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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>Quadratically Regularized OT: Localization Bounds — ICML 2026 Reproduction</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. Replace text or swap for an <img> if your venue allows logos. -->
	<div class="venue-badge">
	<div class="vb-venue">ICML</div>
	<div class="vb-year">2026</div>
	<div class="vb-tag">REPRO</div>
	</div>

	<!-- CENTER: title block -->
	<div class="title-block">
	<h1 class="title">Quadratically Regularized OT: <span class="accent">Localization Bounds</span></h1>
	<div class="subtitle">Independent reproduction — two localization theorems for QOT, both verified by derivation audit + full-scale diagnostic.</div>
	<div class="authors-line">
	<span class="author">Original paper: Long Nguyen-Chi, Nam Nguyen, Binh T. Nguyen</span> ·
	<span class="author">Reproduction: Claude Code agent<sup>✉</sup></span>
	<span class="aff">ICML 2026 Reproducibility Challenge · Hugging Face × AlphaXiv · arXiv:2605.24644</span>
	</div>
	</div>

	<!-- RIGHT: QR only (no lab logo for a reproduction poster). -->
	<div class="right-block">
	<div class="qr-block">
	<img data-color-exempt="logo" src="images/qr.png" alt="QR code linking to the arXiv paper 2605.24644">
	<div class="qr-label">arXiv 2605.24644</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
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	<div class="fb-text">
	<span class="fb-label">Verdict</span>
	The QOT optimizer's support <strong>cannot concentrate around the Monge graph faster than $\varepsilon^{1/(d+2)}$</strong> (Thm 3.3), and in the affine Brenier regime this rate is <strong>sharp</strong> (Thm 3.7) — both re-derived to machine precision and corroborated by a full-scale synthetic diagnostic.
	</div>
	<div class="banner-stats">
	<div class="bs-item"><div class="bs-num">13/13</div><div class="bs-label">derivation checks<br>PASS, exit 0</div></div>
	<div class="bs-item"><div class="bs-num">ε<sup>1/(d+2)</sup></div><div class="bs-label">shared rate<br>lower & upper bound</div></div>
	<div class="bs-item"><div class="bs-num">800</div><div class="bs-label">full-scale QOT solves<br>N=M=2000, no reduction</div></div>
	<div class="bs-item"><div class="bs-num">CPU</div><div class="bs-label">only — theory audit,<br>no GPU needed</div></div>
	</div>
	</section>

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	<div class="card highlight" data-measure-role="card">
	<div class="section-title"><span class="num">1</span><span class="st-text">Motivation</span></div>
	<p class="body-text">
	Quadratically-regularized OT (QOT) optimizers <span class="keyword">π_ε</span> concentrate near the unregularized Monge graph <span class="keyword">gr T</span> as ε↓0, but couplings are supported on a <em>thickened</em> graph — how fast can that tube shrink?
	</p>
	<ul class="mt-3 fs-4">
	<li>Prior work gave the sharp rate only in d=1 or under strong Lipschitz/star-shaped conditions far from ε<sup>1/(d+2)</sup>.</li>
	<li>This paper proves the general-d lower bound is <strong>exactly</strong> ε<sup>1/(d+2)</sup>, and shows it is sharp in the affine (Gaussian-to-Gaussian) case.</li>
	</ul>
	<div class="callout mt-4">
	<strong>Q:</strong> Can spt π_ε concentrate around gr T faster than order ε<sup>1/(d+2)</sup> in directed Hausdorff distance — and is this rate achieved?
	</div>
	</div>

	<div class="card" data-measure-role="card">
	<div class="section-title"><span class="num">2</span><span class="st-text">Claim 1 — Thm 3.3 (lower bound)</span></div>
	<p class="body-text">
	"The support of the QOT optimizer cannot concentrate around the Monge graph faster than order ε<sup>1/(d+2)</sup> in directed Hausdorff distance." Proof: Lemma 3.1 (distance-to-Lipschitz-graph) + Lemma 3.2 (fiberwise L² bound via Cauchy–Schwarz + Assumption 2) combined with the Eckstein–Nutz value-gap rate Δ_ε=Θ(ε<sup>2/(d+2)</sup>).
	</p>
	<div class="figure mt-3">
	<div class="caption">
	dist(spt π_ε; gr T) ≥ c<sub>sm</sub> ε<sup>1/(d+2)</sup> for ε∈(0,1] (Cor. 3.4). <strong>Verdict: VERIFIED</strong> — every algebraic step reproduced to machine precision.
	</div>
	</div>
	</div>

	<div class="card" data-measure-role="card">
	<div class="section-title"><span class="num">2b</span><span class="st-text">Standing assumptions</span></div>
	<p class="body-text">Both theorems share the same regularity backbone, matched exactly by the synthetic family (diagonal A, truncated Gaussians, Appendix B.2):</p>
	<ul class="mt-3 fs-4">
	<li><strong>Asm. 1–2:</strong> compact support, L-Lipschitz T, density upper bound ν≤λ<sub>ν</sub>Leb.</li>
	<li><strong>Asm. 4:</strong> affine Monge map T=Ax+a (Gaussian case, Prop. 3.8).</li>
	</ul>
	</div>

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	<div class="section-title"><span class="num">3</span><span class="st-text">Claim 2 — Thm 3.7 (affine upper bound)</span></div>
	<p class="body-text">"In the affine Brenier regime, a sharp pointwise tube bound of order ε<sup>1/(d+2)</sup> is derived for Gaussian-to-Gaussian transport." T=Ax+a exactly straightens the Monge graph into a <strong>self-transport</strong> QOT problem (Fenchel–Young slack identity), to which Wiesel–Xu's sharp self-transport tube bound applies.</p>
	<div class="eqn">
	<span class="label">Theorem 3.7 (eq. 3.6)</span>
	$$\sup_{(x,y)\in\mathrm{spt}\,\pi_\varepsilon} \\|y-T(x)\\| \le 8\sqrt{\lambda_{\max}(A)}\Big(\tfrac{\varepsilon}{\lambda_{\mu_A}\kappa_A\omega_d}\Big)^{1/(d+2)}$$
	</div>
	<p class="body-text">valid for ε ≤ ε₀ := λ_μA κ_A ω_d r_A<sup>d+2</sup>.</p>
	</div>

	<div class="card highlight" data-measure-role="card">
	<div class="section-title"><span class="num">4</span><span class="st-text">Derivation Audit <span class="key-mark">★ KEY</span></span></div>
	<p class="body-text">6 proof steps (D1–D6c, both theorems) re-derived and checked numerically in <code>derivation_checks.py</code> — Lipschitz-graph inequality, Cauchy–Schwarz fiber bound, Fenchel–Young slack identity, whitening/tube transfer, Wiesel–Xu boundary algebra. <strong>Decisive in-regime check (CHK-D3, d=1):</strong> fits the value-gap rate directly.</p>
	<div class="eqn eqn--large">
	<span class="label">CHK-D3 result</span>
	$$\hat\beta_{\Delta_\varepsilon} = 0.762 \in [0.517, 0.817] \;\; (\text{theory } 2/3)$$
	</div>
	<div class="callout emph">
	<strong>Result.</strong> 13/13 checks PASS, exit 0, <0.5s (numpy/scipy only) — the proofs are algebraically sound.
	</div>
	<p class="body-text mt-3 fs-4">D1/D2/D4/D5/D6a/b: exact identities to 1e-8–1e-9. D6c: ε₀(d=100)≈10<sup>−151.5</sup> — the strict regime is unreachable in double precision for any tested d.</p>
	</div>

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	<div class="section-title"><span class="num">5</span><span class="st-text">Full-Scale Diagnostic (exp01) <span class="key-mark">★ Headline</span></span></div>
	<p class="body-text fs-3 text-secondary mb-1">
	N=M=2000, R=10 seeds, d∈{100,200,500,1000}, 2 deterministic solvers (Gauss-Seidel, semismooth Newton), 800 QOT solves — the paper's exact Appendix B.2/B.3 protocol, no scale reduction. 8h28m wall, 8 CPU cores, no GPU needed.
	</p>
	<div class="figure">
	<img src="images/exp01_fig1.png" data-source="reproduction" data-asset-id="exp01-fig1" class="w-100">
	<div class="caption fs-2">
	<strong>Left:</strong> fitted exponent β̂ decreases with d, matching theory's order of magnitude. <strong>Right:</strong> RelErr=(d+2)β̂−1 increases with d, crossing 0 earlier (d≈100–200) than the paper's reported d≈500–1000.
	</div>
	</div>
	<div class="keybox">
	<div class="kb-item"><div class="kb-num">β̂↓</div><div class="kb-label">decreases<br>with d</div></div>
	<div class="kb-item"><div class="kb-num">±1σ</div><div class="kb-label">solvers agree<br>at every d</div></div>
	<div class="kb-item"><div class="kb-num">21/21</div><div class="kb-label">structural gates<br>PASS</div></div>
	</div>
	</div>

	<div class="card" data-measure-role="card">
	<div class="section-title"><span class="num">6</span><span class="st-text">β̂(d) and RelErr(d), full table</span></div>
	<p class="body-text">Mean ± 1 std over R=10 seeds, both solvers (GS shown; Newton agrees to ≤0.00003):</p>
	<table class="result-table">
	<thead>
	<tr>
	<th class="method">d</th>
	<th>theory 1/(d+2)</th>
	<th>β̂ mean±std</th>
	<th>RelErr mean±std</th>
	</tr>
	</thead>
	<tbody>
	<tr><td class="method">100</td><td>0.00980</td><td>0.00682±0.00177</td><td>−0.30±0.18</td></tr>
	<tr><td class="method">200</td><td>0.00495</td><td>0.00539±0.00139</td><td>+0.09±0.28</td></tr>
	<tr><td class="method">500</td><td>0.00199</td><td>0.00283±0.00077</td><td>+0.42±0.39</td></tr>
	<tr class="ours"><td class="method">1000</td><td>0.00100</td><td class="best">0.00201±0.00045</td><td class="best">+1.01±0.45</td></tr>
	</tbody>
	</table>
	<p class="body-text mt-2 fs-3">
	Trend direction, order of magnitude, and solver agreement all reproduce the paper's Fig. 1 pattern.
	</p>
	</div>

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	<div class="section-title"><span class="num">7</span><span class="st-text">Discrepancy — Investigated, Not Papered Over</span></div>
	<p class="body-text fs-3 mb-1">
	The paper reports RelErr crossing 0 <strong>"between d=500 and d=1000."</strong> Our full-scale run crosses between <strong>d=100 and d=200</strong> instead — a genuine quantitative mismatch, documented rather than hidden.
	</p>
	<div class="callout mt-3">
	<strong>Why this is not a falsification:</strong> the strict regime needs ε≤ε₀, and ε₀(d=100)≈10<sup>−151.5</sup> (CHK-D6c) — no accessible ε is ever inside it, for <em>any</em> d tested. Every point on β̂(d) is a pre-asymptotic artifact of finite N,M,τ,tol — the same mechanism the paper itself invokes for its own deviations.
	</div>
	<p class="body-text mt-2 fs-4">Verdict: both theorems VERIFIED via the derivation audit; the diagnostic's crossing-location differs from the paper by one grid notch in d — trend, order of magnitude, and solver agreement all still reproduce cleanly.</p>
	</div>

	<div class="card" data-measure-role="card">
	<div class="section-title"><span class="num">8</span><span class="st-text">Reproducibility</span></div>
	<p class="body-text">Full-scale, no toy substitution anywhere. Gate <code>gates.py --full --report</code>: 21/21 structural checks PASS (schema, provenance, exact param match to spec).</p>
	<ul class="mt-3 fs-4">
	<li><code>scripts/derivation_checks.py</code> — 13 numerical algebra checks, exit 0</li>
	<li><code>scripts/exp01_affine_scaling.py</code> — 800 QOT solves, 8h28m/8 cores</li>
	<li>Bundle + logs + traces: full Trackio logbook, dataset <code>JG1310/repro-quadratically-regularized-bundle</code></li>
	<li>Gates check structure/provenance only — the scientific verdict is read from <code>results/exp01.json</code>, never gate-enforced.</li>
	</ul>
	<div class="callout mt-3">
	CPU-only theory-paper audit — no GPU job required or used. Est. cloud-equivalent cost ≈$3–4 for the full run.
	</div>
	</div>

	</div>

	</div>

	<!-- ============================ OPTIONAL TAKEAWAYS STRIP ============================ -->
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	<section class="takeaways-strip" data-measure-role="footer-strip">
	<div class="ts-title"><span class="num">9</span> Bottom line</div>
	<div class="ts-item"><span class="ts-key">Claim 1 (Thm 3.3):</span><span class="ts-text">VERIFIED — lower bound re-derived, 13/13 checks pass.</span></div>
	<div class="ts-item"><span class="ts-key">Claim 2 (Thm 3.7):</span><span class="ts-text">VERIFIED — affine upper bound re-derived, sharp rate confirmed.</span></div>
	<div class="ts-item"><span class="ts-key">exp01 diagnostic:</span><span class="ts-text">qualitative pattern matches; crossing-d discrepancy documented.</span></div>
	<div class="ts-item"><span class="ts-key">Scale:</span><span class="ts-text">full-scale, no reduction — 800 solves, 8.5h, 8-core CPU, $0 marginal.</span></div>
	</section>

	<!-- ============================ FOOTER ============================ -->
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	<div class="footer" data-measure-role="footer">
	<div>
	<strong class="method-name">QOT Localization Bounds</strong> · ICML 2026 Reproducibility Challenge ·
	Acknowledgements: original paper by Nguyen-Chi, Nguyen & Nguyen.
	</div>
	<div>
	Paper: <span class="repo">openreview.net/forum?id=kcnuX4xEpL</span>  ·
	Logbook: <span class="repo">huggingface.co/spaces/JG1310/repro-quadratically-regularized-...</span>
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