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	<!DOCTYPE html>
	<!--
	Vendored + adapted into posterly from ARIS (Auto-claude-code-research-in-sleep),
	skill paper-poster-html. Origin: https://github.com/wanshuiyin/Auto-claude-code-research-in-sleep
	Upstream tokenization: MIT (c) 2026 wanshuiyin -- ../LICENSES/aris-MIT.txt
	This is the tokenized form of posterly's own neutral template; the class names and
	content are posterly's own (c) 2026 Ruishuo Chen. As an adapted derivative this file
	ships as part of posterly under AGPL-3.0. See ../NOTICE.md.
	-->
	<!--
	============================================================
	TEMPLATE: landscape_4col (ARIS fork)
	CANVAS: 60in × 36in landscape (ICML / NeurIPS / generic landscape)
	LAYOUT: header → optional banner → 4 body columns → optional takeaways → footer
	USE WHEN: standard ML conference poster with 3-5 content cards per column,
	mix of figures + equations + small tables.
	HOW TO USE:
	1. Copy this file to your working directory as `poster.html`.
	2. Edit the DESIGN TOKENS in `:root` to your lab/venue colors.
	3. Replace TODO content placeholders (search for "TODO").
	4. Run `python tools/run_gates.py poster.html --tokens design_tokens.json` (the default driver: preflight -> style -> measure -> polish; design_tokens.json is the design-direction pack written at lock time -- SKILL.md Step 2.5) to align columns and check style/structure.
	5. Run `python tools/render_preview.py poster.html` to produce the PDF.
	NOTE: as shipped, this scaffold passes `preflight` (structure) but is
	EXPECTED to FAIL `measure`/`polish` -- figures are commented out and copy
	is TODO stubs, so columns only fill the top of the canvas. Those two gates
	judge a FILLED poster; balance them once you've added real content. See
	templates/README.md ("Scaffolds, not finished posters").
	MEASURE ROLES: every layout-critical element carries `data-measure-role`.
	Generic measurement scripts depend on these — do not remove.

	CANVAS RETARGETING: to print at another size, change the canvas in exactly
	TWO places that must stay in sync -- the `@page { size: ... }` rule and the
	`.poster { width/height }` (also the `@media print .poster { width/height }`).
	Example: ICLR 2026 main conference uses 185cm 90cm landscape (official print
	service spec) -- set @page to `185cm 90cm`, width to `1850 * var(--u)` and
	height to `900 * var(--u)` (and the print-override .poster to `185cm`/`90cm`).

	Adapted from posterly (MIT, © 2026 Ruishuo Chen) — see LICENSES/ & NOTICE.md;
	ARIS modifications: flat de-gradient, --fs token scale, zero-inline-style
	utilities, data-source/data-color-exempt contracts.
	============================================================
	-->
	<html lang="en">
	<head>
	<meta charset="UTF-8">
	<meta name="viewport" content="width=device-width, initial-scale=1.0">
	<meta name="generator" content="posterly">
	<title>Flatness-Aware SGLD — Reproduction — ICML 2026</title>

	<!-- MathJax v3 for inline equations. CDN by default when the file is
	opened by hand; the posterly check tools (measure/polish/pack/…)
	intercept this request and serve the skill's bundled copy
	(assets/mathjax/tex-svg.js, MathJax 3.2.2), so gates render math
	deterministically even offline. To make a hand-opened poster
	offline too, copy that bundle next to the poster and point the
	<script> `src` at it. -->
	<script>
	window.MathJax = {
	tex: {
	inlineMath: [['$', '$'], ['\$', '\$']],
	displayMath: [['$$', '$$'], ['\\[', '\\]']],
	packages: {'[+]': ['ams']}
	},
	svg: { fontCache: 'global' }
	};
	</script>
	<script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-svg.js"></script>

	<style>
	/* =========================================================
	CANVAS — 60" × 36" landscape
	========================================================= */
	@page { size: 60in 36in; margin: 0; }

	:root {
	/* ===== DESIGN TOKENS ===== */

	/* Primary accent (header underline, card-highlight bar, section nums, .keyword) */
	--accent: #2D5F8B;
	--accent-deep: #1F4566;
	--accent-light: #E8F1F8;
	--accent-soft: #D7E5F0;
	--accent-ink: #FFFFFF; /* ink ON the accent (chips, callout, thead) */

	/* Emphasis register for "ours / best" (table .ours row, "★" callouts).
	Default is the warm gold; a redesign may choose any register that
	keeps 4.5:1 with --emph-ink (see templates/THEMES.md Mechanism 1). */
	--emph: #C9A24A;
	--emph-soft: #FFF7E0;
	--emph-ink: #14314A; /* ink ON the emphasis fill. NOT var(--accent-deep):
	#1F4566 on the default gold measures 4.16:1 (< AA);
	#14314A measures 5.58:1 (>= 4.5:1). */

	/* Text */
	--text-primary: #1A1A1A;
	--text-secondary: #555555;
	--text-muted: #888888;

	/* Backgrounds */
	--bg-page: #F6F2F0;
	--bg-card: #FFFFFF;
	--bg-card-tint: #FAFAFB;
	--bg-emphasis: var(--accent-light);

	/* Borders */
	--border-soft: #D8D8D8;
	--border-strong: var(--accent);

	/* Screen-only dark mat behind the poster (the off-canvas viewport
	background). Print resets html/body to white, so this never reaches
	paper; kept as a token so no color literal lives outside this block. */
	--bg-viewport: #2B2B2B;

	/* Base unit. Print: 1mm; screen preview: 1.6px (~3.78px = 1mm at 96dpi). */
	--u: 1.6px;

	/* Font-size scale (9 archetypes). Every `font-size` in this template
	references one of these. `calc(var(--fs-N) * k)` is permitted ONLY for
	a COMPONENTS.md-defined variant (e.g. `.eqn--large`). */
	--fs-1: calc(9 * var(--u)); /* micro label */
	--fs-2: calc(10 * var(--u)); /* small caption */
	--fs-3: calc(11 * var(--u)); /* caption / table */
	--fs-4: calc(12 * var(--u)); /* body text */
	--fs-5: calc(13 * var(--u)); /* equation / emphasis */
	--fs-6: calc(15 * var(--u)); /* subtitle */
	--fs-7: calc(16 * var(--u)); /* section title */
	--fs-8: calc(22 * var(--u)); /* banner number */
	--fs-9: calc(32 * var(--u)); /* main title */

	/* Fonts. Override at :root if your venue mandates a specific family. */
	--font-serif: "Charter", "Source Serif Pro", "Georgia", serif;
	--font-sans: "Inter", "Helvetica Neue", sans-serif;


	/* Shadows & watermark ink — tokenized so rules 1/3 stay literal-free
	(the faint radial page tint is the ONE allowed literal outside this
	block; rule 5 validates it WHEN ENABLED -- the style gate ships with rules 4-5 off). */
	--shadow-screen: 0 0 60px rgba(0, 0, 0, 0.5);
	--shadow-card: 0 calc(2 * var(--u)) calc(6 * var(--u)) rgba(45, 95, 139, 0.05);
	--ornament-ink: rgba(45, 95, 139, 0.06);
	/* Identity mark ink — the corner-signature ⊕ adopts the LOCAL accent so it
	reads native (the woven-signature instead inherits its host's ink). The
	glyph GEOMETRY, not the hue, is the posterly signature. Re-theme with the
	palette like any other token. */
	--ps-mark-ink: var(--accent-deep);
	/* Corner-radius scale. Every border-radius calc() multiplies by this;
	1 = the shipped soft look, 0 = square/flat (see templates/THEMES.md). */
	--rs: 1;
	/* Figure mount: the ground behind paper figures (transparent PNGs sit on
	it) and the keyline around them (.figure img / .ff-fig img). Re-theme
	with the Axis 6 frame decision so figures sit mounted in the design
	instead of pasted on it; --fig-frame: transparent = frameless mount.
	QR backgrounds stay literal white -- that's scannability, not styling. */
	--fig-bg: white;
	--fig-frame: var(--border-soft);
	/* ===== END DESIGN TOKENS ===== */
	}

	/* =========================================================
	RESET + BASE
	========================================================= */
	* { box-sizing: border-box; margin: 0; padding: 0; }

	/* =========================================================
	BASE DEFENSES -- wrap & ink safety. KEEP this block in ANY
	skeleton, including a fully custom one (copy it over and
	EXTEND the selector lists with your own prose/display
	classes -- a hand-rolled skeleton that drops it strands
	single-word widows and ragged titles; polish warns as
	TEXT-WRAP).
	- pretty: fills each line, protects the last-line orphan
	(single-word widow). Safe on all prose.
	- balance: evens CENTERED display text only -- never pair
	it with left-aligned multi-sentence prose (SKILL.md wrap
	rules).
	- Any inline class you add that paints a background (a
	.mark highlight, a keyword chip) must declare its own
	`color` (Gate G -- never inherit ink across a ground
	change) plus `box-decoration-break: clone;
	-webkit-box-decoration-break: clone;` so a wrapped
	highlight keeps its padding on both fragments (then it
	never needs  -gluing to hold one line).
	========================================================= */
	p, li, dd, figcaption,
	.body-text, .caption, .callout, .section-title { text-wrap: pretty; }
	.title { text-wrap: balance; }
	/* Case-safety: lowercase Greek under text-transform:uppercase paints
	as its capital (α -> Α, a Latin-A lookalike -- "SHARPEN α>1" prints
	as "SHARPEN A>1"). Wrap the symbol: <span class="tt-none">α</span>.
	polish flags misses as SYMBOL-CASE. */
	.tt-none { text-transform: none; }

	html, body {
	background: var(--bg-viewport);
	font-family: var(--font-serif);
	color: var(--text-primary);
	-webkit-font-smoothing: antialiased;
	}

	/* =========================================================
	UTILITY CLASSES — replace inline `style=` in the body markup.
	The template (and any finished poster) must carry ZERO `style=`
	attributes. Use these classes instead. Two documented exceptions
	exist ONLY as examples and may appear in comments, never live:
	- a logo/seal SVG flagged data-color-exempt="logo"
	- a paper figure `<img ... data-source="paper" style="width: NN%">`
	(AR width tweak; prefer the .w-NN classes below where possible)
	========================================================= */

	/* Poster container — exact print dimensions; data-measure-role="poster" so poster_check.py
	can verify the canvas size. */
	.poster {
	width: calc(1524 * var(--u));
	height: calc(914 * var(--u));
	background: var(--bg-page);
	/* Page tint kept (radial only, all color stops alpha <= 0.06 per the
	de-gradient policy: linear-gradient is banned, this low-alpha radial
	wash is the single allowed exception). */
	background-image:
	radial-gradient(ellipse at top left, rgba(45, 95, 139, 0.06), transparent 40%),
	radial-gradient(ellipse at bottom right, rgba(201, 162, 74, 0.05), transparent 50%);
	margin: 20px auto;
	padding: calc(10 * var(--u)) calc(14 * var(--u));
	display: grid;
	grid-template-columns: minmax(0, 1fr); /* column-axis twin of the minmax(0,1fr) body-row defense: without it the implicit `auto` column grows to a wide child's max-content and a full-width row overflows the canvas (measure's canvas-overflow gate). Keep on any custom skeleton. */
	grid-template-rows: auto auto minmax(0, 1fr) auto auto; /* header \| banner \| body \| takeaways \| footer. minmax(0,1fr) (not bare 1fr): an over-tall body compresses instead of pushing takeaways/footer off-canvas */
	gap: calc(6 * var(--u));
	box-shadow: var(--shadow-screen);
	position: relative;
	overflow: hidden;
	}

	/* Decorative top bar — flat solid accent (de-gradient). */
	.poster::before {
	content: "";
	position: absolute; top: 0; left: 0; right: 0;
	height: calc(8 * var(--u));
	background: var(--accent);
	}

	/* =========================================================
	HEADER
	========================================================= */
	.header {
	display: grid;
	grid-template-columns: 1fr minmax(50%, auto) 1fr; /* equal side tracks (1fr) -> the title track is centred on the poster, not just between the side blocks; the centre track is floored at 50% so a one-line title still fills it (else a short title measures narrow and trips a false HEADER/TITLE-SQUEEZED). Best-effort: a side block wide enough to clamp its 1fr track can still pull the title off-centre. */
	align-items: center;
	gap: calc(16 * var(--u));
	padding: calc(2 * var(--u)) calc(4 * var(--u)) calc(5 * var(--u));
	border-bottom: calc(2 * var(--u)) solid var(--accent);
	}

	.venue-badge {
	justify-self: start; /* anchor to the far-left edge so the centre track stays centred */
	display: flex; flex-direction: column;
	align-items: center; justify-content: center;
	min-width: calc(95 * var(--u));
	text-align: center;
	border-right: calc(1 * var(--u)) solid var(--border-soft);
	padding-right: calc(12 * var(--u));
	}
	.venue-badge .vb-venue {
	font-family: var(--font-sans);
	font-weight: 800;
	font-size: var(--fs-9);
	color: var(--accent-deep);
	line-height: 1;
	letter-spacing: -0.5px;
	}
	.venue-badge .vb-year {
	font-family: var(--font-sans);
	font-size: var(--fs-5);
	color: var(--text-secondary);
	margin-top: calc(3 * var(--u));
	letter-spacing: 1.2px;
	}
	.venue-badge .vb-tag {
	font-family: var(--font-sans);
	font-size: var(--fs-2);
	color: var(--accent);
	font-weight: 700;
	margin-top: calc(2 * var(--u));
	letter-spacing: 1.2px;
	}

	.title-block { text-align: center; min-width: 0; }
	.title {
	font-family: var(--font-sans);
	font-weight: 800;
	font-size: var(--fs-9);
	line-height: 1.05;
	color: var(--accent-deep);
	letter-spacing: -0.5px;
	}
	.title .accent { color: var(--emph); }
	.subtitle {
	font-family: var(--font-sans);
	font-weight: 500;
	font-size: var(--fs-6);
	color: var(--text-secondary);
	margin-top: calc(2 * var(--u));
	font-style: italic;
	}
	.authors-line {
	font-family: var(--font-sans);
	font-size: var(--fs-4);
	color: var(--accent);
	font-weight: 600;
	margin-top: calc(3 * var(--u));
	}
	.authors-line .author { margin: 0 calc(4 * var(--u)); }
	.authors-line sup { font-size: 0.7em; color: var(--accent); }
	.authors-line .aff {
	color: var(--text-secondary);
	font-weight: 400;
	display: block;
	margin-top: calc(2 * var(--u));
	font-size: var(--fs-4);
	}

	.right-block {
	justify-self: end; /* keep logo+QR in the far-right corner */
	display: flex; align-items: center;
	gap: calc(10 * var(--u));
	}
	.qr-block { display: flex; flex-direction: column; align-items: center; gap: calc(2 * var(--u)); }
	.qr-block img {
	width: calc(85 * var(--u));
	height: calc(85 * var(--u));
	border: calc(2 * var(--u)) solid var(--accent);
	border-radius: calc(4 * var(--u) * var(--rs));
	background: white;
	padding: calc(2 * var(--u));
	}
	.qr-label {
	font-family: var(--font-sans);
	font-size: var(--fs-3);
	color: var(--accent);
	font-weight: 600;
	}
	/* Optional logo slot — drop your lab logo here. Keep its rendered height
	close to the QR (~85u) so the header doesn't grow disproportionately.
	A raster logo needs no exemption. An inline-SVG logo with brand colors
	must carry data-color-exempt="logo" so style_check skips its fills, e.g.:
	<svg data-color-exempt="logo" ...> ... </svg> */
	.logo-slot img { height: calc(85 * var(--u)); width: auto; max-width: calc(360 * var(--u)); object-fit: contain; display: block; }
	/* Logo size classes on .logo-slot -- pick from the file's aspect ratio (SKILL.md Gate E).
	tall/square sit at the QR-matched slot height; wide is intentionally shorter (~68% of the
	QR) and width-capped so a long wordmark doesn't out-mass the title. Do NOT combine these
	with logo-stack (that row is width-normalized, below). */
	.logo-slot.logo-tall img,
	.logo-slot.logo-square img { height: calc(85 * var(--u)); }
	.logo-slot.logo-wide img { height: calc(58 * var(--u)); max-width: calc(300 * var(--u)); }
	/* Logo chip: a solid backing that keeps a transparent / edge-white logo legible on a
	colored/dark header (white chip) or a light header (.logo-chip-dark); the padding + radius
	fold a stray white box into a deliberate rounded tile (SKILL.md Gate E, Background). */
	.logo-chip {
	display: inline-flex; align-items: center; justify-content: center;
	background: var(--bg-card); border-radius: calc(3 * var(--u) * var(--rs));
	padding: calc(3 * var(--u)) calc(5 * var(--u));
	}
	.logo-chip.logo-chip-dark { background: var(--text-primary); }
	/* logo-row: institution logos in the header right block (REAL logos the
	user provided — never fabricate a seal). Each img MUST carry
	data-color-exempt="logo"; height pairs with the QR. */
	.logo-row { display: flex; align-items: center; gap: calc(5 * var(--u)); }
	.logo-row img { height: calc(68 * var(--u)); width: auto; display: block; }
	/* --boxed variant: each logo in a labeled tile (logo + institution name) —
	more presence at poster distance; white tile + soft border, no new hues. */
	.logo-row .lr-item {
	display: flex; flex-direction: column; align-items: center;
	gap: calc(2 * var(--u));
	background: var(--bg-card);
	border: 1px solid var(--border-soft);
	border-radius: calc(3 * var(--u) * var(--rs));
	padding: calc(4 * var(--u)) calc(6 * var(--u));
	}
	.logo-row .lr-item img { height: calc(58 * var(--u)); }
	.logo-row .lr-label {
	font-family: var(--font-sans); font-weight: 600; font-size: var(--fs-1);
	color: var(--text-secondary); text-align: center; line-height: 1.15;
	}
	/* logo-stack variant: WIDE WORDMARKS (AR >= ~2) normalized to EQUAL WIDTH
	and stacked vertically, left-aligned. Use when two wide wordmarks of
	different aspect ratio read unbalanced height-matched in a row (equal
	width lets the less-wide mark grow taller and aligns a clean block).
	NOT for a square seal or tall mark (equal width blows it up — keep those
	height-matched). Do NOT also apply the logo-wide/tall/square classes. */
	.logo-row.logo-stack { flex-direction: column; align-items: flex-start; gap: calc(8 * var(--u)); }
	.logo-row.logo-stack img { width: calc(170 * var(--u)); height: auto; }
	/* venue badge may carry the official venue logo above its text line */
	.venue-badge img { height: calc(62 * var(--u)); width: auto; display: block; margin: 0 auto calc(2 * var(--u)); }
	.venue-badge .vb-title { font-family: var(--font-sans); font-weight: 800; font-size: var(--fs-5); color: var(--accent-deep); letter-spacing: 0.5px; }

	/* =========================================================
	OPTIONAL FRAMEWORK BANNER (delete this whole section if you don't need it)
	========================================================= */
	.framework-banner {
	display: flex;
	align-items: center;
	gap: calc(16 * var(--u));
	/* Flat solid emphasis fill (de-gradient). */
	background: var(--bg-emphasis);
	border: calc(1 * var(--u)) solid var(--border-soft);
	border-left: calc(6 * var(--u)) solid var(--accent);
	border-radius: calc(6 * var(--u) * var(--rs));
	padding: calc(6 * var(--u)) calc(14 * var(--u));
	}
	.framework-banner img { height: calc(120 * var(--u)); width: auto; display: block; }
	.framework-banner .banner-stats {
	flex: 1;
	display: grid;
	grid-template-columns: 1fr 1fr;
	gap: calc(6 * var(--u));
	}
	.framework-banner .bs-item {
	background: white;
	border: calc(1 * var(--u)) solid var(--border-soft);
	border-left: calc(3 * var(--u)) solid var(--accent);
	border-radius: calc(3 * var(--u) * var(--rs));
	padding: calc(2 * var(--u)) calc(8 * var(--u));
	text-align: center;
	}
	.framework-banner .bs-num {
	font-family: var(--font-sans);
	font-weight: 800;
	font-size: var(--fs-8);
	color: var(--accent);
	line-height: 1;
	}
	.framework-banner .bs-label {
	font-family: var(--font-sans);
	font-size: var(--fs-3);
	color: var(--text-secondary);
	margin-top: calc(2 * var(--u));
	line-height: 1.2;
	}
	.framework-banner .fb-text {
	flex: 1.6;
	font-family: var(--font-serif);
	font-size: var(--fs-6);
	line-height: 1.5;
	text-wrap: pretty; /* prose: fill each line, protect last-line orphan. NOT balance — on multi-sentence prose balance shortens+hyphenates line 1 (crammed-left, gap-right). */
	text-align: center;
	}
	.framework-banner .fb-text strong { color: var(--accent-deep); }
	.framework-banner .fb-label {
	display: inline-block;
	background: var(--accent);
	color: var(--accent-ink);
	font-family: var(--font-sans);
	font-size: var(--fs-5);
	font-weight: 700;
	padding: calc(2 * var(--u)) calc(8 * var(--u));
	border-radius: calc(4 * var(--u) * var(--rs));
	text-transform: uppercase;
	letter-spacing: 1px;
	vertical-align: middle;
	line-height: 1;
	position: relative;
	top: calc(-1 * var(--u));
	}

	/* Method-overview figure for the banner (banner-figure component,
	catalogued in COMPONENTS.md). Usually CAPTIONLESS -- the banner text
	block beside it is the figure's explanation. width:min-content collapses
	the slot to the IMAGE (a captionless figure's slot IS the image); if a
	short caption is used it wraps at the image box and can NEVER set the
	flex-item width; margin-inline centres a block image (text-align does
	not); overflow-wrap guards a long unbreakable token. Do NOT hand-roll a
	.fb-fig wrapper or a bare <img class="w-100"> here. */
	.framework-banner .banner-figure {
	flex: 0 0 auto;
	width: min-content;
	margin: 0;
	text-align: center;
	}
	.framework-banner .banner-figure img {
	height: calc(120 * var(--u));
	width: auto;
	display: block;
	margin-inline: auto;
	}
	.framework-banner .banner-figure figcaption {
	width: 100%;
	margin-top: calc(2 * var(--u));
	font-family: var(--font-sans);
	font-size: var(--fs-2);
	line-height: 1.2;
	color: var(--text-secondary);
	text-align: center;
	text-wrap: pretty;
	overflow-wrap: anywhere;
	}
	.framework-banner .banner-figure figcaption strong { color: var(--accent-deep); }

	/* =========================================================
	BODY: 4 columns, variable-width hint via fr values
	========================================================= */
	.body-grid {
	display: grid;
	grid-template-columns: 1fr 1.05fr 1.05fr 1fr;
	gap: calc(10 * var(--u));
	/* No overflow:hidden here — it would clip card shadows (pitfall #7).
	.poster already clips at the page boundary; min-height:0 is the
	grid-blowout guard. */
	min-height: 0;
	}
	.column {
	display: flex; flex-direction: column;
	gap: calc(6 * var(--u));
	min-height: 0;
	height: 100%;
	padding-bottom: calc(4 * var(--u)); /* shadow breathing room above takeaways */
	}

	/* =========================================================
	CARDS
	========================================================= */
	.card {
	background: var(--bg-card);
	border-radius: calc(5 * var(--u) * var(--rs));
	padding: calc(4 * var(--u)) calc(9 * var(--u));
	border: calc(1 * var(--u)) solid var(--border-soft);
	box-shadow: var(--shadow-card);
	position: relative;
	}
	.card.tinted { background: var(--bg-card-tint); }
	.card.card--compact { padding: calc(3 * var(--u)) calc(6 * var(--u)); } /* predefined variant: tighter padding (fix (f)) */
	.card.highlight {
	border-left: calc(6 * var(--u)) solid var(--accent);
	/* Flat emphasis tint (de-gradient). */
	background: var(--bg-emphasis);
	}

	.section-title {
	font-family: var(--font-sans);
	font-weight: 700;
	font-size: var(--fs-7);
	color: var(--accent-deep);
	margin-bottom: calc(3 * var(--u));
	display: flex; align-items: center;
	gap: calc(5 * var(--u));
	}
	/* Title text + any ★ marker share ONE .st-text span, so the heading wraps as
	natural text (hanging indent) instead of flex-wrapping atomic items: the
	number badge is never stranded on its own line and the ★ never widows. */
	.section-title .st-text { flex: 1; min-width: 0; line-height: 1.18; }
	/* Graceful fallback if a title is NOT wrapped in .st-text: float the badge so
	bare inline text still wraps beside it rather than dropping below it. */
	.section-title:not(:has(.st-text)) { display: block; line-height: 1.18; }
	.section-title:not(:has(.st-text)) .num { float: left; margin-right: calc(5 * var(--u)); }
	.section-title .num {
	display: inline-flex; align-items: center; justify-content: center;
	width: calc(22 * var(--u)); height: calc(22 * var(--u));
	background: var(--accent); color: var(--accent-ink);
	border-radius: 50%;
	font-size: var(--fs-5); font-weight: 700;
	flex-shrink: 0;
	}
	/* Inline "★ KEY" / "★ Headline" marker beside a section title. */
	.section-title .key-mark { color: var(--emph); font-size: var(--fs-3); }

	.body-text, .card p, .card li {
	font-family: var(--font-serif);
	font-size: var(--fs-4);
	line-height: 1.3;
	color: var(--text-primary);
	}
	.card ul, .card ol { padding-left: calc(18 * var(--u)); }
	.card li { margin-bottom: calc(2 * var(--u)); }

	.keyword { color: var(--accent); font-weight: 700; }
	.keyword-emph { color: var(--emph); font-weight: 700; }
	.highlight-text {
	background: var(--bg-emphasis);
	padding: 0 calc(3 * var(--u));
	border-radius: calc(2 * var(--u) * var(--rs));
	}

	/* Equation block */
	.eqn {
	background: var(--bg-emphasis);
	border-left: calc(3 * var(--u)) solid var(--accent);
	padding: calc(4 * var(--u)) calc(10 * var(--u));
	margin: calc(4 * var(--u)) 0;
	font-size: var(--fs-5);
	overflow-x: hidden;
	}
	/* Predefined variant for a larger / more legible display equation.
	`calc(var(--fs-N) * k)` is allowed here because this is a
	COMPONENTS.md-registered variant (DESIGN_FINAL §A / §10 f). */
	.eqn--large { font-size: calc(var(--fs-5) * 1.25); }
	.eqn .label {
	display: block;
	font-family: var(--font-sans);
	font-size: var(--fs-2);
	color: var(--accent);
	font-weight: 600;
	margin-bottom: calc(2 * var(--u));
	text-transform: uppercase;
	letter-spacing: 1px;
	}

	/* Callout: solid accent for primary, solid emphasis fill for theorems / "★ key" strips */
	.callout {
	background: var(--accent);
	color: var(--accent-ink);
	padding: calc(5 * var(--u)) calc(10 * var(--u));
	border-radius: calc(4 * var(--u) * var(--rs));
	font-size: var(--fs-4);
	margin: calc(4 * var(--u)) 0;
	}
	.callout strong { color: var(--emph); }
	/* Flat solid emphasis fill (de-gradient). */
	.callout.emph {
	background: var(--emph);
	color: var(--emph-ink); /* route through the token — accent-deep here would dodge the register swap AND fail AA on the default gold */
	}
	.callout.emph strong { color: var(--emph-ink); }

	/* Figure container */
	.figure { margin: calc(4 * var(--u)) 0; text-align: center; }
	.figure img:not([class*="w-"]) { width: 100%; }
	.figure--wide img { width: 100%; } /* predefined variant: force full card width over any .w-NN (fix (f)) */
	.figure img {
	border-radius: calc(4 * var(--u) * var(--rs));
	border: calc(1 * var(--u)) solid var(--fig-frame);
	background: var(--fig-bg);
	}
	.figure .caption {
	font-family: var(--font-sans);
	font-size: var(--fs-3);
	color: var(--text-secondary);
	margin-top: calc(3 * var(--u));
	line-height: 1.3;
	text-align: left;
	}
	.figure .caption strong { color: var(--accent-deep); }

	/* Float a figure BESIDE text in a TEXT-RICH card: text wraps to its side
	and then below it. The clearfix grows the card to contain the float;
	mark the <img> data-fig-layout="beside-text" so the AR gates honour the
	intentionally small width. Use ONLY when there is enough text to fill the
	figure's height -- a text-sparse card should center .figure instead. A
	short-text float leaves an L-shaped void below the text, which polish's
	FIG/BESIDE-TEXT-VOID flags. */
	.fig-wrap::after { content: ""; display: table; clear: both; }
	.ff-fig {
	float: right;
	width: 48%; max-width: 58%; min-width: 38%;
	margin: calc(1 * var(--u)) 0 calc(3 * var(--u)) calc(11 * var(--u));
	text-align: center;
	}
	.ff-fig.left {
	float: left;
	margin: calc(1 * var(--u)) calc(11 * var(--u)) calc(3 * var(--u)) 0;
	}
	.ff-fig img {
	display: block;
	width: 100%;
	border-radius: calc(4 * var(--u) * var(--rs));
	border: calc(1 * var(--u)) solid var(--fig-frame);
	background: var(--fig-bg);
	}
	.ff-fig .caption {
	font-family: var(--font-sans);
	font-size: var(--fs-3);
	color: var(--text-secondary);
	margin-top: calc(3 * var(--u));
	line-height: 1.3;
	text-align: center;
	}

	/* Result table with .ours row highlighted with the emphasis tint */
	.result-table {
	width: 100%;
	border-collapse: collapse;
	font-family: var(--font-sans);
	font-size: var(--fs-3);
	margin-top: calc(3 * var(--u));
	}
	.result-table th, .result-table td {
	padding: calc(2 * var(--u)) calc(4 * var(--u));
	text-align: center;
	border-bottom: calc(1 * var(--u)) solid var(--border-soft);
	}
	.result-table thead th {
	background: var(--accent); color: var(--accent-ink);
	font-weight: 600; font-size: var(--fs-2);
	}
	.result-table tbody tr.group-row td {
	background: var(--bg-emphasis); font-weight: 700;
	text-align: left;
	color: var(--accent-deep);
	padding-left: calc(8 * var(--u));
	border-bottom: calc(2 * var(--u)) solid var(--accent);
	}
	.result-table tbody tr.ours td { background: var(--emph-soft); font-weight: 700; }
	.result-table tbody tr.ours td:first-child { color: var(--accent-deep); }
	/* Greyed reference row (was inline color:#888). */
	.result-table tbody tr.reference td { color: var(--text-muted); }
	.result-table .method { text-align: left; padding-left: calc(8 * var(--u)); }
	.result-table .best { color: var(--accent); font-weight: 700; }

	/* 3-up stat box */
	.keybox {
	display: grid;
	grid-template-columns: repeat(3, 1fr);
	gap: calc(4 * var(--u));
	margin: calc(4 * var(--u)) 0 0;
	}
	.keybox .kb-item {
	background: var(--bg-emphasis);
	border-top: calc(2 * var(--u)) solid var(--accent);
	padding: calc(3 * var(--u));
	text-align: center;
	/* The grid stretches every tile to the tallest one; center the content
	vertically so a 1-line tile's number aligns with a 2-line neighbour's
	instead of top-ragged (SKILL Layout pitfall — stat-tile alignment). */
	display: flex; flex-direction: column; justify-content: center;
	border-radius: 0 0 calc(3 * var(--u) * var(--rs)) calc(3 * var(--u) * var(--rs));
	}
	.kb-item .kb-num {
	font-family: var(--font-sans);
	font-weight: 800;
	font-size: var(--fs-6);
	color: var(--accent);
	line-height: 1;
	}
	.kb-item .kb-label {
	font-family: var(--font-sans);
	font-size: var(--fs-1);
	color: var(--text-secondary);
	margin-top: calc(2 * var(--u));
	line-height: 1.1;
	/* Reserve two label lines (2lh tracks this line-height) so 1-line
	and 2-line labels occupy the same height and the big numbers
	align across tiles. Keep labels <= 2 lines (COMPONENTS §keybox). */
	min-height: 2lh;
	}

	/* Optional takeaways strip (delete the section if not needed) */
	.takeaways-strip {
	display: grid;
	grid-template-columns: auto repeat(4, 1fr);
	align-items: center;
	gap: calc(10 * var(--u));
	/* Flat solid emphasis fill (de-gradient). */
	background: var(--bg-emphasis);
	border: calc(1 * var(--u)) solid var(--border-soft);
	border-radius: calc(5 * var(--u) * var(--rs));
	padding: calc(8 * var(--u)) calc(12 * var(--u));
	}
	.takeaways-strip .ts-title {
	font-family: var(--font-sans);
	font-size: var(--fs-6);
	font-weight: 800;
	color: var(--accent-deep);
	display: flex; align-items: center; gap: calc(6 * var(--u));
	}
	.takeaways-strip .ts-title .num {
	display: inline-flex; align-items: center; justify-content: center;
	width: calc(22 * var(--u)); height: calc(22 * var(--u));
	background: var(--accent); color: var(--accent-ink);
	border-radius: 50%;
	font-size: var(--fs-5); font-weight: 700;
	}
	.takeaways-strip .ts-item {
	border-left: calc(3 * var(--u)) solid var(--accent);
	padding-left: calc(8 * var(--u));
	line-height: 1.3;
	text-align: center;
	text-wrap: balance;
	}
	.takeaways-strip .ts-key {
	font-family: var(--font-sans);
	font-size: var(--fs-3);
	font-weight: 700;
	color: var(--accent);
	text-transform: uppercase;
	letter-spacing: 1px;
	}
	.takeaways-strip .ts-text {
	font-family: var(--font-serif);
	font-size: var(--fs-4);
	margin-left: calc(4 * var(--u));
	}

	/* Footer */
	.footer {
	grid-column: 1 / -1;
	display: flex; justify-content: space-between; align-items: baseline;
	/* When the two blocks can't sit side by side they stack (wrap) instead of
	overflowing; row-gap keeps the stacked blocks legible. */
	flex-wrap: wrap; gap: calc(2 * var(--u)) calc(10 * var(--u));
	padding-top: calc(8 * var(--u));
	border-top: calc(1 * var(--u)) solid var(--border-soft);
	font-family: var(--font-sans);
	font-size: var(--fs-5);
	color: var(--text-muted);
	}
	.footer .method-name { color: var(--accent-deep); }
	/* Long repo URL / email breaks mid-token rather than overflowing the edge. */
	.footer .repo { color: var(--accent); font-weight: 600; overflow-wrap: anywhere; }

	/* Optional watermark */
	.ornament {
	position: absolute;
	right: calc(20 * var(--u));
	bottom: calc(20 * var(--u));
	font-family: var(--font-sans);
	font-size: var(--fs-9);
	color: var(--ornament-ink);
	font-weight: 900;
	letter-spacing: 4px;
	pointer-events: none;
	user-select: none;
	}

	/* Identity mark — corner-signature (Axis 8 identity accessory, ALWAYS ON).
	Sits in the poster's outer-padding "safe zone" (bottom-right gutter),
	clear of the content box; glyph-only ⊕. This SUPERSEDES the legacy
	.ornament text watermark above -- leave that disabled (enabling both
	just duplicates the corner mark). See COMPONENTS.md. */
	.ps-sprite { position: absolute; width: 0; height: 0; overflow: hidden; }
	.corner-sig {
	position: absolute;
	right: calc(4 * var(--u));
	bottom: calc(3 * var(--u));
	width: calc(7 * var(--u));
	height: calc(7 * var(--u));
	color: var(--ps-mark-ink);
	opacity: 0.5;
	pointer-events: none;
	user-select: none;
	}
	.corner-sig svg { display: block; width: 100%; height: 100%; }
	/* Dark / colored ground: swap to a light ink so the mark doesn't vanish.
	Add `on-dark` to .corner-sig on an Axis-2 dark-base poster. */
	.corner-sig.on-dark { color: var(--accent-light); }

	/* Identity mark — woven-signature: ONE ⊕ riding EXISTING content (a best /
	target / ★ marker, an inline bullet, a terminal sentence period, the
	wordmark "o"). Sized to the HOST TEXT so it reads as an inline glyph, not
	a block, and inherits the host ink so it blends. Placed per-poster (never
	on a data character); aria-hidden; data-color-exempt="logo". See
	COMPONENTS.md. */
	[data-ps-mark="woven"] {
	display: inline-block;
	width: 0.82em;
	height: 0.82em;
	vertical-align: -0.14em;
	line-height: 0;
	}
	[data-ps-mark="woven"] svg { display: block; width: 100%; height: 100%; }

	/* =========================================================
	DENSITY COMPONENTS (catalogued — see COMPONENTS.md).
	Guardrail: no component-local color semantics. Distinction is
	carried by labels, order and typography — never by new hues.
	========================================================= */
	/* equation-stack: compact multi-row formula stack (denser than one big .eqn) */
	.equation-stack { margin: calc(2 * var(--u)) 0; }
	.equation-stack .eqn { margin: calc(2 * var(--u)) 0; padding: calc(2 * var(--u)) calc(8 * var(--u)); }

	/* eqn-anatomy: term-by-term anatomy grid (2x2 default; --row variant = 1x4) */
	.eqn-anatomy {
	display: grid; grid-template-columns: 1fr 1fr;
	gap: calc(2.5 * var(--u)); margin: calc(3 * var(--u)) 0;
	}
	.eqn-anatomy.eqn-anatomy--row { grid-template-columns: repeat(4, 1fr); }
	.eqn-anatomy .ea-item {
	background: var(--bg-card-tint);
	border: 1px solid var(--border-soft);
	border-left: calc(2 * var(--u)) solid var(--accent);
	border-radius: calc(2 * var(--u) * var(--rs));
	padding: calc(2 * var(--u)) calc(4 * var(--u));
	font-size: var(--fs-2); line-height: 1.3;
	}
	.eqn-anatomy .ea-tag {
	display: inline-block; font-family: var(--font-sans); font-weight: 700;
	font-size: var(--fs-1); color: var(--accent-deep);
	background: var(--accent-light); border-radius: calc(1.5 * var(--u) * var(--rs));
	padding: calc(0.5 * var(--u)) calc(2.5 * var(--u));
	margin-right: calc(1.5 * var(--u));
	}

	/* flow-strip: labeled pipeline, ALL steps the same accent; ONLY the
	final step may carry the emphasis top bar. Not an "algorithm" unless the
	paper has one — see COMPONENTS.md. */
	.flow-strip { display: flex; align-items: stretch; gap: calc(1.5 * var(--u)); margin: calc(3 * var(--u)) 0; }
	.flow-strip .step {
	flex: 1; background: var(--accent-light);
	border: 1px solid var(--accent-soft); border-radius: calc(2 * var(--u) * var(--rs));
	padding: calc(2 * var(--u)) calc(2.5 * var(--u));
	text-align: center; font-size: var(--fs-2); line-height: 1.25;
	}
	.flow-strip .step .step-name {
	display: block; font-family: var(--font-sans); font-weight: 700;
	font-size: var(--fs-1); color: var(--accent-deep);
	letter-spacing: 0.5px; text-transform: uppercase;
	margin-bottom: calc(1 * var(--u));
	}
	.flow-strip .step--final { border-top: calc(1.5 * var(--u)) solid var(--emph); background: var(--emph-soft); }
	.flow-strip .arrow {
	align-self: center; color: var(--accent);
	font-family: var(--font-sans); font-weight: 700; font-size: var(--fs-4);
	flex: 0 0 auto;
	}

	/* figure--duo: two paper figures sharing one caption; each img MUST be
	a data-source="paper" asset and carry .w-45/.w-50 (42-48% each). */
	.figure--duo { display: flex; gap: calc(3 * var(--u)); align-items: flex-start; justify-content: center; }
	.figure--duo img { max-width: 48%; min-width: 0; } /* hard cap: duo contract is 42-48% each; an oversized w-NN cannot overflow the card */

	/* result-table derived column (emph-soft = DERIVED arithmetic; label it) */
	.result-table th.derived, .result-table td.derived {
	background: var(--emph-soft); font-family: var(--font-sans); font-weight: 700;
	}

	/* keybox 4-up variant */
	.keybox.keybox--4 { grid-template-columns: repeat(4, 1fr); }

	/* algo: compact numbered procedure — ONLY when the paper itself states
	an explicit algorithm/procedure; never invent steps. */
	ol.algo { padding-left: calc(16 * var(--u)); }
	ol.algo li { margin-bottom: calc(1.5 * var(--u)); font-size: var(--fs-3); line-height: 1.3; }

	/* claim-pills: provenance mini-table (numeric-heavy posters only) */
	.claim-pills { width: 100%; border-collapse: collapse; font-family: var(--font-sans); font-size: var(--fs-2); }
	.claim-pills td {
	border-bottom: 1px solid var(--border-soft);
	padding: calc(1.5 * var(--u)) calc(3 * var(--u));
	}
	.claim-pills .cp-id {
	font-weight: 700; color: var(--accent-deep); background: var(--accent-light);
	border-radius: calc(1.5 * var(--u) * var(--rs)); padding: 0 calc(2.5 * var(--u) * var(--rs)); white-space: nowrap;
	}
	.claim-pills .cp-fact { color: var(--accent-deep); font-weight: 700; }
	.claim-pills .cp-derived { color: var(--emph); font-weight: 700; }

	/* Grid-blowout guards: (1) columns may shrink below min-content so a
	wide equation can never widen its track and squeeze siblings;
	(2) over-wide display math scales down to fit its box instead of
	blowing out the card (SVG keeps aspect via height:auto). */
	.body-grid > .column { min-width: 0; }
	.card { min-width: 0; }
	.eqn mjx-container > svg { max-width: 100%; height: auto; }

	/* ── Utility classes — defined LAST so equal-specificity ties
	resolve in the utility's favor (source order). ── */
	.fs-1 { font-size: var(--fs-1); }
	.fs-2 { font-size: var(--fs-2); }
	.fs-3 { font-size: var(--fs-3); }
	.fs-4 { font-size: var(--fs-4); }
	.fs-5 { font-size: var(--fs-5); }
	.fs-6 { font-size: var(--fs-6); }
	.fs-7 { font-size: var(--fs-7); }
	.fs-8 { font-size: var(--fs-8); }
	.fs-9 { font-size: var(--fs-9); }

	.mt-1 { margin-top: calc(1 * var(--u)); }
	.mt-2 { margin-top: calc(2 * var(--u)); }
	.mt-3 { margin-top: calc(3 * var(--u)); }
	.mt-4 { margin-top: calc(4 * var(--u)); }
	.mt-5 { margin-top: calc(5 * var(--u)); }
	.mt-6 { margin-top: calc(6 * var(--u)); }

	.mb-1 { margin-bottom: calc(1 * var(--u)); }
	.mb-2 { margin-bottom: calc(2 * var(--u)); }
	.mb-3 { margin-bottom: calc(3 * var(--u)); }
	.mb-4 { margin-bottom: calc(4 * var(--u)); }

	.w-45 { width: 45%; }
	.w-50 { width: 50%; }
	.w-55 { width: 55%; }
	.w-60 { width: 60%; }
	.w-65 { width: 65%; }
	.w-70 { width: 70%; }
	.w-75 { width: 75%; }
	.w-80 { width: 80%; }
	.w-85 { width: 85%; }
	.w-90 { width: 90%; }
	.w-95 { width: 95%; }
	.w-100 { width: 100%; }

	.text-secondary { color: var(--text-secondary); }
	.text-muted { color: var(--text-muted); }
	.nowrap { white-space: nowrap; }
	.text-center { text-align: center; }

	/* =========================================================
	PRINT OVERRIDE — KEEP LAST so it wins source-order ties.
	========================================================= */
	@media print {
	html, body { background: white; }
	.poster { margin: 0; box-shadow: none; width: 60in; height: 36in; page-break-after: avoid; }
	:root { --u: 1mm; }
	}
	</style>
	</head>
	<body>

	<div class="poster" data-measure-role="poster"
	data-posterly-contract="identity-v1" data-ps-identity="on">

	<!-- Identity sprite: the posterly ⊕ registration glyph, defined ONCE.
	Zero-size + absolute so it never enters the grid or shifts layout.
	data-color-exempt="logo" -- it IS the posterly logo mark, which keeps
	style_check rule 11 (no decorative SVG) and the rule-4 hue gate honest.
	Symbol id/viewBox/paths are byte-identical across all templates. -->
	<svg class="ps-sprite" data-color-exempt="logo" aria-hidden="true"><defs>
	<symbol id="psReg" viewBox="0 0 100 100">
	<circle cx="50" cy="50" r="25" fill="none" stroke="currentColor" stroke-width="8"/>
	<line x1="50" y1="7" x2="50" y2="93" stroke="currentColor" stroke-width="8"/>
	<line x1="7" y1="50" x2="93" y2="50" stroke="currentColor" stroke-width="8"/>
	</symbol>
	</defs></svg>

	<!-- =========================== HEADER =========================== -->
	<header class="header" data-measure-role="header">
	<!-- LEFT: venue badge. -->
	<div class="venue-badge">
	<div class="vb-venue">ICML</div>
	<div class="vb-year">2026</div>
	<div class="vb-tag">REPRODUCTION</div>
	</div>

	<!-- CENTER: title block -->
	<div class="title-block">
	<h1 class="title">Flatness-Aware SGLD: <span class="accent">Reproduction</span></h1>
	<div class="subtitle">Theory verified to machine precision; the per-step efficiency claim reproduces; the large-scale accuracy & flatness gains are not visible at CPU-affordable reduced scale.</div>
	<div class="authors-line">
	<span class="author">Independent reproduction agent (Claude Code)</span> ·
	<span class="author">ICML 2026 Reproducibility Challenge</span>
	<span class="aff">Paper: arXiv 2510.02174 · OpenReview Kw0nhxeTeI · CPU-only (numpy + Modal 32-core)</span>
	</div>
	</div>

	<!-- RIGHT: QR to the published logbook. -->
	<div class="right-block">
	<div class="qr-block">
	<img data-color-exempt="logo" src="qr.png" alt="QR code linking to the published Trackio reproduction logbook">
	<div class="qr-label">Full logbook</div>
	</div>
	</div>
	</header>

	<!-- ===================== OPTIONAL FRAMEWORK BANNER =====================
	Delete this entire <section> if your poster doesn't need a banner.
	(To do so, also remove `auto` from the grid-template-rows above.) -->
	<section class="framework-banner" data-measure-role="banner">
	<!-- OPTIONAL method-overview figure on the left. The banner text block
	beside it IS its explanation, so it usually needs NO caption. Use the
	captionless banner-figure component (NOT a bare <img class="w-100"> --
	the `.framework-banner img { width:auto }` rule wins over .w-100):
	<figure class="banner-figure">
	<img src="assets/paper_figures/framework.png" data-source="paper"
	data-asset-id="framework" alt="Method overview">
	</figure>
	A figcaption here just duplicates the banner text and a long one
	stretches the figure slot (polish: BANNER/IMAGE-SLOT). If you truly
	need panel labels, bake them into the image or keep them to ONE short
	<figcaption> line (the component bounds it to the image width). -->
	<div class="fb-text">
	<span class="fb-label">CLAIM SCORECARD</span>
	<strong>fSGLD</strong> = SGLD on the smoothing surrogate $g_\sigma$; the coupling $\sigma=\beta^{-(1+\eta)/4}$ makes its invariant measure target the Hessian-trace regularized Gibbs law, biasing toward flat minima at one gradient per step. We audit all three theorems exactly, and run all three experiments at the largest reduced scale the CPU-only budget allows, reporting each verdict on its own claim page.
	</div>
	<div class="banner-stats">
	<div class="bs-item"><div class="bs-num">3/3</div><div class="bs-label">theorems<br>verified</div></div>
	<div class="bs-item"><div class="bs-num">1.92×</div><div class="bs-label">SAM cost<br>vs fSGLD ✓</div></div>
	<div class="bs-item"><div class="bs-num">4.00</div><div class="bs-label">O(σ⁴) slope<br>(machine prec.)</div></div>
	<div class="bs-item"><div class="bs-num">CPU</div><div class="bs-label">no GPU<br>6s + 4.5h</div></div>
	</div>
	</section>

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	<div class="card highlight" data-measure-role="card" data-logbook-target="executive-summary">
	<div class="section-title"><span class="num">1</span><span class="st-text">What was reproduced</span></div>
	<p class="body-text">
	<span class="keyword">fSGLD</span> runs SGLD on the randomized-smoothing surrogate $g_\sigma(\theta)=\mathbb{E}_\varepsilon[u(\theta+\varepsilon)]$. Since $g_\sigma = u + \tfrac{\sigma^2}{2}\,\mathrm{tr}\,H + O(\sigma^4)$, its invariant measure targets the <span class="keyword">Hessian-trace regularized</span> Gibbs law, biasing toward flat minima at one gradient per step.
	</p>
	<table class="result-table">
	<thead><tr><th class="method">Claim</th><th>Kind</th><th>Verdict</th></tr></thead>
	<tbody>
	<tr class="ours"><td class="method">1 · 2 · 3</td><td>theory</td><td class="best">verified</td></tr>
	<tr><td class="method">4</td><td>expt</td><td>efficiency ✓ / acc ✗</td></tr>
	<tr class="reference"><td class="method">5</td><td>expt</td><td>verified (reduced)</td></tr>
	<tr><td class="method">6</td><td>expt</td><td>not repro (reduced)</td></tr>
	</tbody>
	</table>
	<div class="callout mt-4">
	Theory verifies to machine precision and the efficiency claim reproduces, but the large-scale accuracy and flatness gains do not appear at CPU-affordable scale.
	</div>
	</div>

	<div class="card" data-measure-role="card" data-logbook-target="claim-1-coupling-invariant-measure-converges-in-kl-and-w2-to-the-hessian-trace-target-prop-3-1-true-prop-3-4">
	<div class="section-title"><span class="num">2</span><span class="st-text">Claim 1 — coupling ⇒ target</span></div>
	<p class="body-text">
	<strong>Prop 3.4</strong> (claim mislabels "3.1"): under $\sigma=\beta^{-(1+\eta)/4}$, $W_2(\pi^{\mathrm{fSGLD}}_\beta,\pi^\star_{\beta,\sigma})\!\to\!0$. The surrogate residual $g_\sigma-v$ is <span class="keyword">exactly $3\sigma^4$</span>.
	</p>
	<div class="eqn">
	<span class="label">Remainder scaling (Part A)</span>
	$$\text{slope}\,\log\|g_\sigma\!-\!v\|\,/\log\sigma = 4.0000$$
	</div>
	<ul class="fs-4">
	<li>Coupling kills remainder: $\beta\!\cdot\!\mathbb{E}[R]\sim\beta^{-\eta}$, slope $=-0.1000=-\eta$.</li>
	<li>KL/$W_2$ shrink at large $\beta$; flatness bias directional.</li>
	</ul>
	<table class="result-table">
	<thead><tr><th class="method">Check</th><th>Predicted</th><th>Measured</th></tr></thead>
	<tbody>
	<tr class="ours"><td class="method">D1a slope</td><td>4</td><td class="best">4.0000</td></tr>
	<tr class="ours"><td class="method">D1b slope</td><td>−η = −0.1</td><td class="best">−0.1000</td></tr>
	<tr><td class="method">KL log-slope</td><td>< 0</td><td>−0.165</td></tr>
	<tr class="reference"><td class="method">flat-basin mass</td><td>π⋆ > e^(−βu)</td><td>4e-9 ≫ 2e-13</td></tr>
	</tbody>
	</table>
	</div>

	</div>

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	<div class="card" data-measure-role="card" data-logbook-target="claim-2-non-asymptotic-w1-bound-and-o-1-4-w2-rate-for-fsgld-thm-3-2-cor-3-3-true-thm-3-5-cor-d-16">
	<div class="section-title"><span class="num">3</span><span class="st-text">Claim 2 — discretization rates</span></div>
	<p class="body-text"><strong>Thm 3.5 / Cor D.16</strong> (claim mislabels "3.2/3.3"): $W_1$ discretization is $O(\lambda^{1/2})$, $W_2$ is $O(\lambda^{1/4})$, above an invariant-measure floor $D$.</p>
	<div class="eqn">
	<span class="label">Fitted exponents, Part C</span>
	$$p_{W_1}=2.18\;(\geq 0.40),\quad q_{W_2}=3.46\;(\geq 0.20)$$
	</div>
	<p class="body-text fs-4">Measured error decays within the $O(\sqrt\lambda)$ / $O(\lambda^{1/4})$ envelopes; the $k$-sweep shows the predicted exponential-in-$k$ mixing saturating at floor $D$.</p>
	<table class="result-table">
	<thead><tr><th class="method">λ</th><th>W₁</th><th>W₂</th></tr></thead>
	<tbody>
	<tr><td class="method">0.001</td><td>0.0036</td><td>0.0042</td></tr>
	<tr><td class="method">0.008</td><td>0.0227</td><td>0.0278</td></tr>
	<tr><td class="method">0.032</td><td>0.1648</td><td>0.3301</td></tr>
	</tbody>
	</table>
	<p class="body-text fs-3">The $k$-sweep W₁ falls 20k→0.049, 100k→0.026, 500k→0.024, then holds at floors W₁ 0.0165, W₂ 0.0507.</p>
	</div>

	<div class="card highlight" data-measure-role="card" data-logbook-target="claim-3-excess-risk-bounds-for-the-hessian-trace-objective-toward-flat-minima-thm-3-5-true-thm-3-8">
	<div class="section-title"><span class="num">4</span><span class="st-text">Claim 3 — excess risk</span></div>
	<p class="body-text"><strong>Thm 3.8</strong> (claim mislabels "3.5"): $\mathbb{E}[v(\theta_k)]-\inf v \leq D_1 e^{-\dot c\lambda k/4}+D_2\lambda^{1/4}+D_3$.</p>
	<div class="eqn eqn--large">
	<span class="label">Floor-subtracted rate</span>
	$$r_{\text{excess}}=9.97\;\;(\geq 0.20),\;\text{decreasing}$$
	</div>
	<div class="callout emph">
	<strong>Reading.</strong> $v=u+\tfrac{\sigma^2}{2}\mathrm{tr}\,H$ penalizes curvature, so minimizing $v$ = converging toward <em>flat</em> minima. Excess risk sits at a finite-$\beta$ floor, exactly as Thm 3.8 predicts.
	</div>
	<p class="body-text fs-3 mt-2">$\mathbb{E}[v(\theta_k)]-\inf v$ across λ: 0.317, 0.316, 0.314, 0.308, 0.306, 0.323 — floor $D_3 \approx 0.312$. The λ-dependent part is small and in-envelope; the finite-β floor dominates — exactly as expected, since the paper stresses fSGLD operates at finite β and is not an annealing scheme.</p>
	</div>

	</div>

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	<div class="card highlight" data-measure-role="card" data-logbook-target="claim-4-resnet-34-cifar-100n-fsgld-vs-sam-accuracy-and-per-iteration-speed-table-1-paper-table-3">
	<div class="section-title"><span class="num">5</span><span class="st-text">Claim 4 — ResNet-34 scratch</span></div>
	<p class="body-text fs-3 text-secondary mb-1">
	Reduced scale: SmallCNN 0.5M, 15k CIFAR-100N subset (real 40.2% human noise), 30 ep, 3 seeds, fixed β=1e8.
	</p>
	<p class="body-text fs-4"><strong>Efficiency ✓ (architecture-independent):</strong> SAM needs 2 gradient evals, fSGLD needs 1 — measured cost ratio matches the paper's mechanism (paper 41.3/23.7 ≈ 1.74×).</p>
	<div class="keybox">
	<div class="kb-item"><div class="kb-num">1.92×</div><div class="kb-label">SAM /<br>fSGLD ✓</div></div>
	<div class="kb-item"><div class="kb-num">1.03×</div><div class="kb-label">fSGLD /<br>SGD ✓</div></div>
	<div class="kb-item"><div class="kb-num">40.0%</div><div class="kb-label">fSGLD acc<br>< SAM 41.5 ✗</div></div>
	</div>
	<table class="result-table">
	<thead><tr><th class="method">Optimizer</th><th>acc</th><th>s/iter</th></tr></thead>
	<tbody>
	<tr><td class="method">SGD</td><td>41.50</td><td>0.592</td></tr>
	<tr class="reference"><td class="method">SAM</td><td>41.45</td><td>1.173</td></tr>
	<tr class="ours"><td class="method">fSGLD</td><td>39.97</td><td class="best">0.611</td></tr>
	</tbody>
	</table>
	<p class="body-text mt-2 fs-3"><strong>Accuracy ✗ (reduced):</strong> fSGLD 39.97% < SAM 41.45%; the paper's fSGLD > SAM (61.51 vs 59.18) needs the full ResNet and a tuned β. <em>Extraction defect: the claim's 62.02% appears nowhere in the paper.</em></p>
	</div>

	<div class="card" data-measure-role="card" data-logbook-target="claim-5-vit-b-16-fine-tune-cifar-100n-fsgld-vs-sam-table-2-paper-table-4">
	<div class="section-title"><span class="num">6</span><span class="st-text">Claim 5 — fine-tune regime</span></div>
	<p class="body-text">Emulated fine-tuning (clean-pretrain → noisy-finetune SmallCNN; ViT-B/16 backbone not downloadable on CPU budget). Same optimizer that lost from scratch <em>wins</em> here.</p>
	<ul class="mt-3">
	<li><strong>fSGLD 46.95%</strong> > SAM 46.39% > SGD 46.24%.</li>
	<li>Paper ordering (fSGLD 75.67 > SAM 74.66) reproduces.</li>
	</ul>
	<table class="result-table">
	<thead><tr><th class="method">Optimizer</th><th>fine-tune acc</th></tr></thead>
	<tbody>
	<tr><td class="method">SGD</td><td>46.24</td></tr>
	<tr class="reference"><td class="method">SAM</td><td>46.39</td></tr>
	<tr class="ours"><td class="method">fSGLD</td><td class="best">46.95</td></tr>
	</tbody>
	</table>
	<div class="callout mt-3">
	Flatness bias helps most when resisting noisy-label overfitting from a good pretrained basin — exactly the regime with the smaller fSGLD−SAM gap in the paper.
	</div>
	</div>

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	<div class="card highlight" data-measure-role="card" data-logbook-target="claim-6-coupling-ablation-and-hessian-flatness-spectrum-sec-4-3-fig-3-paper-sec-4-4-4-5-fig-1-2">
	<div class="section-title"><span class="num">7</span><span class="st-text">Claim 6 — ablation + flatness</span></div>
	<p class="body-text fs-2 text-secondary mb-1">
	Hessian spectrum of the trained models (Hutchinson trace + power-iteration $\lambda_{\text{top}}$). Paper: fSGLD flatter than SGD.
	</p>
	<table class="result-table">
	<thead>
	<tr>
	<th class="method">Optimizer</th>
	<th>tr H ↓</th>
	<th>λ<sub>top</sub> ↓</th>
	</tr>
	</thead>
	<tbody>
	<tr><td class="method">SGD</td><td>984.9</td><td>95.8</td></tr>
	<tr class="reference"><td class="method">SAM (flattest)</td><td>831.8</td><td>87.5</td></tr>
	<tr class="ours"><td class="method">fSGLD</td><td>3042.9</td><td>331.7</td></tr>
	</tbody>
	</table>
	<p class="body-text mt-2 fs-3">
	<strong>Reversed at reduced scale:</strong> fSGLD's final iterate is ~3× <em>sharper</em> than SGD, not flatter; SAM is the flattest.
	</p>
	<p class="body-text fs-2 text-secondary mb-1">Coupling ablation — accuracy vs implied η (β fixed):</p>
	<table class="result-table">
	<thead><tr><th class="method">η</th><th>−0.5</th><th>0.1</th><th>0.5</th><th>0.9</th><th>1.5</th></tr></thead>
	<tbody>
	<tr><td class="method">acc %</td><td>1.0</td><td>40.0</td><td class="best">41.4</td><td class="best">41.5</td><td>41.4</td></tr>
	</tbody>
	</table>
	<p class="body-text fs-3">Negative η diverges (✓ coupling sign matters), but the prescribed η=0.1 is not the peak and η=1.5 does not drop — the strict interior-(0,1) optimum is not reproduced here.</p>
	</div>

	<div class="card" data-measure-role="card" data-logbook-target="conclusion">
	<div class="section-title"><span class="num">8</span><span class="st-text">Scope & verdicts</span></div>
	<ul class="fs-4">
	<li><strong>Claims 1–3</strong> (theory) — VERIFIED, machine precision.</li>
	<li><strong>Claim 4</strong> — efficiency ✓ / accuracy ✗ (reduced).</li>
	<li><strong>Claim 5</strong> — VERIFIED (reduced, fine-tune).</li>
	<li><strong>Claim 6</strong> — NOT reproduced (reduced).</li>
	</ul>
	<table class="result-table">
	<thead><tr><th class="method">Axis</th><th>This repro</th><th>Paper</th></tr></thead>
	<tbody>
	<tr><td class="method">Model</td><td>SmallCNN 0.5M</td><td>ResNet-34 / ViT-B/16</td></tr>
	<tr><td class="method">Data</td><td>15k subset</td><td>50k + WebVision</td></tr>
	<tr><td class="method">Epochs</td><td>30 / 15</td><td>150 / 75</td></tr>
	<tr class="ours"><td class="method">Hardware</td><td class="best">CPU</td><td>A100 GPUs</td></tr>
	</tbody>
	</table>
	<div class="callout mt-3 fs-3">
	Cost: theory ≈ \$0 (numpy, 6.3 s); experiments ≈ \$6 est. (~4.5 h on a 32-core Modal CPU). Data is real, sha256-verified CIFAR-100N with genuine human label noise; no GPU was used, and every derivation, script and log ships in the linked bundle.
	</div>
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	<div class="ts-title"><span class="num">9</span> Takeaways <span data-ps-mark="woven" data-color-exempt="logo" aria-hidden="true"><svg viewBox="0 0 100 100"><use href="#psReg"/></svg></span></div>
	<div class="ts-item"><span class="ts-key">Theory holds.</span><span class="ts-text">Surrogate identity & coupling exact to 1e-12; all 3 theorems verified.</span></div>
	<div class="ts-item"><span class="ts-key">Efficiency holds.</span><span class="ts-text">SAM ≈ 1.9× fSGLD per-step cost — architecture-independent.</span></div>
	<div class="ts-item"><span class="ts-key">Scale matters.</span><span class="ts-text">From-scratch accuracy & flatness gains vanish at CPU-affordable scale.</span></div>
	<div class="ts-item"><span class="ts-key">Claim text buggy.</span><span class="ts-text">62.02% not in paper; every theorem/table/figure mis-numbered.</span></div>
	</section>

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	<div>
	<strong class="method-name">Flatness-Aware SGLD — Reproduction</strong> · ICML 2026 Reproducibility Challenge ·
	Independent numerical audit + reduced-scale CPU experiments.
	</div>
	<div>
	Paper: <span class="repo">arxiv.org/abs/2510.02174</span>  ·
	Logbook: <span class="repo">hf.co/spaces/JG1310/repro-flatness-aware-stochastic-gradient-langevin-dynamics</span>
	</div>
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