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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: portrait_2col (ARIS fork of posterly portrait_2col_neutral)
	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.

	CANVAS: 24in × 36in PORTRAIT (ICLR / CVPR / many workshops default) DEFAULT.
	CANVAS RETARGETING: change BOTH the @page `size` (one place, in the
	CANVAS block below) AND the .poster `width`/`height` (one place, in
	the .poster rule). Keep them in sync.
	NOTE — ICLR workshops commonly mandate a 61×91 cm PORTRAIT board
	(A0-ish portrait). To retarget to that:
	@page { size: 610mm 910mm; } /* or 61cm 91cm */
	.poster { width: calc(610 * var(--u)); height: calc(910 * var(--u)); }
	(1 print --u = 1mm, so 610/910 u == 610/910 mm == 61×91 cm — almost
	identical to the shipped 24×36in default, which is why portrait is a
	good fit for ICLR-workshop boards.)
	LAYOUT: header → body 2 columns → footer
	(NO framework banner, NO takeaways strip — vertical space is precious)
	USE WHEN: venue mandates portrait orientation, sub-A0/A1 size;
	reader scans top-to-bottom in column 1, then jumps to column 2.
	HOW TO USE:
	1. Copy this file to your working directory as `poster.html`.
	2. Edit the DESIGN TOKENS in `:root` (or swap a tokens/*.json palette) —
	the token block is the ONLY place color/font-size literals may live.
	3. Replace TODO content placeholders (search for "TODO").
	4. Run `python3 tools/run_gates.py poster.html` (the default driver: preflight -> style -> measure -> polish) to align columns and check style/structure.
	5. Run `python3 tools/render_preview.py poster.html` to produce the PDF.
	STYLE CONTRACT (ARIS source gate, tools/style_check.py):
	- NO inline `style=` attributes anywhere in the markup. Use the utility
	classes (.fs-N, .mt-N, .mb-N, .w-NN, .text-secondary, .nowrap,
	.text-center) defined in the UTILITIES block below.
	Sole exceptions: the internal SVG of a `data-color-exempt="logo"` element,
	and a `style="width: NN%"` on a `data-source="paper"` <img> (AR width tweak).
	- All font-size values reference a `--fs-1..9` token (or a COMPONENTS.md
	predefined `calc(var(--fs-N) * k)` variant such as `.eqn--large`).
	- No `linear-gradient`; the only gradient allowed is the .poster radial
	tint (every color stop alpha ≤ 0.06).
	MEASURE ROLES: every layout-critical element carries `data-measure-role`.
	Generic measurement scripts depend on these — do not remove.
	============================================================
	-->
	<html lang="en">
	<head>
	<meta charset="UTF-8">
	<meta name="viewport" content="width=device-width, initial-scale=1.0">
	<title>Exactly Computing do-Shapley Values — ICML 2026 Reproduction</title>

	<!-- MathJax v3 for inline equations. CDN by default; to go offline,
	download a MathJax v3 release into assets/mathjax/ and change the <script>
	`src` below to "assets/mathjax/es5/tex-svg.js". -->
	<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 — 24" × 36" PORTRAIT
	24in × 25.4 mm/in = 609.6 mm ≈ 610 u (width)
	36in × 25.4 mm/in = 914.4 mm ≈ 914 u (height)
	RETARGET: change @page size here AND .poster width/height below.
	ICLR workshop = 610mm 910mm (61×91 cm) portrait — see header note.
	========================================================= */
	@page { size: 24in 45in; margin: 0; }

	:root {
	/* ===== DESIGN TOKENS ===== */
	/* The ONLY place color / font-size literals are allowed (style_check
	rule 1 + 8 locate this block by the comment pair). Override token
	VALUES here only — never restyle component rules with new literals.
	To rebrand, edit these or load a tokens/.json palette. /

	/* Primary accent (header underline, card-highlight bar, section nums, .keyword) */
	--accent: #24689B;
	--accent-deep: #17496F;
	--accent-light: #E7F1F8;
	--accent-soft: #D4E7F3;

	/* Secondary accent for "ours / best" emphasis (table .ours row, "★" callouts) */
	--gold: #B44738;
	--gold-soft: #FBEDEB;

	/* 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-preview backdrop only — @media print overrides body to white.
	Tokenized so no color literal escapes the token block (style_check rule 1). */
	--bg-screen: #2b2b2b;

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

	/* Font-size scale (9 steps). Every font-size in this template references
	one of these. Off-scale px drift is forbidden (style_check rule 8);
	`calc(var(--fs-N) * k)` is allowed ONLY for COMPONENTS.md variants.
	The posterly portrait original used a few off-scale sizes (28u title,
	17u section title, 14u subtitle/eqn/callout); they are snapped to the
	nearest scale token here so the template stays at exactly 9 steps. */
	--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.
	Whitelist (style_check rule 7): serif body, sans heads, mono code. */
	--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);
	/* ===== END DESIGN TOKENS ===== */
	}

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

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

	/* Poster container — exact print dimensions; data-measure-role="poster" so
	poster_check.py can verify the canvas size.
	Width : 610u ≈ 24in.
	Height : 914u ≈ 36in.
	RETARGET: width/height must track @page size (see CANVAS block). */
	.poster {
	width: calc(610 * var(--u));
	height: calc(1143 * var(--u));
	background: var(--bg-page);
	/* The only gradient the catalog sanctions (rule 5 enforces it only when re-enabled): a low-alpha radial tint
	(every color stop alpha ≤ 0.06). NOT a linear-gradient. */
	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(12 * var(--u));
	display: grid;
	/* header \| body \| footer — no banner row, no takeaways row */
	grid-template-rows: auto 1fr auto;
	gap: calc(6 * var(--u));
	box-shadow: var(--shadow-screen);
	position: relative;
	overflow: hidden;
	}

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

	/* =========================================================
	HEADER — same 3-col structure as landscape sibling
	(venue badge \| title-block \| logo+QR), but compressed vertically
	by ~1u of padding since portrait is narrower.
	========================================================= */
	.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(14 * var(--u));
	padding: calc(1 * var(--u)) calc(3 * var(--u)) calc(4 * 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(44 * var(--u));
	text-align: center;
	border-right: calc(1 * var(--u)) solid var(--border-soft);
	padding-right: calc(4 * var(--u));
	}
	/* Portrait venue mark: snapped 28u -> --fs-9 (32u). */
	.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-4);
	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; }
	/* Portrait title snapped 28u -> --fs-9 (32u). The reader is closer and the
	canvas is narrower, so the type still feels appropriately scaled. */
	.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(--gold); }
	/* subtitle snapped 14u -> --fs-6 (15u). */
	.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-5);
	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-3);
	}

	.right-block {
	justify-self: end; /* keep logo+QR in the far-right corner */
	display: flex; align-items: center;
	gap: calc(8 * var(--u));
	}
	.qr-block { display: flex; flex-direction: column; align-items: center; gap: calc(2 * var(--u)); }
	.qr-block img {
	width: calc(75 * var(--u));
	height: calc(75 * var(--u));
	border: calc(2 * var(--u)) solid var(--accent);
	border-radius: calc(4 * var(--u));
	background: white;
	padding: calc(2 * var(--u));
	}
	.qr-label {
	font-family: var(--font-sans);
	font-size: var(--fs-2);
	color: var(--accent);
	font-weight: 600;
	}
	/* Optional logo slot — drop your lab logo here. Keep rendered height ~75u to
	match QR; otherwise the header grows and steals vertical space.
	LOGO CONTRACT: a logo whose own SVG/PNG carries brand colors must be
	marked `data-color-exempt="logo"` so style_check exempts it from the
	palette / inline-style gates, e.g.:
	<div class="logo-slot"><img src="assets/logos/lab.png" alt="Lab"
	data-color-exempt="logo"></div>
	Inline-SVG logos carry the attribute on the <svg> element itself. */
	.logo-slot img { height: calc(75 * var(--u)); width: auto; max-width: calc(300 * 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(75 * var(--u)); }
	.logo-slot.logo-wide img { height: calc(51 * var(--u)); max-width: calc(125 * var(--u)); } /* <22% of the ~586u portrait header */
	/* 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));
	padding: calc(3 * var(--u)) calc(4 * 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(4 * var(--u)); }
	.logo-row img { height: calc(58 * var(--u)); width: auto; display: block; }
	.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));
	padding: calc(3 * var(--u)) calc(5 * var(--u));
	}
	.logo-row .lr-item img { height: calc(48 * 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.
	Width stays under the LOGO/WIDE 22% cap (portrait header ~586u). */
	.logo-row.logo-stack { flex-direction: column; align-items: flex-start; gap: calc(6 * var(--u)); }
	.logo-row.logo-stack img { width: calc(118 * var(--u)); height: auto; }
	/* The narrow portrait header can't fit a logo-stack BESIDE the QR (the row
	would blow the right-block width gate); when a stack is present, drop the
	right-block to a column so the logos sit ABOVE the QR. */
	.right-block:has(.logo-stack) { flex-direction: column; align-items: flex-end; }
	/* venue badge may carry the official venue logo above its text line */
	.venue-badge img { height: calc(46 * 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; }

	/* =========================================================
	BODY: 2 columns (portrait posters live and die on column balance)

	CRITICAL: Last-card bottoms of both columns must align —
	run poster_check.py measure poster.html — spread < 5 px required
	(< 3 px ideal). Gap to footer in 30–50 px.
	========================================================= */
	.body-grid {
	display: grid;
	grid-template-columns: 1fr 1fr;
	gap: calc(12 * var(--u));
	/* No overflow:hidden here — it would clip card shadows (pitfall #6).
	.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(5 * var(--u));
	min-height: 0;
	height: 100%;
	padding-bottom: calc(4 * var(--u)); /* shadow breathing room above footer */
	}
	.column:first-child .card { flex: 1; }
	.column:last-child { gap: calc(3.5 * var(--u)); }
	.column:first-child .card { padding: calc(3 * var(--u)) calc(7 * var(--u)); }
	.column:first-child .body-text,
	.column:first-child .card p,
	.column:first-child .card li { font-size: var(--fs-3); }
	.column:first-child .eqn { font-size: var(--fs-3); padding: calc(2 * var(--u)) calc(7 * var(--u)); margin: calc(2 * var(--u)) 0; }
	.column:first-child .result-table { font-size: var(--fs-3); }
	.column:first-child .result-table th,
	.column:first-child .result-table td { padding: calc(1 * var(--u)) calc(3 * var(--u)); }
	.column:first-child .callout { font-size: var(--fs-4); padding: calc(3 * var(--u)) calc(8 * var(--u)); margin: calc(3 * var(--u)) 0; }
	.column:last-child .card:last-child { padding-bottom: calc(12 * var(--u)); }

	/* =========================================================
	CARDS — identical tokens to sibling
	========================================================= */
	.card {
	background: var(--bg-card);
	border-radius: calc(5 * var(--u));
	padding: calc(4 * var(--u)) calc(8 * 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(7 * var(--u)); } /* predefined variant: tighter padding (fix (f)) */
	/* .highlight de-gradiented: flat emphasis tint instead of a linear-gradient. */
	.card.highlight {
	border-left: calc(6 * var(--u)) solid var(--accent);
	background: var(--bg-emphasis);
	}

	/* section-title snapped 17u -> --fs-7 (16u). */
	.section-title {
	font-family: var(--font-sans);
	font-weight: 700;
	font-size: var(--fs-6);
	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(24 * var(--u)); height: calc(24 * var(--u));
	background: var(--accent); color: white;
	border-radius: 50%;
	font-size: var(--fs-6); font-weight: 700;
	flex-shrink: 0;
	}
	/* Inline "★ key/headline" tag inside a section title (replaces inline styles). */
	.section-title .tag-key {
	color: var(--gold);
	font-size: var(--fs-4);
	}

	/* Body text — snapped 13u -> --fs-5. The 2-column portrait gives each column
	enough width to carry a meatier line height comfortably. */
	.body-text, .card p, .card li {
	font-family: var(--font-serif);
	font-size: var(--fs-4);
	line-height: 1.35;
	color: var(--text-primary);
	}
	.card ul, .card ol { padding-left: calc(20 * var(--u)); }
	.card li { margin-bottom: calc(2 * var(--u)); }

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

	/* Equation block — snapped 14u -> --fs-5 (13u). */
	.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-4);
	overflow-x: hidden;
	}
	/* COMPONENTS.md predefined variant: a larger equation when a formula is the
	hero of a card. The `calc(var(--fs-N) * k)` form is allowed ONLY for
	registered variants like this one (style_check rule 8 / §12.5 nit 1). */
	.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 gold for theorems / "★ key" strips
	(gold .callout de-gradiented to a flat fill). Snapped 13u -> --fs-5. */
	.callout {
	background: var(--accent);
	color: white;
	padding: calc(5 * var(--u)) calc(10 * var(--u));
	border-radius: calc(4 * var(--u));
	font-size: var(--fs-5);
	margin: calc(4 * var(--u)) 0;
	}
	.callout strong { color: white; }
	.callout.gold {
	background: var(--gold-soft);
	border-left: calc(6 * var(--u)) solid var(--gold);
	color: var(--accent-deep);
	}
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	<div class="poster" data-measure-role="poster">
	<header class="header" data-measure-role="header">
	<div class="venue-badge"><div class="vb-venue">ICML</div><div class="vb-year">2026</div><div class="vb-tag">REPRODUCTION</div></div>
	<div class="title-block">
	<h1 class="title"><span class="accent">Exactly Computing</span> do-Shapley Values</h1>
	<div class="subtitle">Direct pseudocode implementation, 108 exact graph audits, 600 ADMGs, and boundary sampling</div>
	<div class="authors-line"><span class="author">Independent reproduction</span> · <span class="aff">Paper #18355 · OpenReview Peim0KY6ty · seed 18355</span></div>
	</div>
	<div class="right-block"><div class="qr-label">6 claims<br>CPU only</div></div>
	</header>

	<div class="body-grid" data-measure-role="body">
	<div class="column" data-measure-role="column">
	<div class="card highlight" data-measure-role="card" data-logbook-target="claim-1-do-shapley-values-can-be-computed-exactly-in-o-r-d-e-t-time-where" data-logbook-label="Claim 1 evidence">
	<div class="section-title"><span class="num">1</span><span class="st-text">Compress the coalition powerset</span></div>
	<p class="body-text">Interventions with the same basis and closure have one value. The Shapley sum can therefore be regrouped <span class="nowrap">without approximation.</span></p>
	<div class="eqn"><span class="label">Class decomposition</span>$$\phi_i=\sum_{c=1}^{r}\nu(c)w_i(c)$$</div>
	<table class="result-table"><thead><tr><th class="method">Exact graph audits</th><th>$d$</th><th>Max error</th></tr></thead><tbody>
	<tr><td class="method">108</td><td>5–10</td><td class="best">$5.88\times10^{-15}$</td></tr>
	</tbody></table>
	<p class="body-text mt-3">Every compressed answer was checked against the literal $2^d$ Shapley definition using independent random class values.</p>
	</div>

	<div class="card" data-measure-role="card" data-logbook-target="claim-2-theorem-5-1-shows-the-do-shapley-value-phi-i-is-identifiable-if" data-logbook-label="Claim 2 evidence">
	<div class="section-title"><span class="num">2</span><span class="st-text">Singletons diagnose identifiability</span></div>
	<p class="body-text">The appendix's Boolean ID recursion was implemented on directed and bidirected observed graphs.</p>
	<div class="eqn"><span class="label">Theorem 5.1 consequence</span>$$\bigwedge_{j=1}^{d}\operatorname{ID}(\{j\},Y)\iff\bigwedge_{S\subseteq[d]}\operatorname{ID}(S,Y)$$</div>
	<table class="result-table"><thead><tr><th class="method">Random ADMGs</th><th>Coalition queries</th><th>Violations</th></tr></thead><tbody>
	<tr><td class="method">600</td><td>8,204</td><td class="best">0</td></tr>
	</tbody></table>
	<p class="body-text mt-3">406 graphs contained a non-identifiable coalition; the bow-arc negative control was also correctly rejected.</p>
	</div>

	<div class="card" data-measure-role="card" data-logbook-target="claim-3-lemma-3-1-establishes-that-for-any-closed-set-with-a" data-logbook-label="Claim 3 evidence">
	<div class="section-title"><span class="num">3</span><span class="st-text">Closed-set removal powers traversal</span></div>
	<p class="body-text">For every closed set generated in all 108 graph instances, each basis element was removed and closure was <span class="nowrap">recomputed from scratch.</span></p>
	<table class="result-table"><thead><tr><th class="method">Audit</th><th>Failures</th></tr></thead><tbody>
	<tr><td class="method">Lemma removal property</td><td class="best">0</td></tr>
	<tr><td class="method">Algorithm 2 vs brute classes</td><td class="best">0</td></tr>
	<tr><td class="method">Duplicate class representatives</td><td class="best">0</td></tr>
	</tbody></table>
	<div class="callout"><strong>Consequence:</strong> the traversal visits one closure per class instead of scanning the powerset.</div>
	<p class="body-text mt-3">The independent comparator enumerated every coalition, canonicalized its basis–closure pair, and required exact set equality with Algorithm 2's output.</p>
	</div>

	<div class="card highlight" data-measure-role="card" data-logbook-target="claim-4-algorithm-3-boundary-sampling-guarantees-discovery-of-min-m-r-distinct" data-logbook-label="Claim 4 evidence">
	<div class="section-title"><span class="num">4</span><span class="st-text">Every boundary query is distinct</span></div>
	<p class="body-text">Algorithm 3 was tested at budgets below, at, and above $r$ on 90 independent DAGs.</p>
	<div class="eqn"><span class="label">Observed invariant</span>$$\|\mathcal C\|=\min(m,r)$$</div>
	<table class="result-table"><thead><tr><th class="method">Budget checks</th><th>Duplicate / count failures</th></tr></thead><tbody>
	<tr><td class="method">450</td><td class="best">0</td></tr>
	</tbody></table>
	<p class="body-text mt-3">Each graph was checked at five budgets spanning one query, fractional lattice coverage, exactly $r$, and beyond $r$.</p>
	<div class="callout"><strong>Queue exhaustion:</strong> at $m\geq r$, traversal returned each class exactly once.</div>
	</div>
	</div>

	<div class="column" data-measure-role="column">
	<div class="card highlight" data-measure-role="card" data-logbook-target="claim-5-the-doregressionmsr-estimator-consistently-outperforms-baseline-variants-and-exhibits-a" data-logbook-label="Claim 5 evidence">
	<div class="section-title"><span class="num">5</span><span class="st-text">Structural phase transition <span class="tag-key fs-3 nowrap">★ HEADLINE</span></span></div>
	<p class="body-text text-secondary fs-3 mb-1">Reduced diagnostic on a $d=9$, $r=133$ graph; 160 trials per budget ratio.</p>
	<div class="figure"><img class="w-75" src="results/summary.png" alt="Exact class computation, boundary discovery, reduced phase transition, and sparse graph scaling"><div class="caption fs-2"><strong>Bottom:</strong> boundary sampling spends every query on a new class and switches to the exact sum at $m=r$.</div></div>
	<div class="callout callout--tight"><strong>At $m=r$:</strong> boundary exact in 160/160 trials; random coalitions exact in 0/160.</div>
	</div>

	<div class="card" data-measure-role="card" data-logbook-target="claim-5-the-doregressionmsr-estimator-consistently-outperforms-baseline-variants-and-exhibits-a" data-logbook-label="Claim 5 scope">
	<div class="section-title"><span class="num">5a</span><span class="st-text">What was—and was not—rerun</span></div>
	<table class="result-table"><thead><tr><th class="method">$m/r$</th><th>Boundary error</th><th>Random diagnostic</th></tr></thead><tbody>
	<tr><td class="method">0.25</td><td class="best">0.0947</td><td>0.3989</td></tr>
	<tr><td class="method">0.50</td><td class="best">0.0304</td><td>0.3968</td></tr>
	<tr><td class="method">0.75</td><td class="best">0.00725</td><td>0.3364</td></tr>
	<tr><td class="method">1.00</td><td class="best">0</td><td>0.3404</td></tr>
	</tbody></table>
	<p class="body-text mt-2 fs-3">These are partial class sums, not doRegressionMSR. The named estimator superiority remains Figure-5/source evidence because no author <span class="nowrap">implementation was linked.</span></p>
	</div>

	<div class="card" data-measure-role="card" data-logbook-target="claim-6-real-world-causal-structures-tend-to-be-sparse-so-the-number" data-logbook-label="Claim 6 evidence">
	<div class="section-title"><span class="num">6</span><span class="st-text">Sparse graphs compress dramatically</span></div>
	<table class="result-table"><thead><tr><th class="method">$d=12$ family</th><th>Median $r$</th><th>$r/2^d$</th></tr></thead><tbody>
	<tr><td class="method">Chain</td><td>13</td><td class="best">0.0032</td></tr>
	<tr><td class="method">Sparse shortcuts</td><td>155.5</td><td>0.0380</td></tr>
	<tr><td class="method">Moderate shortcuts</td><td>946.5</td><td>0.2311</td></tr>
	</tbody></table>
	<p class="body-text mt-2 fs-3">This independently verifies the sparsity mechanism over 648 synthetic DAGs. It does not replace the paper's 156 learned TALENT structures, whose machine-readable graphs were not released.</p>
	<p class="body-text fs-2">Synthetic audit supports the mechanism; the real-data trend remains source-only.</p>
	</div>

	<div class="card" data-measure-role="card" data-logbook-target="conclusion" data-logbook-label="Scope and rerun">
	<div class="section-title"><span class="num">7</span><span class="st-text">Reproducibility boundary</span></div>
	<ul><li><strong>Exact:</strong> Claims 1, 3, and 4 directly rerun from pseudocode.</li><li><strong>ID consequence:</strong> Claim 2 checked <span class="nowrap">over 600 ADMGs.</span></li><li><strong>Partial:</strong> Claim 5 named estimator and Claim 6 real-data endpoint remain source-only.</li><li><strong>Compute:</strong> 16.6 s, CPU only, seed 18355.</li></ul>
	<div class="callout gold mt-3"><strong>Bottom line:</strong> four claims directly supported; two empirical claims honestly bounded.</div>
	</div>
	</div>
	</div>

	<div class="footer" data-measure-role="footer"><div><strong class="method-name">DO-SHAPLEY</strong> · ICML 2026 · exact algorithm audit</div><div>OpenReview: <span class="repo">Peim0KY6ty</span>  ·  arXiv: <span class="repo">2602.07203</span></div></div>
	<div class="ornament">CLASSES · LATTICES · REPRODUCTION</div>
	</div>
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