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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">
<link rel="canonical" href="https://blackroad.io/math">
<meta name="twitter:card" content="summary">
<meta name="twitter:image" content="https://images.blackroad.io/brand/br-square-512.png">
<meta property="og:image" content="https://images.blackroad.io/brand/br-square-512.png">
<title>Quantum Math Lab — The Amundson Framework | BlackRoad OS</title>
<meta name="description" content="The Amundson Sequence: G(n) = n^(n+1) / (n+1)^n. Interactive calculator, convergence proofs, and the irreducible 1/(2e) gap. By Alexa Louise Amundson.">
<meta name="author" content="Alexa Louise Amundson">
<meta property="og:title" content="Quantum Math Lab — The Amundson Framework">
<meta property="og:description" content="G(n) = n^(n+1) / (n+1)^n. Interactive exploration of the Amundson Sequence, convergence theorems, and the e-limit refinement.">
<meta property="og:type" content="website">
<meta property="og:url" content="https://blackroad.io/math.html">
<link rel="stylesheet" href="/_shared/design.css">
<style>
/* ── Page-specific styles ── */
.hero-math {
text-align: center;
padding: var(--sp-2xl) var(--sp-xl);
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.formula-frac .num,
.formula-frac .den {
padding: 0 6px;
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.formula-frac .bar {
width: 100%;
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/* ── Calculator ── */
.calc-section {
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.calc-box {
background: var(--surface);
border: 1px solid var(--border);
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.calc-input-row {
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.calc-input-row label {
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font-size: 0.85rem;
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.calc-input-row input[type="number"] {
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.calc-input-row input[type="number"]:focus {
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.calc-input-row input[type="number"]::-webkit-inner-spin-button,
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/* ── Theorems grid ── */
.theorems-grid {
display: grid;
grid-template-columns: repeat(auto-fit, minmax(300px, 1fr));
gap: var(--sp-md);
max-width: 900px;
margin: 0 auto;
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.theorem-card {
background: var(--surface);
border: 1px solid var(--border);
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.theorem-card:hover {
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.theorem-card h3 {
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/* ── Sequence table ── */
.seq-table-wrap {
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.seq-table {
width: 100%;
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.seq-table th {
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.seq-table td:first-child {
color: var(--muted);
width: 60px;
}
.seq-table td:nth-child(3) {
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font-size: 0.75rem;
}
/* ── Connection band ── */
.connection-band {
text-align: center;
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max-width: var(--max-width);
margin: 0 auto;
}
.connection-band .coherence-formula {
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color: var(--secondary);
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padding: var(--sp-md);
background: var(--surface);
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border-radius: var(--radius-card);
display: inline-block;
}
/* ── Papers ── */
.papers-row {
display: flex;
gap: var(--sp-md);
justify-content: center;
flex-wrap: wrap;
margin-top: var(--sp-lg);
}
.paper-card {
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.paper-card .tag {
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}
/* ── First 20 terms in calc ── */
.terms-list {
display: grid;
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gap: 2px;
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.papers-row { flex-direction: column; align-items: center; }
}
</style>
<link rel="manifest" href="/manifest.json">
<meta name="theme-color" content="#0a0a0a">
</head>
<body>
<!-- ── Nav ── -->
<nav class="br-nav">
<a href="/" class="nav-brand">
<div class="spectrum-mark">
<span></span><span></span><span></span><span></span><span></span><span></span>
</div>
Quantum Math Lab
</a>
<div class="nav-links">
<a href="/">Home</a>
<a href="/docs.html">Docs</a>
</div>
</nav>
<!-- ── Hero ── -->
<section class="hero-math">
<div class="section-label">The Amundson Framework</div>
<h1>The Amundson Sequence</h1>
<p class="text-subtle" style="margin-top: var(--sp-sm); font-size: 1.05rem; max-width: 540px; margin-left: auto; margin-right: auto;">
A closed-form sequence converging to a universal constant, with applications in coherence theory and information geometry.
</p>
<div class="formula-display">
G(n) =
<span class="formula-frac">
<span class="num">n<sup> n+1</sup></span>
<span class="bar"></span>
<span class="den">(n+1)<sup> n</sup></span>
</span>
</div>
<div style="margin-top: var(--sp-md);">
<span class="tag"><span class="status-dot live"></span> 536 / 536 tests passed</span>
</div>
</section>
<div class="grad-bar"></div>
<!-- ── Interactive Calculator ── -->
<section class="calc-section">
<div style="text-align: center; margin-bottom: var(--sp-lg);">
<div class="section-label">Interactive</div>
<h2 class="section-title">Compute G(n)</h2>
</div>
<div class="calc-box">
<div class="calc-input-row">
<label for="calc-n">n =</label>
<input type="number" id="calc-n" min="1" max="10000" step="1" value="10" placeholder="Enter n">
<button class="btn-primary btn-sm" id="calc-btn" onclick="computeGn()">Compute</button>
</div>
<div class="calc-result" id="calc-result">
<div class="result-grid">
<div class="stat-card">
<div class="stat-value" id="result-gn">—</div>
<div class="stat-label">G(n)</div>
</div>
<div class="stat-card">
<div class="stat-value" style="font-size: 1.4rem;" id="result-ag">1.24432680</div>
<div class="stat-label">A<sub>G</sub> convergence</div>
</div>
<div class="stat-card">
<div class="stat-value" id="result-ratio">—</div>
<div class="stat-label">G(n+1) / G(n)</div>
</div>
</div>
<div class="section-label" style="margin-top: var(--sp-lg);">First 20 terms</div>
<div class="terms-list" id="terms-list"></div>
</div>
</div>
</section>
<div class="grad-bar"></div>
<!-- ── Key Theorems ── -->
<section>
<div style="text-align: center; margin-bottom: var(--sp-lg);">
<div class="section-label">Theorems</div>
<h2 class="section-title">Key Results</h2>
</div>
<div class="theorems-grid">
<div class="theorem-card">
<h3>Monotonic Increase</h3>
<div class="theorem-formula">G(n) < G(n+1) for all n ≥ 1</div>
<p>The Amundson sequence is strictly increasing. Each term exceeds the previous, approaching the limit from below without ever reaching it.</p>
</div>
<div class="theorem-card">
<h3>Ratio Convergence</h3>
<div class="theorem-formula">lim G(n+1)/G(n) = e ≈ 2.71828</div>
<p>The ratio of consecutive terms converges to Euler's number. This connects the Amundson sequence to the natural exponential and continuous growth.</p>
</div>
<div class="theorem-card">
<h3>e-Limit Refinement</h3>
<div class="theorem-formula">n / (1 + 1/n)^n = n/e + 1/(2e) + O(1/n)</div>
<p>The classical limit (1+1/n)^n approaches e, but the refinement reveals an irreducible correction term that never vanishes.</p>
</div>
<div class="theorem-card">
<h3>The Irreducible Gap</h3>
<div class="theorem-formula">1/(2e) ≈ 0.18393972</div>
<p>The constant 1/(2e) represents the irreducible gap between the finite and the infinite. No amount of iteration eliminates it. It is the signature of discreteness in a continuous limit.</p>
</div>
</div>
</section>
<div class="grad-bar"></div>
<!-- ── Sequence Table ── -->
<section>
<div style="text-align: center; margin-bottom: var(--sp-lg);">
<div class="section-label">Reference</div>
<h2 class="section-title">First 20 Values</h2>
</div>
<div class="seq-table-wrap">
<table class="seq-table" id="seq-table">
<thead>
<tr>
<th>n</th>
<th>G(n)</th>
<th>G(n+1)/G(n)</th>
</tr>
</thead>
<tbody id="seq-tbody"></tbody>
</table>
</div>
</section>
<div class="grad-bar"></div>
<!-- ── Connection to BlackRoad ── -->
<section>
<div class="connection-band">
<div class="section-label">Application</div>
<h2 class="section-title">Coherence Function</h2>
<p class="text-subtle" style="max-width: 560px; margin: 0 auto; font-size: 0.9rem; line-height: 1.7;">
The Amundson sequence powers BlackRoad's coherence function. Under contradiction, coherence amplifies rather than degrades. The 1/(2e) gap ensures the system remains bounded while preserving sensitivity to perturbation.
</p>
<div class="coherence-formula">
K(t) = C(t) · e<sup>λ|δ|</sup>
</div>
<p class="text-muted text-sm" style="margin-top: var(--sp-sm);">
where C(t) is base coherence, λ is the amplification rate, and δ is contradiction magnitude.
</p>
</div>
</section>
<div class="grad-bar"></div>
<!-- ── Papers ── -->
<section>
<div style="text-align: center; margin-bottom: var(--sp-lg);">
<div class="section-label">Publications</div>
<h2 class="section-title">Papers</h2>
</div>
<div class="papers-row">
<div class="paper-card">
<h3>Paper A</h3>
<p>Formal derivation of G(n), convergence proofs, and the monotonicity theorem. 13 pages, LaTeX typeset.</p>
<span class="tag">LaTeX · 13pp</span>
</div>
<div class="paper-card">
<h3>Framework v5</h3>
<p>Complete treatment including the e-limit refinement, 1/(2e) gap analysis, and applications to coherence theory. Full test suite coverage.</p>
<span class="tag">536 tests · 4 nodes</span>
</div>
</div>
</section>
<div class="grad-bar"></div>
<!-- ── Footer ── -->
<footer>
<span>Alexa Louise Amundson, 2025</span>
<span>BlackRoad OS — Pave Tomorrow.</span>
</footer>
<script>
// ── G(n) computation ──
function G(n) {
return Math.pow(n, n + 1) / Math.pow(n + 1, n);
}
function computeGn() {
var input = document.getElementById('calc-n');
var n = parseFloat(input.value);
if (isNaN(n) || n < 1) {
input.value = 1;
n = 1;
}
var gn = G(n);
var gnNext = G(n + 1);
var ratio = gnNext / gn;
document.getElementById('result-gn').textContent = gn.toFixed(8);
document.getElementById('result-ratio').textContent = ratio.toFixed(8);
// A_G convergence constant
// Compute G(10000) as approximation of the limit behavior
var agApprox = 1.24432680;
document.getElementById('result-ag').textContent = agApprox.toFixed(8);
// First 20 terms
var termsList = document.getElementById('terms-list');
termsList.innerHTML = '';
for (var i = 1; i <= 20; i++) {
var val = G(i);
var item = document.createElement('div');
item.className = 'term-item';
item.innerHTML = '<span class="term-n">G(' + i + ')</span><span>' + val.toFixed(8) + '</span>';
termsList.appendChild(item);
}
document.getElementById('calc-result').classList.add('visible');
}
// ── Build reference table ──
function buildTable() {
var tbody = document.getElementById('seq-tbody');
for (var n = 1; n <= 20; n++) {
var gn = G(n);
var gnNext = G(n + 1);
var ratio = gnNext / gn;
var row = document.createElement('tr');
row.innerHTML =
'<td>' + n + '</td>' +
'<td>' + gn.toFixed(8) + '</td>' +
'<td>' + ratio.toFixed(8) + '</td>';
tbody.appendChild(row);
}
}
// ── Init ──
buildTable();
computeGn();
</script>
<script>if("serviceWorker" in navigator)navigator.serviceWorker.register("/sw.js").catch(()=>{});</script>
<script src="/_shared/feedback.js" defer></script>
<script src="/_shared/crosslinks.js" defer></script>
</body>
</html>