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<!doctype html>
<html lang="en">
<head>
<meta charset="UTF-8" />
<!-- Favicons / site icon -->
<link rel="icon" type="image/png" href="../../icon.png" />
<link rel="icon" type="image/png" sizes="16x16" href="../../icon.png" />
<link rel="icon" type="image/png" sizes="32x32" href="../../icon.png" />
<link rel="apple-touch-icon" sizes="180x180" href="../../icon.png" />
<link rel="shortcut icon" href="../../icon.png" />
<meta name="viewport" content="width=device-width, initial-scale=1.0" />
<meta
name="description"
content="Rational Certificate Complexity: A computational taxonomy of mathematical constants based on bit-length cost of convergence engines."
/>
<meta name="author" content="" />
<title>
Rational Certificate Complexity: A Computational Taxonomy of Mathematical Constants
</title>
<!-- CSS will be added here -->
<style>
/* ============================================ */
/* CSS RESET & BASE STYLES */
/* ============================================ */
*,
*::before,
*::after {
box-sizing: border-box;
margin: 0;
padding: 0;
}
:root {
/* Color palette - academic, mathematical */
--color-bg: #fafaf7;
--color-bg-elevated: #ffffff;
--color-bg-dark: #1a1d2e;
--color-ink: #1c1f2e;
--color-ink-muted: #4a4f66;
--color-ink-faint: #7a809a;
--color-border: #e4e2dc;
--color-border-strong: #c8c5b8;
/* Accent colors */
--color-accent: #6b4ea8; /* deep violet - mathematical */
--color-accent-hover: #553d8a;
--color-accent-light: #ede8f7;
--color-gold: #b8893a; /* algebraic/optimal highlight */
--color-gold-light: #f7efd9;
--color-rust: #b85c3a; /* transcendental/expensive */
--color-teal: #2d7a7a; /* framework accent */
/* Typography */
--font-serif:
'Iowan Old Style', 'Palatino Linotype', Palatino, 'URW Palladio L', Georgia, serif;
--font-sans: -apple-system, BlinkMacSystemFont, 'Inter', 'Segoe UI', Roboto, sans-serif;
--font-mono: 'JetBrains Mono', 'SF Mono', Menlo, Consolas, 'Courier New', monospace;
/* Spacing */
--space-xs: 0.5rem;
--space-sm: 1rem;
--space-md: 1.5rem;
--space-lg: 2.5rem;
--space-xl: 4rem;
--space-2xl: 6rem;
/* Layout */
--content-width: 720px;
--content-width-wide: 980px;
/* Effects */
--shadow-sm: 0 1px 2px rgba(28, 31, 46, 0.05);
--shadow-md: 0 4px 16px rgba(28, 31, 46, 0.08);
--shadow-lg: 0 12px 40px rgba(28, 31, 46, 0.12);
--radius-sm: 4px;
--radius-md: 8px;
--radius-lg: 12px;
--transition: 200ms ease;
}
html {
scroll-behavior: smooth;
font-size: 17px;
}
body.article-page {
font-family: var(--font-serif);
line-height: 1.7;
color: var(--color-ink);
background: var(--color-bg);
-webkit-font-smoothing: antialiased;
-moz-osx-font-smoothing: grayscale;
}
/* ============================================ */
/* HEADER / NAVIGATION */
/* ============================================ */
.site-header {
position: sticky;
top: 0;
z-index: 100;
background: rgba(250, 250, 247, 0.92);
backdrop-filter: saturate(180%) blur(12px);
-webkit-backdrop-filter: saturate(180%) blur(12px);
border-bottom: 1px solid var(--color-border);
}
.site-nav {
max-width: var(--content-width-wide);
margin: 0 auto;
padding: 0.875rem 1.5rem;
display: flex;
align-items: center;
justify-content: space-between;
gap: 1rem;
font-family: var(--font-sans);
}
.nav-brand .brand-link {
font-weight: 700;
font-size: 1rem;
color: var(--color-ink);
text-decoration: none;
letter-spacing: -0.01em;
border-bottom: 2px solid var(--color-accent);
padding-bottom: 1px;
}
.nav-menu {
display: flex;
list-style: none;
gap: 0.25rem;
flex-wrap: wrap;
}
.nav-link {
display: inline-block;
padding: 0.4rem 0.75rem;
font-size: 0.85rem;
color: var(--color-ink-muted);
text-decoration: none;
border-radius: var(--radius-sm);
transition:
background var(--transition),
color var(--transition);
}
.nav-link:hover {
background: var(--color-accent-light);
color: var(--color-accent);
}
/* ============================================ */
/* HERO SECTION */
/* ============================================ */
.hero-section {
background: linear-gradient(180deg, #fafaf7 0%, #f3f1ea 100%);
padding: var(--space-2xl) 1.5rem var(--space-xl);
border-bottom: 1px solid var(--color-border);
position: relative;
overflow: hidden;
}
.hero-section::before {
content: 'Σ';
position: absolute;
top: -3rem;
right: -2rem;
font-size: 28rem;
font-family: var(--font-serif);
color: rgba(107, 78, 168, 0.04);
line-height: 1;
pointer-events: none;
user-select: none;
}
.hero-content {
max-width: var(--content-width);
margin: 0 auto;
position: relative;
z-index: 1;
}
.article-eyebrow {
font-family: var(--font-sans);
font-size: 0.8rem;
font-weight: 600;
text-transform: uppercase;
letter-spacing: 0.12em;
color: var(--color-accent);
margin-bottom: var(--space-md);
}
.article-title {
font-size: clamp(2.25rem, 5vw, 3.75rem);
font-weight: 700;
line-height: 1.1;
letter-spacing: -0.02em;
color: var(--color-ink);
margin-bottom: var(--space-sm);
}
.article-subtitle {
font-size: clamp(1.1rem, 2.2vw, 1.4rem);
font-style: italic;
color: var(--color-ink-muted);
font-weight: 400;
margin-bottom: var(--space-lg);
line-height: 1.4;
}
.article-meta {
font-family: var(--font-sans);
font-size: 0.85rem;
color: var(--color-ink-faint);
display: flex;
gap: 0.5rem;
align-items: center;
flex-wrap: wrap;
}
.meta-separator {
color: var(--color-border-strong);
}
/* ============================================ */
/* MAIN ARTICLE */
/* ============================================ */
.article-main {
max-width: var(--content-width);
margin: 0 auto;
padding: var(--space-xl) 1.5rem;
}
.article-section {
margin-bottom: var(--space-2xl);
scroll-margin-top: 5rem;
}
.section-heading {
font-size: clamp(1.6rem, 3.2vw, 2.1rem);
font-weight: 700;
line-height: 1.25;
letter-spacing: -0.015em;
color: var(--color-ink);
margin-bottom: var(--space-lg);
padding-bottom: var(--space-sm);
border-bottom: 2px solid var(--color-border);
position: relative;
}
.section-heading::before {
content: '';
position: absolute;
bottom: -2px;
left: 0;
width: 3.5rem;
height: 2px;
background: var(--color-accent);
}
.subsection {
margin-top: var(--space-lg);
margin-bottom: var(--space-lg);
}
.subsection-heading {
font-size: 1.3rem;
font-weight: 700;
color: var(--color-ink);
margin-bottom: var(--space-md);
letter-spacing: -0.01em;
}
.section-paragraph {
margin-bottom: var(--space-md);
color: var(--color-ink);
font-size: 1.05rem;
}
.lead-paragraph {
font-size: 1.18rem;
line-height: 1.65;
color: var(--color-ink);
}
.section-paragraph em {
font-style: italic;
color: var(--color-accent);
}
.section-paragraph strong {
font-weight: 700;
}
/* ============================================ */
/* MATH BLOCKS */
/* ============================================ */
.math-block {
background: var(--color-bg-dark);
color: #f3f1ea;
padding: 1.1rem 1.4rem;
border-radius: var(--radius-md);
overflow-x: auto;
margin: var(--space-md) 0;
border-left: 3px solid var(--color-accent);
box-shadow: var(--shadow-sm);
}
.math-code {
font-family: var(--font-mono);
font-size: 0.95rem;
line-height: 1.6;
color: #f3f1ea;
white-space: pre;
}
.highlight-math {
border-left-color: var(--color-gold);
background: linear-gradient(135deg, #1a1d2e 0%, #2b2238 100%);
}
/* ============================================ */
/* LISTS */
/* ============================================ */
.example-list,
.feature-list,
.conclusion-list,
.finding-list {
list-style: none;
margin: var(--space-md) 0;
padding-left: 0;
}
.example-item,
.feature-item,
.conclusion-item,
.finding-item {
padding: 0.5rem 0 0.5rem 1.5rem;
position: relative;
font-size: 1.02rem;
line-height: 1.6;
}
.example-item::before,
.feature-item::before,
.conclusion-item::before,
.finding-item::before {
content: '▸';
position: absolute;
left: 0;
color: var(--color-accent);
font-weight: bold;
}
.example-item {
font-family: var(--font-mono);
font-size: 0.95rem;
}
.numbered-list,
.component-list {
list-style: none;
counter-reset: item;
margin: var(--space-md) 0;
padding-left: 0;
}
.numbered-item,
.component-item {
counter-increment: item;
padding: 0.6rem 0 0.6rem 2.5rem;
position: relative;
font-size: 1.02rem;
line-height: 1.6;
}
.numbered-item::before,
.component-item::before {
content: counter(item);
position: absolute;
left: 0;
top: 0.6rem;
width: 1.75rem;
height: 1.75rem;
background: var(--color-accent);
color: white;
border-radius: 50%;
display: flex;
align-items: center;
justify-content: center;
font-family: var(--font-sans);
font-size: 0.8rem;
font-weight: 700;
}
.component-name {
color: var(--color-accent);
}
/* Check list */
.check-list {
list-style: none;
margin: var(--space-md) 0;
padding-left: 0;
}
.check-item {
padding: 0.6rem 0 0.6rem 2rem;
position: relative;
font-size: 1.02rem;
line-height: 1.6;
}
.check-item.check-yes::before {
content: '✓';
position: absolute;
left: 0;
color: var(--color-teal);
font-weight: bold;
font-size: 1.2rem;
}
.check-item.check-no::before {
content: '✗';
position: absolute;
left: 0;
color: var(--color-rust);
font-weight: bold;
font-size: 1.2rem;
}
/* ============================================ */
/* CALLOUTS */
/* ============================================ */
.callout {
background: var(--color-bg-elevated);
border-left: 4px solid var(--color-accent);
padding: var(--space-md) var(--space-md);
margin: var(--space-lg) 0;
border-radius: 0 var(--radius-md) var(--radius-md) 0;
box-shadow: var(--shadow-sm);
}
.callout-title {
font-family: var(--font-sans);
font-size: 0.8rem;
font-weight: 700;
text-transform: uppercase;
letter-spacing: 0.1em;
color: var(--color-accent);
margin-bottom: var(--space-sm);
}
.callout-text {
margin-bottom: var(--space-sm);
font-size: 1rem;
color: var(--color-ink);
}
.callout-text:last-child {
margin-bottom: 0;
}
.callout-caveats {
background: linear-gradient(135deg, var(--color-accent-light) 0%, #f5f0ff 100%);
border-left-color: var(--color-accent);
}
.dual-callout {
display: grid;
grid-template-columns: 1fr;
gap: var(--space-md);
margin: var(--space-lg) 0;
}
@media (min-width: 720px) {
.dual-callout {
grid-template-columns: 1fr 1fr;
}
}
.callout-shown {
border-left-color: var(--color-teal);
background: #effaf7;
}
.callout-shown .callout-title {
color: var(--color-teal);
}
.callout-not-shown {
border-left-color: var(--color-rust);
background: #fbf2ee;
}
.callout-not-shown .callout-title {
color: var(--color-rust);
}
/* ============================================ */
/* BLOCKQUOTES */
/* ============================================ */
.central-claim,
.definition-blockquote {
font-family: var(--font-serif);
font-size: 1.2rem;
line-height: 1.5;
font-style: italic;
color: var(--color-ink);
padding: var(--space-md) var(--space-lg);
margin: var(--space-lg) 0;
border-left: 4px solid var(--color-gold);
background: var(--color-gold-light);
border-radius: 0 var(--radius-md) var(--radius-md) 0;
position: relative;
}
.central-claim::before,
.definition-blockquote::before {
content: '\201C';
position: absolute;
top: -0.5rem;
left: 0.5rem;
font-size: 3.5rem;
color: var(--color-gold);
font-family: Georgia, serif;
opacity: 0.3;
line-height: 1;
}
/* ============================================ */
/* SERIES CARDS */
/* ============================================ */
.series-card {
background: var(--color-bg-elevated);
border: 1px solid var(--color-border);
border-radius: var(--radius-md);
padding: var(--space-md);
margin: var(--space-md) 0;
transition:
transform var(--transition),
box-shadow var(--transition);
border-top: 3px solid var(--color-accent);
}
.series-card:hover {
transform: translateY(-2px);
box-shadow: var(--shadow-md);
}
.series-card-leibniz {
border-top-color: var(--color-rust);
}
.series-card-nilakantha {
border-top-color: var(--color-accent);
}
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border-top-color: var(--color-teal);
}
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font-size: 1.15rem;
font-weight: 700;
color: var(--color-ink);
margin-bottom: 0.25rem;
}
.series-description {
font-family: var(--font-sans);
font-size: 0.85rem;
color: var(--color-ink-faint);
font-style: italic;
margin-bottom: var(--space-sm);
}
.series-card .math-block {
margin: 0;
}
/* ============================================ */
/* RESULT STATEMENTS */
/* ============================================ */
.result-statement {
font-family: var(--font-mono);
font-size: 1rem;
background: var(--color-accent-light);
padding: var(--space-sm) var(--space-md);
border-radius: var(--radius-sm);
border-left: 3px solid var(--color-accent);
margin: var(--space-md) 0;
color: var(--color-ink);
}
.result-optimal {
background: var(--color-gold-light);
border-left-color: var(--color-gold);
font-weight: 600;
}
/* ============================================ */
/* CLASSIFICATION TABLE */
/* ============================================ */
.table-wrapper {
overflow-x: auto;
margin: var(--space-md) 0;
border-radius: var(--radius-md);
box-shadow: var(--shadow-sm);
border: 1px solid var(--color-border);
}
.complexity-table {
width: 100%;
border-collapse: collapse;
background: var(--color-bg-elevated);
font-family: var(--font-sans);
font-size: 0.95rem;
}
.table-head {
background: var(--color-bg-dark);
color: #f3f1ea;
}
.table-header {
padding: 0.9rem 1rem;
text-align: left;
font-weight: 600;
font-size: 0.85rem;
text-transform: uppercase;
letter-spacing: 0.05em;
}
.table-row {
border-top: 1px solid var(--color-border);
transition: background var(--transition);
}
.table-row:hover {
background: #faf6ee;
}
.table-cell {
padding: 0.9rem 1rem;
vertical-align: top;
}
.cell-iteration,
.cell-bitlength {
font-family: var(--font-mono);
font-size: 0.9rem;
color: var(--color-ink-muted);
}
.table-row-highlight {
background: var(--color-gold-light);
}
.table-row-highlight:hover {
background: #f3eacd;
}
.table-row-highlight .cell-bitlength {
color: var(--color-gold);
font-weight: 600;
}
/* ============================================ */
/* FRAMEWORK GRID */
/* ============================================ */
.framework-grid {
display: grid;
grid-template-columns: 1fr;
gap: var(--space-md);
margin: var(--space-lg) 0;
}
@media (min-width: 720px) {
.framework-grid {
grid-template-columns: repeat(3, 1fr);
}
}
.framework-card {
background: var(--color-bg-elevated);
padding: var(--space-md);
border-radius: var(--radius-md);
border: 1px solid var(--color-border);
transition:
transform var(--transition),
box-shadow var(--transition);
}
.framework-card:hover {
transform: translateY(-3px);
box-shadow: var(--shadow-md);
}
.framework-card-classical {
border-top: 3px solid var(--color-rust);
}
.framework-card-diophantine {
border-top: 3px solid var(--color-teal);
}
.framework-card-ko-friedman {
border-top: 3px solid var(--color-accent);
}
.framework-name {
font-size: 1.05rem;
font-weight: 700;
color: var(--color-ink);
margin-bottom: var(--space-sm);
line-height: 1.3;
}
.framework-description {
font-size: 0.95rem;
color: var(--color-ink-muted);
line-height: 1.6;
}
/* ============================================ */
/* COMPLEXITY CLASS HIERARCHY */
/* ============================================ */
.class-hierarchy {
display: grid;
grid-template-columns: 1fr;
gap: var(--space-md);
margin: var(--space-lg) 0;
}
.complexity-class {
background: var(--color-bg-elevated);
padding: var(--space-md) var(--space-md) var(--space-md) 5rem;
border-radius: var(--radius-md);
box-shadow: var(--shadow-sm);
position: relative;
border: 1px solid var(--color-border);
overflow: hidden;
}
.class-label {
position: absolute;
top: 0;
left: 0;
width: 4rem;
height: 100%;
display: flex;
align-items: center;
justify-content: center;
font-family: var(--font-sans);
font-size: 1.4rem;
font-weight: 700;
color: white;
}
.class-rc1 {
border-left: 4px solid var(--color-gold);
}
.class-rc1 .class-label {
background: var(--color-gold);
}
.class-rc2 {
border-left: 4px solid var(--color-accent);
}
.class-rc2 .class-label {
background: var(--color-accent);
}
.class-rc3 {
border-left: 4px solid var(--color-rust);
}
.class-rc3 .class-label {
background: var(--color-rust);
}
.class-name {
font-size: 1.05rem;
font-weight: 700;
margin-bottom: 0.25rem;
color: var(--color-ink);
}
.class-formula {
font-family: var(--font-mono);
font-size: 0.9rem;
color: var(--color-ink-muted);
background: var(--color-bg);
padding: 0.4rem 0.7rem;
border-radius: var(--radius-sm);
display: inline-block;
margin-bottom: var(--space-sm);
}
.class-examples {
font-size: 0.95rem;
color: var(--color-ink-muted);
line-height: 1.5;
}
/* ============================================ */
/* SPECIAL PARAGRAPHS */
/* ============================================ */
.closing-paragraph {
font-size: 1.15rem;
font-style: italic;
color: var(--color-ink-muted);
border-top: 1px solid var(--color-border);
padding-top: var(--space-md);
margin-top: var(--space-lg);
}
.subsection-optimal .result-statement {
background: linear-gradient(135deg, var(--color-gold-light), #fff8e2);
border-left-width: 4px;
}
/* ============================================ */
/* FOOTER */
/* ============================================ */
.site-footer {
background: var(--color-bg-dark);
color: #c8c5b8;
padding: var(--space-xl) 1.5rem var(--space-md);
margin-top: var(--space-2xl);
}
.footer-content {
max-width: var(--content-width-wide);
margin: 0 auto;
display: grid;
grid-template-columns: 1fr;
gap: var(--space-lg);
padding-bottom: var(--space-lg);
border-bottom: 1px solid rgba(255, 255, 255, 0.08);
}
@media (min-width: 720px) {
.footer-content {
grid-template-columns: 2fr 1fr 1fr;
}
}
.footer-heading {
font-family: var(--font-sans);
font-size: 0.8rem;
font-weight: 700;
text-transform: uppercase;
letter-spacing: 0.12em;
color: #f3f1ea;
margin-bottom: var(--space-sm);
}
.footer-text {
font-size: 0.95rem;
line-height: 1.6;
color: #a8a59a;
}
.footer-tag-list {
list-style: none;
display: flex;
flex-wrap: wrap;
gap: 0.4rem;
}
.footer-tag {
font-family: var(--font-sans);
font-size: 0.78rem;
padding: 0.3rem 0.7rem;
background: rgba(255, 255, 255, 0.06);
border: 1px solid rgba(255, 255, 255, 0.1);
border-radius: 999px;
color: #c8c5b8;
}
.footer-bottom {
max-width: var(--content-width-wide);
margin: 0 auto;
padding-top: var(--space-md);
text-align: center;
}
.footer-copyright {
font-family: var(--font-sans);
font-size: 0.8rem;
color: #6a6e7c;
}
/* ============================================ */
/* RESPONSIVE ADJUSTMENTS */
/* ============================================ */
@media (max-width: 640px) {
html {
font-size: 16px;
}
.nav-menu {
gap: 0;
}
.nav-link {
padding: 0.4rem 0.5rem;
font-size: 0.78rem;
}
.hero-section {
padding: var(--space-xl) 1.25rem var(--space-lg);
}
.article-main {
padding: var(--space-lg) 1.25rem;
}
.article-section {
margin-bottom: var(--space-xl);
}
.math-block {
padding: 0.9rem 1rem;
}
.math-code {
font-size: 0.85rem;
}
.complexity-class {
padding-left: 4rem;
}
.class-label {
width: 3rem;
font-size: 1.1rem;
}
}
@media (max-width: 480px) {
.nav-brand .brand-link {
font-size: 0.9rem;
}
.article-title {
font-size: 2rem;
}
}
/* ============================================ */
/* SELECTION & FOCUS */
/* ============================================ */
::selection {
background: var(--color-accent);
color: white;
}
a:focus-visible,
button:focus-visible {
outline: 2px solid var(--color-accent);
outline-offset: 2px;
}
</style>
</head>
<body class="article-page">
<!-- ============================================ -->
<!-- HEADER / NAVIGATION -->
<!-- ============================================ -->
<header class="site-header">
<nav class="site-nav">
<div class="nav-brand">
<a href="../../index.html" class="brand-link">RC Complexity</a>
</div>
<ul class="nav-menu">
<li class="nav-item">
<a href="../../index.html" class="nav-link">← Home</a>
</li>
<li class="nav-item">
<a href="#core-idea" class="nav-link">Core Idea</a>
</li>
<li class="nav-item">
<a href="#background" class="nav-link">Background</a>
</li>
<li class="nav-item">
<a href="#framework" class="nav-link">Framework</a>
</li>
<li class="nav-item">
<a href="#classification" class="nav-link">Classification</a>
</li>
<li class="nav-item">
<a href="#conclusion" class="nav-link">Conclusion</a>
</li>
</ul>
</nav>
</header>
<!-- ============================================ -->
<!-- HERO / ARTICLE TITLE -->