
<div class="color-swatch" style="background:#7B2CBF;" title="#7B2CBF"></div>
<div class="color-swatch" style="background:#E0AAFF;" title="#E0AAFF"></div>
<div class="color-swatch" style="background:#C77DFF;" title="#C77DFF"></div>
<div class="color-swatch" style="background:#3c096c;" title="#3c096c"></div>
<div class="color-swatch" style="background:#10002B;" title="#10002B"></div>
</div>
</div>
</div>
<div class="footer">
<p><strong>Exploring the Ruliad, one β-reduction at a time.</strong></p>
<p style="margin-top:10px;">
<a href="https://github.com/Abraxas1010/ruliad-lambda">GitHub</a> •
<a href="https://writings.stephenwolfram.com/2025/09/the-ruliology-of-lambdas/">Wolfram's Article</a> •
MIT License
</p>
<img src="visualizations/ruliad_footer.svg" alt="Ruliad Pattern" width="300" style="margin-top:20px;opacity:0.7;">
</div>
</body>
</html>