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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="An exploration of how x + sin(x) reveals the architecture of mathematical continua, cubic convergence, and rational certificate engines for π."
/>
<meta name="author" content="Mathematical Analysis" />
<meta
name="keywords"
content="mathematics, pi, convergence, fixed-point, sine, Taylor series, rational arithmetic"
/>
<title>
The Simplest Increment: How x + sin(x) Reveals the Architecture of Mathematical Continua
</title>
<!-- CSS will be added here -->
<style>
/* ============================================================
MATHEMATICAL CONTINUA - COMPLETE STYLESHEET
A modern, academic design for mathematical content
============================================================ */
/* ============================================================
1. CSS CUSTOM PROPERTIES (DESIGN TOKENS)
============================================================ */
:root {
/* Color Palette */
--color-bg-primary: #0d1117;
--color-bg-secondary: #161b22;
--color-bg-tertiary: #1c2128;
--color-bg-card: #21262d;
--color-bg-elevated: #2d333b;
--color-accent-primary: #58a6ff;
--color-accent-secondary: #79c0ff;
--color-accent-gold: #e3b341;
--color-accent-green: #3fb950;
--color-accent-purple: #bc8cff;
--color-accent-orange: #f78166;
--color-accent-teal: #39d353;
--color-text-primary: #e6edf3;
--color-text-secondary: #8b949e;
--color-text-muted: #6e7681;
--color-text-inverse: #0d1117;
--color-border-default: #30363d;
--color-border-muted: #21262d;
--color-border-accent: #388bfd;
/* Typography */
--font-sans: 'Inter', 'Segoe UI', system-ui, -apple-system, sans-serif;
--font-serif: 'Georgia', 'Times New Roman', serif;
--font-mono: 'JetBrains Mono', 'Fira Code', 'Cascadia Code', 'Consolas', monospace;
--font-size-xs: 0.75rem;
--font-size-sm: 0.875rem;
--font-size-base: 1rem;
--font-size-md: 1.125rem;
--font-size-lg: 1.25rem;
--font-size-xl: 1.5rem;
--font-size-2xl: 1.875rem;
--font-size-3xl: 2.25rem;
--font-size-4xl: 3rem;
--font-size-5xl: 3.75rem;
--font-weight-normal: 400;
--font-weight-medium: 500;
--font-weight-semibold: 600;
--font-weight-bold: 700;
--font-weight-extrabold: 800;
--line-height-tight: 1.25;
--line-height-snug: 1.375;
--line-height-normal: 1.5;
--line-height-relaxed: 1.625;
--line-height-loose: 1.75;
--line-height-reading: 1.85;
/* Spacing */
--space-1: 0.25rem;
--space-2: 0.5rem;
--space-3: 0.75rem;
--space-4: 1rem;
--space-5: 1.25rem;
--space-6: 1.5rem;
--space-8: 2rem;
--space-10: 2.5rem;
--space-12: 3rem;
--space-16: 4rem;
--space-20: 5rem;
--space-24: 6rem;
--space-32: 8rem;
/* Border Radius */
--radius-sm: 4px;
--radius-md: 8px;
--radius-lg: 12px;
--radius-xl: 16px;
--radius-2xl: 24px;
--radius-full: 9999px;
/* Shadows */
--shadow-sm: 0 1px 3px rgba(0, 0, 0, 0.4);
--shadow-md: 0 4px 12px rgba(0, 0, 0, 0.5);
--shadow-lg: 0 8px 24px rgba(0, 0, 0, 0.6);
--shadow-xl: 0 16px 48px rgba(0, 0, 0, 0.7);
--shadow-glow-blue: 0 0 20px rgba(88, 166, 255, 0.15);
--shadow-glow-gold: 0 0 20px rgba(227, 179, 65, 0.15);
/* Transitions */
--transition-fast: 150ms ease;
--transition-base: 250ms ease;
--transition-slow: 400ms ease;
/* Layout */
--max-width-content: 780px;
--max-width-wide: 1100px;
--nav-height: 64px;
}
/* ============================================================
2. RESET & BASE STYLES
============================================================ */
*,
*::before,
*::after {
box-sizing: border-box;
margin: 0;
padding: 0;
}
html {
scroll-behavior: smooth;
scroll-padding-top: calc(var(--nav-height) + var(--space-6));
font-size: 16px;
-webkit-text-size-adjust: 100%;
}
body {
font-family: var(--font-sans);
font-size: var(--font-size-base);
font-weight: var(--font-weight-normal);
line-height: var(--line-height-normal);
color: var(--color-text-primary);
background-color: var(--color-bg-primary);
min-height: 100vh;
overflow-x: hidden;
-webkit-font-smoothing: antialiased;
-moz-osx-font-smoothing: grayscale;
}
img,
video {
max-width: 100%;
height: auto;
display: block;
}
a {
color: var(--color-accent-primary);
text-decoration: none;
transition: color var(--transition-fast);
}
a:hover {
color: var(--color-accent-secondary);
text-decoration: underline;
}
ul,
ol {
list-style: none;
}
button {
cursor: pointer;
border: none;
background: none;
font-family: inherit;
}
code {
font-family: var(--font-mono);
}
sub,
sup {
font-size: 0.75em;
line-height: 0;
position: relative;
vertical-align: baseline;
}
sup {
top: -0.5em;
}
sub {
bottom: -0.25em;
}
/* Focus styles for accessibility */
:focus-visible {
outline: 2px solid var(--color-accent-primary);
outline-offset: 3px;
border-radius: var(--radius-sm);
}
/* ============================================================
3. SITE HEADER & NAVIGATION
============================================================ */
.site-header {
position: sticky;
top: 0;
z-index: 100;
background-color: rgba(13, 17, 23, 0.92);
backdrop-filter: blur(12px);
-webkit-backdrop-filter: blur(12px);
border-bottom: 1px solid var(--color-border-default);
height: var(--nav-height);
}
.site-nav {
display: flex;
align-items: center;
justify-content: space-between;
max-width: var(--max-width-wide);
margin: 0 auto;
padding: 0 var(--space-6);
height: 100%;
gap: var(--space-6);
}
.site-nav__brand {
display: flex;
align-items: center;
gap: var(--space-3);
flex-shrink: 0;
text-decoration: none;
}
.site-nav__logo-mark {
display: flex;
align-items: center;
justify-content: center;
width: 36px;
height: 36px;
background: linear-gradient(
135deg,
var(--color-accent-primary),
var(--color-accent-purple)
);
border-radius: var(--radius-md);
font-size: var(--font-size-lg);
font-weight: var(--font-weight-bold);
color: white;
flex-shrink: 0;
}
.site-nav__publication-name {
font-size: var(--font-size-sm);
font-weight: var(--font-weight-semibold);
color: var(--color-text-primary);
letter-spacing: 0.02em;
white-space: nowrap;
}
.site-nav__links {
display: flex;
align-items: center;
gap: var(--space-1);
flex-wrap: nowrap;
overflow: hidden;
}
.site-nav__item {
flex-shrink: 0;
}
.site-nav__link {
display: block;
padding: var(--space-2) var(--space-3);
font-size: var(--font-size-sm);
font-weight: var(--font-weight-medium);
color: var(--color-text-secondary);
border-radius: var(--radius-md);
transition:
color var(--transition-fast),
background-color var(--transition-fast);
white-space: nowrap;
text-decoration: none;
}
.site-nav__link:hover {
color: var(--color-text-primary);
background-color: var(--color-bg-elevated);
text-decoration: none;
}
.site-nav__menu-toggle {
display: none;
flex-direction: column;
justify-content: center;
align-items: center;
gap: 5px;
width: 40px;
height: 40px;
border-radius: var(--radius-md);
transition: background-color var(--transition-fast);
flex-shrink: 0;
}
.site-nav__menu-toggle:hover {
background-color: var(--color-bg-elevated);
}
.site-nav__hamburger-bar {
display: block;
width: 20px;
height: 2px;
background-color: var(--color-text-secondary);
border-radius: var(--radius-full);
transition:
transform var(--transition-base),
opacity var(--transition-fast);
}
/* ============================================================
4. HERO SECTION
============================================================ */
.hero-section {
position: relative;
overflow: hidden;
padding: var(--space-20) var(--space-6) var(--space-16);
background: linear-gradient(
180deg,
var(--color-bg-secondary) 0%,
var(--color-bg-primary) 100%
);
border-bottom: 1px solid var(--color-border-default);
}
.hero-section::before {
content: '';
position: absolute;
top: 0;
left: 0;
right: 0;
bottom: 0;
background:
radial-gradient(ellipse 60% 40% at 70% 20%, rgba(88, 166, 255, 0.06) 0%, transparent 70%),
radial-gradient(ellipse 40% 60% at 20% 80%, rgba(188, 140, 255, 0.05) 0%, transparent 70%);
pointer-events: none;
}
.hero-section__inner {
position: relative;
z-index: 1;
max-width: var(--max-width-content);
margin: 0 auto;
}
.hero-section__category-tag {
display: flex;
flex-wrap: wrap;
gap: var(--space-2);
margin-bottom: var(--space-6);
}
.category-tag {
display: inline-block;
padding: var(--space-1) var(--space-3);
font-size: var(--font-size-xs);
font-weight: var(--font-weight-semibold);
letter-spacing: 0.06em;
text-transform: uppercase;
color: var(--color-accent-primary);
background-color: rgba(88, 166, 255, 0.1);
border: 1px solid rgba(88, 166, 255, 0.25);
border-radius: var(--radius-full);
}
.hero-section__title {
font-size: clamp(var(--font-size-2xl), 5vw, var(--font-size-4xl));
font-weight: var(--font-weight-extrabold);
line-height: var(--line-height-tight);
color: var(--color-text-primary);
margin-bottom: var(--space-6);
letter-spacing: -0.02em;
}
.hero-section__title-emphasis {
display: block;
background: linear-gradient(
135deg,
var(--color-accent-primary) 0%,
var(--color-accent-purple) 100%
);
-webkit-background-clip: text;
-webkit-text-fill-color: transparent;
background-clip: text;
margin-top: var(--space-2);
}
.hero-section__lead {
font-size: var(--font-size-lg);
line-height: var(--line-height-relaxed);
color: var(--color-text-secondary);
margin-bottom: var(--space-10);
max-width: 65ch;
}
.hero-section__lead em {
color: var(--color-accent-secondary);
font-style: italic;
}
/* Formula Showcase */
.hero-section__formula-showcase {
margin-bottom: var(--space-10);
}
.formula-showcase__display {
display: flex;
align-items: center;
flex-wrap: wrap;
gap: var(--space-4);
padding: var(--space-6) var(--space-8);
background: var(--color-bg-card);
border: 1px solid var(--color-border-default);
border-left: 4px solid var(--color-accent-primary);
border-radius: var(--radius-lg);
box-shadow: var(--shadow-md), var(--shadow-glow-blue);
margin-bottom: var(--space-3);
}
.formula-showcase__label {
font-size: var(--font-size-xs);
font-weight: var(--font-weight-semibold);
letter-spacing: 0.08em;
text-transform: uppercase;
color: var(--color-text-muted);
flex-shrink: 0;
}
.formula-showcase__expression {
font-family: var(--font-mono);
font-size: clamp(var(--font-size-lg), 3vw, var(--font-size-2xl));
font-weight: var(--font-weight-medium);
color: var(--color-accent-secondary);
flex: 1;
text-align: center;
}
.formula-showcase__converges-to {
font-size: var(--font-size-xl);
font-weight: var(--font-weight-bold);
color: var(--color-accent-gold);
flex-shrink: 0;
}
.formula-showcase__caption {
font-size: var(--font-size-sm);
color: var(--color-text-muted);
text-align: center;
font-style: italic;
}
/* Article Metadata */
.article-meta {
display: flex;
flex-wrap: wrap;
gap: var(--space-4);
padding-top: var(--space-6);
border-top: 1px solid var(--color-border-default);
}
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display: flex;
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gap: var(--space-2);
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}
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color: var(--color-text-muted);
}
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font-weight: var(--font-weight-medium);
}
/* Background Decoration */
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position: absolute;
top: 0;
right: 0;
bottom: 0;
width: 40%;
pointer-events: none;
overflow: hidden;
z-index: 0;
}
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position: absolute;
font-family: var(--font-serif);
font-weight: var(--font-weight-bold);
color: rgba(88, 166, 255, 0.04);
user-select: none;
line-height: 1;
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font-size: 18rem;
top: -2rem;
right: -2rem;
color: rgba(88, 166, 255, 0.05);
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font-size: 5rem;
bottom: 20%;
right: 15%;
color: rgba(188, 140, 255, 0.04);
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top: 30%;
right: 40%;
color: rgba(227, 179, 65, 0.03);
}
.bg-decoration__symbol--integral {
font-size: 12rem;
bottom: -2rem;
right: 5%;
color: rgba(63, 185, 80, 0.03);
}
/* ============================================================
5. TABLE OF CONTENTS
============================================================ */
.table-of-contents {
background-color: var(--color-bg-secondary);
border-bottom: 1px solid var(--color-border-default);
padding: var(--space-8) var(--space-6);
}
.table-of-contents__inner {
max-width: var(--max-width-content);
margin: 0 auto;
}
.table-of-contents__heading {
font-size: var(--font-size-xs);
font-weight: var(--font-weight-semibold);
letter-spacing: 0.1em;
text-transform: uppercase;
color: var(--color-text-muted);
margin-bottom: var(--space-4);
}
.table-of-contents__list {
display: grid;
grid-template-columns: repeat(auto-fill, minmax(280px, 1fr));
gap: var(--space-1);
counter-reset: toc-counter;
}
.table-of-contents__item {
counter-increment: toc-counter;
}
.table-of-contents__link {
display: flex;
align-items: baseline;
gap: var(--space-2);
padding: var(--space-2) var(--space-3);
font-size: var(--font-size-sm);
color: var(--color-text-secondary);
border-radius: var(--radius-md);
transition:
color var(--transition-fast),
background-color var(--transition-fast);
text-decoration: none;
}
.table-of-contents__link::before {
content: counter(toc-counter, decimal-leading-zero);
font-size: var(--font-size-xs);
font-weight: var(--font-weight-semibold);
color: var(--color-text-muted);
font-family: var(--font-mono);
flex-shrink: 0;
}
.table-of-contents__item .table-of-contents__link::before {
content: none;
}
.table-of-contents__list > .table-of-contents__item > .table-of-contents__link::before {
content: counter(toc-counter, decimal-leading-zero);
}
.table-of-contents__link:hover {
color: var(--color-accent-primary);
background-color: rgba(88, 166, 255, 0.06);
text-decoration: none;
}
.table-of-contents__sublist {
margin-left: var(--space-6);
margin-top: var(--space-1);
counter-reset: toc-sub-counter;
}
.table-of-contents__subitem {
counter-increment: toc-sub-counter;
}
.table-of-contents__subitem .table-of-contents__link {
font-size: var(--font-size-xs);
color: var(--color-text-muted);
padding: var(--space-1) var(--space-3);
}
.table-of-contents__subitem .table-of-contents__link::before {
content: none;
}
.table-of-contents__subitem .table-of-contents__link:hover {
color: var(--color-accent-primary);
}
/* ============================================================
6. MAIN ARTICLE LAYOUT
============================================================ */
.article-main {
max-width: var(--max-width-content);
margin: 0 auto;
padding: 0 var(--space-6);
}
.article-body {
padding: var(--space-4) 0;
}
/* ============================================================
7. ARTICLE SECTIONS
============================================================ */
.article-section {
padding: var(--space-16) 0;
border-bottom: 1px solid var(--color-border-muted);
}
.article-section:last-child {
border-bottom: none;
}
.article-section--alternate {
/* Subtle visual differentiation for alternating sections */
}
.article-section__inner {
/* Container for section content */
}
.article-section__heading {
display: flex;
align-items: baseline;
gap: var(--space-4);
font-size: clamp(var(--font-size-xl), 3vw, var(--font-size-2xl));
font-weight: var(--font-weight-bold);
color: var(--color-text-primary);
margin-bottom: var(--space-8);
letter-spacing: -0.01em;
line-height: var(--line-height-tight);
}
.article-section__number {
font-family: var(--font-mono);
font-size: var(--font-size-sm);
font-weight: var(--font-weight-semibold);
color: var(--color-accent-primary);
background-color: rgba(88, 166, 255, 0.1);
border: 1px solid rgba(88, 166, 255, 0.2);
padding: var(--space-1) var(--space-2);
border-radius: var(--radius-sm);
flex-shrink: 0;
letter-spacing: 0.05em;
}
.article-section__content {
/* Content wrapper */
}
.article-section__paragraph {
font-size: var(--font-size-md);
line-height: var(--line-height-reading);
color: var(--color-text-secondary);
margin-bottom: var(--space-6);
max-width: 70ch;
}
.article-section__paragraph--lead {
font-size: var(--font-size-lg);
color: var(--color-text-primary);
font-weight: var(--font-weight-medium);
}
.article-section__paragraph--closing {
font-style: italic;
color: var(--color-text-primary);
font-size: var(--font-size-lg);
border-left: 3px solid var(--color-accent-primary);
padding-left: var(--space-5);
margin-top: var(--space-8);
}
.article-section__paragraph em {
color: var(--color-accent-secondary);
font-style: italic;
}
.article-section__paragraph strong {
color: var(--color-text-primary);
font-weight: var(--font-weight-semibold);
}
/* ============================================================
8. MATH BLOCKS
============================================================ */
.math-block {
margin: var(--space-6) 0;
padding: var(--space-5) var(--space-6);
background-color: var(--color-bg-card);
border: 1px solid var(--color-border-default);
border-radius: var(--radius-lg);
overflow-x: auto;
}
.math-block--featured {
border-left: 4px solid var(--color-accent-primary);
background: linear-gradient(135deg, rgba(88, 166, 255, 0.05) 0%, var(--color-bg-card) 100%);
box-shadow: var(--shadow-md), var(--shadow-glow-blue);
text-align: center;
padding: var(--space-8);
}
.math-block--result {
border-left: 4px solid var(--color-accent-green);
background: linear-gradient(135deg, rgba(63, 185, 80, 0.05) 0%, var(--color-bg-card) 100%);
}
.math-block--inline {
display: inline-block;
margin: var(--space-2) 0;
padding: var(--space-2) var(--space-4);
}
.math-block__expression {
font-family: var(--font-mono);
font-size: var(--font-size-base);
color: var(--color-accent-secondary);
line-height: var(--line-height-relaxed);
white-space: pre-wrap;
word-break: break-word;
}
.math-block--featured .math-block__expression {
font-size: clamp(var(--font-size-lg), 2.5vw, var(--font-size-xl));
}
/* ============================================================
9. CONVERGENCE VISUALIZATION
============================================================ */
.convergence-visual {
margin: var(--space-8) 0;
padding: var(--space-6);
background-color: var(--color-bg-secondary);
border: 1px solid var(--color-border-default);
border-radius: var(--radius-xl);
overflow: hidden;
}
.convergence-visual__heading {
font-size: var(--font-size-sm);
font-weight: var(--font-weight-semibold);
letter-spacing: 0.06em;
text-transform: uppercase;
color: var(--color-text-muted);
margin-bottom: var(--space-5);
}
.convergence-visual__steps {
display: flex;
align-items: center;
gap: var(--space-3);
overflow-x: auto;
padding-bottom: var(--space-2);
flex-wrap: wrap;
}
.convergence-step {
display: flex;
flex-direction: column;
align-items: center;
gap: var(--space-2);
padding: var(--space-4) var(--space-5);
background-color: var(--color-bg-card);
border: 1px solid var(--color-border-default);
border-radius: var(--radius-lg);
min-width: 120px;
flex-shrink: 0;
transition:
transform var(--transition-base),
box-shadow var(--transition-base);
}
.convergence-step:hover {
transform: translateY(-2px);
box-shadow: var(--shadow-md);
}
.convergence-step--seed {
border-color: rgba(88, 166, 255, 0.3);
}
.convergence-step--iter1 {
border-color: rgba(188, 140, 255, 0.3);
}
.convergence-step--iter2 {
border-color: rgba(227, 179, 65, 0.3);
}
.convergence-step--iter3 {
border-color: rgba(63, 185, 80, 0.3);
}
.convergence-step__label {
font-size: var(--font-size-xs);
font-weight: var(--font-weight-semibold);
letter-spacing: 0.05em;
text-transform: uppercase;
color: var(--color-text-muted);
}
.convergence-step__digits {
font-size: var(--font-size-sm);
font-weight: var(--font-weight-bold);
color: var(--color-accent-gold);
}
.convergence-step--seed .convergence-step__digits {
color: var(--color-accent-primary);
}
.convergence-step--iter1 .convergence-step__digits {
color: var(--color-accent-purple);
}
.convergence-step--iter2 .convergence-step__digits {
color: var(--color-accent-gold);
}
.convergence-step--iter3 .convergence-step__digits {
color: var(--color-accent-green);
}
.convergence-step__value {
font-family: var(--font-mono);
font-size: var(--font-size-xs);
color: var(--color-text-secondary);
word-break: break-all;
text-align: center;
}
.convergence-step__arrow {
font-size: var(--font-size-xl);
color: var(--color-text-muted);
flex-shrink: 0;
}
/* ============================================================
10. DERIVATION BLOCK
============================================================ */
.derivation-block {
margin: var(--space-6) 0;
border: 1px solid var(--color-border-default);
border-radius: var(--radius-xl);
overflow: hidden;
}
.derivation-block__step {
padding: var(--space-5) var(--space-6);
border-bottom: 1px solid var(--color-border-muted);
background-color: var(--color-bg-secondary);
transition: background-color var(--transition-fast);
}
.derivation-block__step:last-child {
border-bottom: none;
}
.derivation-block__step:hover {
background-color: var(--color-bg-tertiary);
}
.derivation-block__step--result {
background-color: rgba(63, 185, 80, 0.05);
border-top: 2px solid var(--color-accent-green);
}
.derivation-block__step-label {
display: block;
font-size: var(--font-size-xs);
font-weight: var(--font-weight-semibold);
letter-spacing: 0.05em;
text-transform: uppercase;
color: var(--color-text-muted);
margin-bottom: var(--space-3);
}
.derivation-block__step--result .derivation-block__step-label {
color: var(--color-accent-green);
}
.derivation-block__step .math-block {
margin: 0;
background: transparent;
border: none;
padding: 0;
}
/* ============================================================
11. INSIGHT CALLOUTS
============================================================ */
.insight-callout {
display: flex;
gap: var(--space-4);
margin: var(--space-8) 0;
padding: var(--space-5) var(--space-6);
border-radius: var(--radius-lg);
border: 1px solid;
}
.insight-callout--mechanism {
background-color: rgba(88, 166, 255, 0.06);
border-color: rgba(88, 166, 255, 0.2);
}
.insight-callout--symmetry {
background-color: rgba(188, 140, 255, 0.06);
border-color: rgba(188, 140, 255, 0.2);
}
.insight-callout--classification {
background-color: rgba(227, 179, 65, 0.06);
border-color: rgba(227, 179, 65, 0.2);
}
.insight-callout--distinction {
background-color: rgba(247, 129, 102, 0.06);
border-color: rgba(247, 129, 102, 0.2);
}
.insight-callout--saturation {
background-color: rgba(63, 185, 80, 0.06);
border-color: rgba(63, 185, 80, 0.2);
}
.insight-callout__icon {
font-size: var(--font-size-xl);
flex-shrink: 0;
line-height: 1.4;
}
.insight-callout__content {
flex: 1;
}
.insight-callout__text {
font-size: var(--font-size-base);
line-height: var(--line-height-relaxed);