From 6abd9cc87168d237f8cae73ceec12865d0c65bd1 Mon Sep 17 00:00:00 2001
From: Max Ghenis <mghenis@gmail.com>
Date: Sat, 3 Oct 2026 22:27:38 -0400
Subject: [PATCH 1/9] Correct retired SNAP QC claims in the paper, README,
 FACTS and simulator

FSBEN: Mathematica computes it for USDA from each edited case record with
the QC Minimodel's benefit formula (tech doc, August 2026 posting). The
paper and simulator no longer call it "the agency's own computational
canon", "the agency's own benefit-calculation software" or "a closed
model". The Step 12-13 reconciliation is described as the tech doc gives
it; in Colorado FSBEN ends within $5 of BENFIX in 797 of 856 cases.

Replay: the solver starts from the file's edited inputs and moves only the
ELEMENT1 input in $3 steps. A match (246 of 283) is consistent with
correct arithmetic on a wrong input; 230 moved an input and 16 moved
nothing. A miss does not establish a computation error: 17 of the 37
moved nothing, and of 26 cases with a computational finding 13 reproduce.
The "computation-side upper bound" reading and the disclosure-perturbation
description are withdrawn, matching ANALYSIS.md (#92) and cause_shares.json
(#94). The superseded "33 of 246" is reconciled in FACTS H8.

Colorado's 0.03-point FY 2024 margin now carries its FY 2025 rate, 10.09%.
FACTS C4 (tech-doc errata) is withdrawn: no source records the errata, so
the abstract, introduction and oracle section no longer claim them.

Re-rendered app/public/paper/web (revision 10) and bumped the wrapper's
iframe version. tests/test_retired_claims.py ties every corrected figure
to claims_audit.json, cause_shares.json and the replay rows and keeps the
retired wording out of the living files.

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
---
 README.md                       |   5 +-
 app/public/index.html           |   2 +-
 app/public/paper/index.html     |  21 +-
 app/public/paper/web/index.html |  65 +++---
 app/public/paper/web/index.pdf  | Bin 209689 -> 211731 bytes
 paper/FACTS.md                  |  18 +-
 paper/index.qmd                 | 182 +++++++++------
 tests/test_retired_claims.py    | 402 ++++++++++++++++++++++++++++++++
 8 files changed, 573 insertions(+), 122 deletions(-)
 create mode 100644 tests/test_retired_claims.py

diff --git a/README.md b/README.md
index 4e81c6b..4f9b407 100644
--- a/README.md
+++ b/README.md
@@ -28,8 +28,9 @@ noise alone materially affects tier assignment.
 ## What v0.1 finds (FY 2024 file, all 53 jurisdictions)
 
 - **Tier assignment is noisy.** Several states' tiers are near coin flips
-  under QC sampling variation (Colorado: official rate 9.97%, 0.03 points
-  from the 15% boundary, with a ±0.9-point sampling SD).
+  under QC sampling variation (Colorado: official FY 2024 rate 9.97%, 0.03
+  points below the 15% boundary, with a ±0.9-point sampling SD; its FY 2025
+  rate, 10.09%, is above the boundary).
 - **Simplification options carry large expected values** where they can move
   a state across a boundary: about $609M/yr in combined expected state
   cost-share reduction nationally at 50% category-suppression effectiveness
diff --git a/app/public/index.html b/app/public/index.html
index f4720f9..535225f 100644
--- a/app/public/index.html
+++ b/app/public/index.html
@@ -130,7 +130,7 @@ <h2>Targeted review scenarios</h2>
 
   <section class="chart-card engine-panel" id="engine-panel" hidden>
     <h2>Axiom rules engine vs the FNA QC Minimodel</h2>
-    <p class="sub">The QC file's formula benefit (FSBEN) is recomputed by the QC Minimodel used by the Food and Nutrition Administration (formerly FNS) — a closed model. The Axiom rules engine replays the same reviews through open, per-provision-cited RuleSpec encodings — compared stage by stage below against the file's own recorded values.</p>
+    <p class="sub">Mathematica computes the QC file's formula benefit (FSBEN) for USDA from each edited case record with the benefit formula of the QC Minimodel, one of the SNAP microsimulation models of the Food and Nutrition Administration (formerly FNS). The Axiom rules engine replays the same reviews through open, per-provision-cited RuleSpec encodings — compared stage by stage below against the file's own recorded values.</p>
     <div id="engine-body"></div>
   </section>
 
diff --git a/app/public/paper/index.html b/app/public/paper/index.html
index 8fc0947..12526af 100644
--- a/app/public/paper/index.html
+++ b/app/public/paper/index.html
@@ -76,23 +76,28 @@ <h1>Administrative quality control data as an oracle</h1>
       failures, and the fiscal 2025 rates landing where the sampling-noise
       mechanism says realized years should: 18 of 53 jurisdictions changed
       cost-share tiers, only 7 of those changes beyond two-year sampling
-      noise. This page embeds the revision-7 manuscript (updated 2026-08-13), which
-      carries five rounds of adversarial review, archived unedited, and adds
-      the engine-adoption accounting bounds, the official-vs-file wedge
-      decomposition, and the pre-registered model-repair experiments.
+      noise. This page embeds revision 10 (2026-10-03), which corrects how
+      revision 9 described the QC file's computed benefit and the Colorado
+      error-case replay, adds Colorado's fiscal 2025 rate beside its fiscal
+      2024 margin, and withdraws a claim of errata in the federal technical
+      documentation that no source records; the fact catalog lists each
+      change. The manuscript carries five rounds of adversarial review,
+      archived unedited, and adds the engine-adoption accounting bounds, the
+      official-vs-file wedge decomposition, and the pre-registered
+      model-repair experiments.
     </p>
-    <span class="paper-tag">Revision 9 · 2026-08-16 · Max Ghenis, PolicyEngine</span>
+    <span class="paper-tag">Revision 10 · 2026-10-03 · Max Ghenis, PolicyEngine</span>
   </header>
 
   <nav class="paper-actions" aria-label="Paper actions">
-    <a class="primary" href="web/index.html?v=r9-20260816">Open standalone HTML</a>
+    <a class="primary" href="web/index.html?v=r10-20261003">Open standalone HTML</a>
     <a href="web/index.pdf">Download PDF</a>
     <a href="../">Live simulator</a>
     <a href="https://github.com/PolicyEngine/snap-qc-sim">Code and artifacts</a>
   </nav>
 
   <div class="paper-frame-card">
-    <iframe src="web/index.html?v=r9-20260816"
+    <iframe src="web/index.html?v=r10-20261003"
             title="SNAP payment error working paper (embedded manuscript)"
             loading="lazy"
             sandbox="allow-same-origin allow-popups allow-popups-to-escape-sandbox"
@@ -101,7 +106,7 @@ <h1>Administrative quality control data as an oracle</h1>
 
   <p class="paper-foot">
     <a href="#paper-top">↑ Back to top</a> ·
-    <a href="web/index.html?v=r9-20260816">Open manuscript in a new page</a>
+    <a href="web/index.html?v=r10-20261003">Open manuscript in a new page</a>
   </p>
 </main>
 
diff --git a/app/public/paper/web/index.html b/app/public/paper/web/index.html
index c9db00b..f9c322e 100644
--- a/app/public/paper/web/index.html
+++ b/app/public/paper/web/index.html
@@ -7,7 +7,7 @@
 <meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes">
 
 <meta name="author" content="Max Ghenis">
-<meta name="dcterms.date" content="2026-08-16">
+<meta name="dcterms.date" content="2026-10-03">
 
 <title>Administrative quality control data as an oracle: verifying encoded SNAP rules and simulating payment error under cost sharing</title>
 <style>
@@ -106,12 +106,14 @@
 intermediates, exactly — all 6,081 in-scope cases of the 6,194-case
 official universe (113 documented exclusions), six computation stages
 per case, zero dollar tolerance — a process that surfaced and fixed two
-defects in the encodings and surfaced errata in the federal technical
-documentation. Replaying reconstructed pre-edit case facts through the
-verified engine explains 78.4% of Colorado's above-threshold official
-error cases — and 91.4% of sub-threshold deviations — as correct
-arithmetic on wrong facts. Pricing that decomposition through the
-formula's tiers, elections, and delay clause — removing the error
+defects in the encodings. For 283 of Colorado's 305 error cases, a
+public solver moves the input named by each case's first finding, where
+it can, until its recomputed benefit reaches the issued amount; run on
+those inputs, the verified engine reproduces the issued benefit within
+$5 for 78.4% of the above-threshold official error cases and 91.4% of
+the sub-threshold deviations, which is consistent with correct
+arithmetic on a wrong input. Pricing a cause-code decomposition through
+the formula's tiers, elections, and delay clause — removing the error
 dollars the file's cause codes assign to the computing apparatus, an
 accounting scenario rather than a causal estimate — moves summed
 expected fiscal 2028 bills from $7.7B to $7.6B a year under a strict
@@ -136,10 +138,10 @@
 — implies.
 ">
 <meta name="citation_author" content="Max Ghenis">
-<meta name="citation_publication_date" content="2026-08-16">
-<meta name="citation_cover_date" content="2026-08-16">
+<meta name="citation_publication_date" content="2026-10-03">
+<meta name="citation_cover_date" content="2026-10-03">
 <meta name="citation_year" content="2026">
-<meta name="citation_online_date" content="2026-08-16">
+<meta name="citation_online_date" content="2026-10-03">
 <meta name="citation_language" content="en">
 <meta name="citation_reference" content="citation_title=Technical documentation for the fiscal year 2024 supplemental nutrition assistance program quality control database and the QC minimodel;,citation_author=U.S. Department of Agriculture, Food and Nutrition Service;,citation_publication_date=2025;,citation_cover_date=2025;,citation_year=2025;,citation_fulltext_html_url=https://snapqcdata.net/datafiles;">
 <meta name="citation_reference" content="citation_title=Supplemental nutrition assistance program payment error rates, fiscal year 2024;,citation_author=U.S. Department of Agriculture, Food and Nutrition Service;,citation_publication_date=2025;,citation_cover_date=2025;,citation_year=2025;,citation_fulltext_html_url=https://www.fns.usda.gov/snap/qc/per;">
@@ -202,7 +204,7 @@ <h1 class="title">Administrative quality control data as an oracle: verifying en
                 <div>
             <div class="quarto-title-meta-heading">Published</div>
             <div class="quarto-title-meta-contents">
-              <p class="date">August 16, 2026</p>
+              <p class="date">October 3, 2026</p>
             </div>
           </div>
           
@@ -216,7 +218,7 @@ <h1 class="title">Administrative quality control data as an oracle: verifying en
     <div>
       <div class="abstract">
         <div class="block-title">Abstract</div>
-        <p>Beginning in fiscal year 2028, each state’s Supplemental Nutrition Assistance Program (SNAP) payment error rate sets its share of benefit costs, in steps of five percentage points at rates of 6, 8, and 10 percent. A measurement system built for oversight now moves large sums, which raises two questions this paper treats together: whether the rules that produce benefit amounts can be executed correctly, and how the errors that remain should be measured and simulated. I use the USDA Quality Control (QC) public-use files as a verification oracle for independently encoded SNAP rules: across seven states, the encodings reproduce the file’s benefit-computation chain, from file-recorded intermediates, exactly — all 6,081 in-scope cases of the 6,194-case official universe (113 documented exclusions), six computation stages per case, zero dollar tolerance — a process that surfaced and fixed two defects in the encodings and surfaced errata in the federal technical documentation. Replaying reconstructed pre-edit case facts through the verified engine explains 78.4% of Colorado’s above-threshold official error cases — and 91.4% of sub-threshold deviations — as correct arithmetic on wrong facts. Pricing that decomposition through the formula’s tiers, elections, and delay clause — removing the error dollars the file’s cause codes assign to the computing apparatus, an accounting scenario rather than a causal estimate — moves summed expected fiscal 2028 bills from $7.7B to $7.6B a year under a strict class convention and $6.9B under a broad one, and not monotonically: a state just under the statute’s delay threshold sees its near-term expected bill rise as its measured rate falls. I also fit and validate a distributional model of case-level payment deviations, reporting its validation failures alongside its results, and serve one policy counterfactual from it in the deployed simulator: a standard medical deduction scenario computed on engine-recomputed case parameters, whose model-implied Colorado effect opposes the naive accounting bound in sign. The central finding, from an open Monte Carlo simulator resampling each state’s own QC cases, is about measurement: at regulatory sample sizes, several states face nearly even odds — 46 to 50 percent — of being assigned a different cost tier than their official point rate implies. The fiscal 2025 rates, published in June 2026, land where that mechanism says realized years should: 18 of 53 jurisdictions changed cost-sharing tiers in one year, and only seven of those changes exceed the 95% band that two years of sampling noise — estimated from fiscal 2024 microdata alone — implies.</p>
+        <p>Beginning in fiscal year 2028, each state’s Supplemental Nutrition Assistance Program (SNAP) payment error rate sets its share of benefit costs, in steps of five percentage points at rates of 6, 8, and 10 percent. A measurement system built for oversight now moves large sums, which raises two questions this paper treats together: whether the rules that produce benefit amounts can be executed correctly, and how the errors that remain should be measured and simulated. I use the USDA Quality Control (QC) public-use files as a verification oracle for independently encoded SNAP rules: across seven states, the encodings reproduce the file’s benefit-computation chain, from file-recorded intermediates, exactly — all 6,081 in-scope cases of the 6,194-case official universe (113 documented exclusions), six computation stages per case, zero dollar tolerance — a process that surfaced and fixed two defects in the encodings. For 283 of Colorado’s 305 error cases, a public solver moves the input named by each case’s first finding, where it can, until its recomputed benefit reaches the issued amount; run on those inputs, the verified engine reproduces the issued benefit within $5 for 78.4% of the above-threshold official error cases and 91.4% of the sub-threshold deviations, which is consistent with correct arithmetic on a wrong input. Pricing a cause-code decomposition through the formula’s tiers, elections, and delay clause — removing the error dollars the file’s cause codes assign to the computing apparatus, an accounting scenario rather than a causal estimate — moves summed expected fiscal 2028 bills from $7.7B to $7.6B a year under a strict class convention and $6.9B under a broad one, and not monotonically: a state just under the statute’s delay threshold sees its near-term expected bill rise as its measured rate falls. I also fit and validate a distributional model of case-level payment deviations, reporting its validation failures alongside its results, and serve one policy counterfactual from it in the deployed simulator: a standard medical deduction scenario computed on engine-recomputed case parameters, whose model-implied Colorado effect opposes the naive accounting bound in sign. The central finding, from an open Monte Carlo simulator resampling each state’s own QC cases, is about measurement: at regulatory sample sizes, several states face nearly even odds — 46 to 50 percent — of being assigned a different cost tier than their official point rate implies. The fiscal 2025 rates, published in June 2026, land where that mechanism says realized years should: 18 of 53 jurisdictions changed cost-sharing tiers in one year, and only seven of those changes exceed the 95% band that two years of sampling noise — estimated from fiscal 2024 microdata alone — implies.</p>
       </div>
     </div>
 
@@ -252,7 +254,7 @@ <h2 id="toc-title">Table of contents</h2>
   <li><a href="#disclosure" id="toc-disclosure" class="nav-link" data-scroll-target="#disclosure">Disclosure</a></li>
   <li><a href="#acknowledgments" id="toc-acknowledgments" class="nav-link" data-scroll-target="#acknowledgments">Acknowledgments</a></li>
   </ul>
-<div class="quarto-alternate-notebooks"><h2>Notebooks</h2><ul><li><a href="reviews/round-1/code-cert-skeptic-preview.html"><i class="bi bi-journal-code"></i>Adversarial audit: /Users/maxghenis/.cache/axiom-oracles/v2b/CERT_REPORT.md</a></li><li><a href="reviews/round-1/code-stats-redteam-preview.html"><i class="bi bi-journal-code"></i>Adversarial statistical review: snap-qc-sim (main @ 026a362, post PR #9)</a></li><li><a href="reviews/round-1/referee-citations-preview.html"><i class="bi bi-journal-code"></i>Citation Review — snap-qc-sim paper</a></li><li><a href="reviews/round-1/code-claims-sweep-preview.html"><i class="bi bi-journal-code"></i>Claims audit — snap-qc-sim (main @ 026a362)</a></li><li><a href="snapshot/cert/CERT_REPORT-preview.html"><i class="bi bi-journal-code"></i>CO SNAP QC certification: CERTIFIED</a></li><li><a href="reviews/round-1/referee-domain-preview.html"><i class="bi bi-journal-code"></i>Domain expert review: SNAP program and quality-control specialist</a></li><li><a href="FACTS-preview.html"><i class="bi bi-journal-code"></i>Fact catalog</a></li><li><a href="reviews/round-1/code-js-fidelity-preview.html"><i class="bi bi-journal-code"></i>Fidelity audit: app.js model mode vs analysis/predictive_process.py</a></li><li><a href="reviews/round-4/methodology-preview.html"><i class="bi bi-journal-code"></i>Methodology review — revision-4 delta (verbatim, 2026-08-09)</a></li><li><a href="reviews/round-2/round2-methodology-preview.html"><i class="bi bi-journal-code"></i>Methodology review — round 2 (revised manuscript, commit 27e1b09)</a></li><li><a href="reviews/round-1/referee-methodology-preview.html"><i class="bi bi-journal-code"></i>Methodology review: "Administrative quality control data as an oracle" (paper/index.qmd)</a></li><li><a href="reviews/round-1/referee-neutrality-preview.html"><i class="bi bi-journal-code"></i>Neutrality Review</a></li><li><a href="snapshot/qrf-cross-process-attestation-preview.html"><i class="bi bi-journal-code"></i>QRF benchmark cross-process determinism attestation</a></li><li><a href="reviews/round-4/red-team-preview.html"><i class="bi bi-journal-code"></i>Red-team review — revision-4 delta (verbatim, 2026-08-09)</a></li><li><a href="reviews/round-2/round2-redteam-preview.html"><i class="bi bi-journal-code"></i>Red-Team Review — Round 2 (revised manuscript, commit 27e1b09)</a></li><li><a href="reviews/round-1/referee-redteam-preview.html"><i class="bi bi-journal-code"></i>Red-Team Review: "Administrative quality control data as an oracle" (paper/index.qmd @ bce8eb6, origin/paper)</a></li><li><a href="reviews/round-1/referee-reproducibility-preview.html"><i class="bi bi-journal-code"></i>Reproducibility Review</a></li><li><a href="reviews/round-3/methodology-preview.html"><i class="bi bi-journal-code"></i>Round 3 — methodology review (revision-3 changed regions)</a></li><li><a href="reviews/round-3/red-team-preview.html"><i class="bi bi-journal-code"></i>Round 3 — red-team review (revision-3 changed regions)</a></li><li><a href="reviews/round-3/round-diff-preview.html"><i class="bi bi-journal-code"></i>Round 3 — round-diff review (alignment-pass @ 084c6bb)</a></li><li><a href="reviews/round-1/EDITORIAL-preview.html"><i class="bi bi-journal-code"></i>Round-1 editorial synthesis</a></li><li><a href="reviews/round-5/methodology-preview.html"><i class="bi bi-journal-code"></i>Round-5 methodology referee report</a></li><li><a href="reviews/round-5/neutrality-preview.html"><i class="bi bi-journal-code"></i>Round-5 neutrality and style referee report</a></li><li><a href="reviews/round-5/red-team-preview.html"><i class="bi bi-journal-code"></i>Round-5 red-team referee report</a></li><li><a href="reviews/round-5/SUMMARY-preview.html"><i class="bi bi-journal-code"></i>Round-5 referee summary</a></li><li><a href="reviews/round-5/reproducibility-preview.html"><i class="bi bi-journal-code"></i>Round-5 reproducibility referee report</a></li><li><a href="reviews/round-5/round-diff-preview.html"><i class="bi bi-journal-code"></i>Round-5 revision-difference referee report</a></li><li><a href="reviews/round-6/RESOLUTIONS-preview.html"><i class="bi bi-journal-code"></i>Round-6 resolutions (maintainer adjudication, 2026-08-14)</a></li><li><a href="reviews/round-2/round2-rounddiff-preview.html"><i class="bi bi-journal-code"></i>Round-Diff Review: round-1 resolution audit (paper branch @ 27e1b09)</a></li><li><a href="reviews/round-1/referee-stylistic-preview.html"><i class="bi bi-journal-code"></i>Stylistic review: "Administrative quality control data as an oracle"</a></li><li><a href="reviews/round-6/SUMMARY-preview.html"><i class="bi bi-journal-code"></i>SUMMARY.md</a></li><li><a href="snapshot/labs/amterr/ANALYSIS-preview.html"><i class="bi bi-journal-code"></i>What share of SNAP payment errors could a verified rules engine prevent?</a></li></ul></div></nav>
+</nav>
 </div>
 <div id="quarto-margin-sidebar" class="sidebar margin-sidebar zindex-bottom">
 </div>
@@ -268,8 +270,8 @@ <h2 class="anchored" data-anchor-id="introduction">Introduction</h2>
 <p>Under 7 U.S.C. § 2013(a)(2), as amended by the One Big Beautiful Bill Act (Pub. L. No.&nbsp;119-21, 2025; OBBBA), a state whose SNAP payment error rate reaches 6% will pay 5% of benefit costs beginning in fiscal year 2028; at 8%, the share is 10%; at 10%, it is 15%. The rate that determines the FY2028 share is the state’s fiscal 2025 or 2026 rate, at its election; later years use the third preceding year. Implementation is delayed for the highest-error states: a state whose fiscal 2025 rate times 1.5 reaches 20% — that is, a rate above 13.33% (13.34% at the published two-decimal precision) — starts in FY2029 instead, and a state whose fiscal 2026 rate crosses the same test starts in FY2030.<a href="#fn1" class="footnote-ref" id="fnref1" role="doc-noteref"><sup>1</sup></a> For a mid-sized state, one tier is worth tens of millions of dollars a year; for the largest states, more than a hundred million.</p>
 <p>The error rate is estimated by the Quality Control system: each state re-reviews a probability sample of its own active cases — by formula in 7 C.F.R. § 275.11, between 300 and 2,400 annually, or 300 to 1,020 under an optional reduced schedule that every state elected in fiscal 2024 <span class="citation" data-cites="fns2024techdoc">(<a href="#ref-fns2024techdoc" role="doc-biblioref">U.S. Department of Agriculture, Food and Nutrition Service 2025b</a>)</span> — and federal reviewers re-review a subsample, from which the Food and Nutrition Service (FNS; renamed the Food and Nutrition Administration in 2026 — the fiscal 2025 rate publication carries the new letterhead <span class="citation" data-cites="fna2025per">(<a href="#ref-fna2025per" role="doc-biblioref">U.S. Department of Agriculture, Food and Nutrition Administration 2026</a>)</span>, and this paper keeps FNS, the name on every document the fiscal 2024 data cite) estimates official state rates with adjustments including federal re-review integration. The public-use files that result record, for roughly 45,000 cases a year, what the agency issued and what the review concluded the case should have received, with coded findings for every discrepancy.</p>
 <p>This paper treats those files as a research instrument with three uses, the first in the software-testing sense of an <em>oracle</em>: an external source of ground-truth outputs for given inputs.</p>
-<p>First, as a verification target for rules as code. I test independently encoded SNAP rules — statute, regulations, and state policy manuals encoded as executable rule modules — against the QC file’s recorded benefit computations in seven states, holding the bar at exact reproduction: zero dollar tolerance across six computation stages per case (<a href="#sec-oracle" class="quarto-xref">Section&nbsp;4</a>). The target’s precise character matters and is stated in <a href="#sec-oracle-scope" class="quarto-xref">Section&nbsp;4.1</a>: the file’s benefit chain is the FNS QC Minimodel’s computation on edited, internally consistent case records, so parity certifies agreement with the agency’s own computational canon, not independent adjudication. Reaching it surfaced defects on three sides: two in the encodings under test, errata in the federal technical documentation, and a residue attributable to state issuance systems.</p>
-<p>Second, as raw material for decomposing measured error. QC’s own cause coding, a finding-nature classification, and a replay of reconstructed pre-edit case facts through the verified engine give three nested estimates of how much of the error rate reflects computation failure versus wrong facts correctly processed (<a href="#sec-decompose" class="quarto-xref">Section&nbsp;5</a>).</p>
+<p>First, as a verification target for rules as code. I test independently encoded SNAP rules — statute, regulations, and state policy manuals encoded as executable rule modules — against the QC file’s recorded benefit computations in seven states, holding the bar at exact reproduction: zero dollar tolerance across six computation stages per case (<a href="#sec-oracle" class="quarto-xref">Section&nbsp;4</a>). The target’s precise character matters and is stated in <a href="#sec-oracle-scope" class="quarto-xref">Section&nbsp;4.1</a>: the file’s benefit chain is a computation Mathematica runs for USDA on edited case records with the benefit formula of the FNS QC Minimodel, so parity certifies agreement with the benefit computation recorded in the file; it does not adjudicate case outcomes. Reaching it surfaced defects on two sides: two in the encodings under test, and a residue attributable to state issuance systems.</p>
+<p>Second, as raw material for decomposing measured error. QC’s own cause coding and a finding-nature classification give two nested estimates of how much of the error rate reflects computation failure versus wrong facts correctly processed, and a replay of reconstructed case inputs through the verified engine tests whether one wrong input reproduces each issued benefit (<a href="#sec-decompose" class="quarto-xref">Section&nbsp;5</a>).</p>
 <p>Third, as training and validation data for a case-level model of the error process (<a href="#sec-model" class="quarto-xref">Section&nbsp;6</a>) and the basis for a distributional simulator of measured rates under the cost-sharing formula (<a href="#sec-simulate" class="quarto-xref">Section&nbsp;7</a>), public at policyengine.org/us/snap-payment-error-simulator. The simulator’s central output is a measurement result: near tier boundaries, sampling noise alone gives several states nearly even odds of a different cost tier than their point rate implies.</p>
 <p>The analysis pipeline, simulator, and this manuscript live in one public repository; a fact catalog maps every quantitative claim to a committed artifact (<a href="#sec-availability" class="quarto-xref">Section&nbsp;11</a>). The pipeline is deterministic — independent clean-room reruns reproduce every analysis artifact byte-for-byte, and the headline simulation artifact was independently reproduced from the raw file to four decimals — and the manuscript has been through six rounds of adversarial review, archived unedited in the repository.</p>
 </section>
@@ -315,7 +317,7 @@ <h2 class="anchored" data-anchor-id="sec-qc">The QC system and the money attache
 </tr>
 <tr class="even">
 <td><code>FSBEN</code></td>
-<td>the formula benefit recomputed by the FNS QC Minimodel from the case’s edited inputs</td>
+<td>the formula benefit Mathematica computes for USDA from the case’s edited inputs with the FNS QC Minimodel’s benefit formula</td>
 </tr>
 <tr class="odd">
 <td><code>STATUS</code></td>
@@ -346,7 +348,7 @@ <h2 class="anchored" data-anchor-id="sec-qc">The QC system and the money attache
 </section>
 <section id="sec-oracle" class="level2">
 <h2 class="anchored" data-anchor-id="sec-oracle">The QC file as a verification oracle</h2>
-<p>Rules as code — encoding statutes and regulations as executable, testable logic — has a recognized validation problem: against what do you test the encoding? Unit tests encode the encoder’s own reading of the law, and cross-model comparisons inherit both models’ assumptions; the literature has approached validation by compiling legal text alongside experts <span class="citation" data-cites="merigoux2021catala oecd2020cracking">(<a href="#ref-merigoux2021catala" role="doc-biblioref">Mérigoux, Chataing, et al. 2021</a>; <a href="#ref-oecd2020cracking" role="doc-biblioref">Mohun and Roberts 2020</a>)</span> and, closest to this paper, by testing an independent encoding of the French tax code against the tax authority’s own published test cases <span class="citation" data-cites="merigoux2021compiler">(<a href="#ref-merigoux2021compiler" role="doc-biblioref">Mérigoux, Monat, et al. 2021</a>)</span>. Administrative QC data extends that strategy to a benefits program at case level: tens of thousands of real cases per year, each carrying inputs and the agency’s own computed outputs.</p>
+<p>Rules as code — encoding statutes and regulations as executable, testable logic — has a recognized validation problem: against what do you test the encoding? Unit tests encode the encoder’s own reading of the law, and cross-model comparisons inherit both models’ assumptions; the literature has approached validation by compiling legal text alongside experts <span class="citation" data-cites="merigoux2021catala oecd2020cracking">(<a href="#ref-merigoux2021catala" role="doc-biblioref">Mérigoux, Chataing, et al. 2021</a>; <a href="#ref-oecd2020cracking" role="doc-biblioref">Mohun and Roberts 2020</a>)</span> and, closest to this paper, by testing an independent encoding of the French tax code against the tax authority’s own published test cases <span class="citation" data-cites="merigoux2021compiler">(<a href="#ref-merigoux2021compiler" role="doc-biblioref">Mérigoux, Monat, et al. 2021</a>)</span>. Administrative QC data extends that strategy to a benefits program at case level: tens of thousands of real cases per year, each carrying edited inputs and the benefit chain Mathematica computes for USDA from the edited record.</p>
 <p>I tested encodings of federal SNAP law and seven states’ policy manuals (Arizona, California, Colorado, Georgia, Maryland, New York, Texas), maintained by The Axiom Foundation in its public rulespec-us repository and executed by its deterministic rules engine, against the fiscal 2024 QC file. The file contains 6,194 cases in the official universe for those states; 6,081 are in scope for replay — all 113 exclusions (1.8%) are SSI-CAP (Combined Application Project) standardized-benefit units, whose allotments follow a demonstration schedule outside the computed chain, logged per case by the harness. <a href="#tbl-verify" class="quarto-xref">Table&nbsp;2</a> reports the counts. For each in-scope case the harness asserts six values — gross income, standard deduction, excess shelter deduction, net income, maximum allotment, and benefit — against the file’s recorded values at zero dollar tolerance. These six are the stage outputs the public file records for the computed chain; adjudication-dependent amounts the reviewer establishes as facts — claimed medical expenses, utility allowances, child-support obligations, eligibility findings — enter as inputs rather than assertions. The assertion surface can widen as encodings and file coverage allow (the earned-income deduction, for instance, is mechanical given earnings), and the excluded SSI-CAP units are themselves encodable: their demonstration benefit schedules are published in the technical documentation, so bringing them in scope — for both verification and simulation — is an encoding task, not a data gap.</p>
 <p>All 6,081 in-scope cases match on all six asserted values — 36,486 exact cell comparisons, no mismatches.</p>
 <div id="tbl-verify" class="quarto-float quarto-figure quarto-figure-center anchored">
@@ -427,19 +429,19 @@ <h2 class="anchored" data-anchor-id="sec-oracle">The QC file as a verification o
 <section id="sec-oracle-scope" class="level3">
 <h3 class="anchored" data-anchor-id="sec-oracle-scope">What the parity target is</h3>
 <p>Exactness claims invite overreading, so the scope deserves a precise statement — more precise than an earlier draft of this paper gave it.</p>
-<p>The compared benefit chain is the one recorded in the edited public-use file, whose benefit variable (<code>FSBEN</code>) is computed by the FNS QC Minimodel — the agency’s own benefit-calculation software — and whose editing process reconciles case records until internal identities hold, adjusting deduction fields in a documented sequence until the calculated benefit matches the raw benefit within $5 <span class="citation" data-cites="fns2024techdoc">(<a href="#ref-fns2024techdoc" role="doc-biblioref">U.S. Department of Agriculture, Food and Nutrition Service 2025b</a>)</span>. The comparison also supplies several intermediates directly from the file rather than deriving them from raw facts: the QC-calculated medical and child-support deduction amounts, the recorded utility allowance, and the recorded categorical-eligibility status; eligibility screens receive passing defaults, since every filed case was enrolled.</p>
-<p>Parity therefore certifies the following: that the encoded computation chain — income aggregation, deduction sequencing, net-income arithmetic, allotment lookup, rounding — agrees exactly with the agency’s own computational canon, over the full diversity of 6,081 real case configurations, parameter values, and boundary conditions. It is an agreement between two implementations of the same rules, closer in kind to the cross-model comparisons this method extends than to independent adjudication of case outcomes; what distinguishes it from ordinary cross-model comparison is the counterparty (the agency’s own canon), the case realism, and the zero-tolerance bar. It does not certify independent derivation of every state eligibility and deduction pathway from raw facts. Extending the harness to feed raw facts — recorded medical expenses rather than the computed deduction, energy-assistance facts rather than the utility tier — is the necessary next step before the engine can compute counterfactual intermediates under alternative policy (<a href="#sec-roadmap" class="quarto-xref">Section&nbsp;8</a>). The reviewer-adjudicated quantities in the file (<code>RAWBEN</code>, <code>BENFIX</code>, error amounts and findings) are not the parity target; they are the subject of <a href="#sec-decompose" class="quarto-xref">Section&nbsp;5</a>.</p>
-<p>Holding tolerance at zero is what made the exercise informative. Early runs did not match, and each mismatch class was a finding about somebody’s rules: two defects in the encodings under test (a regulatory dollar literal superseded by statute, and a missing whole-dollar rounding step — both fixed in the public repository, with the oracle as the detector), errata in the technical documentation’s variable descriptions (reported upstream), and — once the encodings reproduced the file’s chain exactly — a residue in <em>issued</em> amounts attributable to the issuing side, which <a href="#sec-decompose" class="quarto-xref">Section&nbsp;5</a> takes up. The corrections were global rule fixes, not case-specific accommodations; still, the exactness result is post-correction on the same case set, a reuse the certification protocol mitigates but does not eliminate. A confirmatory run on the fiscal 2025 file at its release is the committed next test.</p>
+<p>The compared benefit chain is the one recorded in the edited public-use file. Its benefit variable (<code>FSBEN</code>) is a constructed variable: Mathematica, which edits the file for USDA under contract, computes it from each edited case record with the benefit formula of the QC Minimodel, one of FNS’s SNAP microsimulation models; the file’s other chain values (gross income, the standard and excess shelter deductions, net income, maximum allotment) are constructed variables too <span class="citation" data-cites="fns2024techdoc">(<a href="#ref-fns2024techdoc" role="doc-biblioref">U.S. Department of Agriculture, Food and Nutrition Service 2025b</a>)</span>. Where the calculated benefit is more than $5 from the issued benefit adjusted for the reviewer’s error amount (Step 13a lists the exact match conditions), the editing tries a documented sequence of changes (setting the dependent care deduction to dependent care costs, setting the utility amount in the excess shelter deduction to a standard allowance or zero, and zeroing the medical deduction of standard medical deduction demonstration participants) and keeps a change only if it produces a match <span class="citation" data-cites="fns2024techdoc">(<a href="#ref-fns2024techdoc" role="doc-biblioref">U.S. Department of Agriculture, Food and Nutrition Service 2025b, III</a>, Steps 12–13)</span>. The sequence does not always close the gap: in Colorado, <code>FSBEN</code> ends within $5 of <code>BENFIX</code> in 797 of 856 cases. The comparison also supplies several intermediates directly from the file rather than deriving them from raw facts: the QC-calculated medical and child-support deduction amounts, the recorded utility allowance, and the recorded categorical-eligibility status; eligibility screens receive passing defaults, since every filed case was enrolled.</p>
+<p>Parity therefore certifies the following: that the encoded computation chain — income aggregation, deduction sequencing, net-income arithmetic, allotment lookup, rounding — agrees exactly with the benefit computation recorded in the edited file, over the full diversity of 6,081 real case configurations, parameter values, and boundary conditions. It is an agreement between two implementations of the same rules, closer in kind to the cross-model comparisons this method extends than to independent adjudication of case outcomes; what distinguishes it from ordinary cross-model comparison is the counterparty (a computation Mathematica runs for USDA on the edited case records), the case realism, and the zero-tolerance bar. It does not certify independent derivation of every state eligibility and deduction pathway from raw facts. Extending the harness to feed raw facts — recorded medical expenses rather than the computed deduction, energy-assistance facts rather than the utility tier — is the necessary next step before the engine can compute counterfactual intermediates under alternative policy (<a href="#sec-roadmap" class="quarto-xref">Section&nbsp;8</a>). The reviewer-adjudicated quantities in the file (<code>RAWBEN</code>, <code>BENFIX</code>, error amounts and findings) are not the parity target; they are the subject of <a href="#sec-decompose" class="quarto-xref">Section&nbsp;5</a>.</p>
+<p>Holding tolerance at zero is what made the exercise informative. Early runs did not match, and each mismatch class was a finding about somebody’s rules: two defects in the encodings under test (a regulatory dollar literal superseded by statute, and a missing whole-dollar rounding step — both fixed in the public repository, with the oracle as the detector) and, once the encodings reproduced the file’s chain exactly, a residue in <em>issued</em> amounts attributable to the issuing side, which <a href="#sec-decompose" class="quarto-xref">Section&nbsp;5</a> takes up. The corrections were global rule fixes, not case-specific accommodations; still, the exactness result is post-correction on the same case set, a reuse the certification protocol mitigates but does not eliminate. A confirmatory run on the fiscal 2025 file at its release is the committed next test.</p>
 <p>The toolchain is certified for reproduction: a pinned engine build and encoding-repository commit reproduce the Colorado suite — 856 cases, 5,136 cells including the 856 benefit cells — under recorded hashes for every input, with an independent audit rebuilding the engine to a byte-identical binary and re-running the comparison to byte-identical evidence. Throughput is 230 cases per second end to end on a laptop, so verification at this scale is not compute-bound for this workload; details and timing calibration are in the repository’s certification report.</p>
 </section>
 </section>
 <section id="sec-decompose" class="level2">
 <h2 class="anchored" data-anchor-id="sec-decompose">Decomposing measured error: computation versus facts</h2>
-<p>If a verified engine issued every benefit, how much measured error would disappear? The QC file supports three nested answers for Colorado fiscal 2024, summarized in <a href="#tbl-decompose" class="quarto-xref">Table&nbsp;3</a>: 305 of 856 sampled cases carry recorded payment deviations (<code>RAWBEN</code> ≠ <code>BENFIX</code>) — but only 110 of the 305 are official errors above the $56 threshold; the other 195 are sub-threshold deviations the official rate excludes. Weighted, the 305 carry $112.6M per year in deviation dollars on $1.268B of issuance, of which $18.5M (16%) is sub-threshold: the deviation-based 8.88% file rate therefore combines an official-definition file rate of 7.42% with sub-threshold dollars, while the official 9.97% additionally incorporates the federal re-review adjustment and the excluded ineligible-case error of <a href="#sec-qc" class="quarto-xref">Section&nbsp;3</a>. All three layers exclude, by construction of the file, that ineligible-case error.</p>
+<p>If a verified engine issued every benefit, how much measured error would disappear? The QC file supports three layers of evidence for Colorado fiscal 2024, summarized in <a href="#tbl-decompose" class="quarto-xref">Table&nbsp;3</a>: 305 of 856 sampled cases carry recorded payment deviations (<code>RAWBEN</code> ≠ <code>BENFIX</code>) — but only 110 of the 305 are official errors above the $56 threshold; the other 195 are sub-threshold deviations the official rate excludes. Weighted, the 305 carry $112.6M per year in deviation dollars on $1.268B of issuance, of which $18.5M (16%) is sub-threshold: the deviation-based 8.88% file rate therefore combines an official-definition file rate of 7.42% with sub-threshold dollars, while the official 9.97% additionally incorporates the federal re-review adjustment and the excluded ineligible-case error of <a href="#sec-qc" class="quarto-xref">Section&nbsp;3</a>. All three layers exclude, by construction of the file, that ineligible-case error.</p>
 <div id="tbl-decompose" class="quarto-float quarto-figure quarto-figure-center anchored">
 <figure class="quarto-float quarto-float-tbl figure">
 <figcaption class="quarto-float-caption-top quarto-float-caption quarto-float-tbl" id="tbl-decompose-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
-Table&nbsp;3: Three nested decompositions of Colorado fiscal 2024 measured error. Layers 1–2 quote the committed cause-share artifact (<code>analysis/cause_shares.json</code>, all 53 jurisdictions, conventions and code-to-class maps serialized in the file); layer 3 is a share of filtered cases. Any-presence attribution counts a case’s dollars toward every cause class present in its slots, so layer-1 classes overlap. Layer 2 partitions cases; its system-side test reuses layer 1’s broad computing-apparatus cause set — the artifact also carries a wider agency-responsibility convention under which the system-side shares are several times larger (Colorado system-caused input 31% rather than 6.6%), answering a responsibility question rather than this table’s computation question. Cause and nature coding reflect state reviewer judgment under handbook guidance <span class="citation" data-cites="fns310handbook">(<a href="#ref-fns310handbook" role="doc-biblioref">U.S. Department of Agriculture, Food and Nutrition Service 2023</a>)</span> and carry cross-state coding variation; fiscal 2024 also brought minor revisions to the finding-code sets.
+Table&nbsp;3: Three decompositions of Colorado fiscal 2024 measured error. Layers 1–2 quote the committed cause-share artifact (<code>analysis/cause_shares.json</code>, all 53 jurisdictions, conventions and code-to-class maps serialized in the file); layer 3 is a share of filtered cases. Any-presence attribution counts a case’s dollars toward every cause class present in its slots, so layer-1 classes overlap. Layer 2 partitions cases; its system-side test reuses layer 1’s broad computing-apparatus cause set — the artifact also carries a wider agency-responsibility convention under which the system-side shares are several times larger (Colorado system-caused input 31% rather than 6.6%), answering a responsibility question rather than this table’s computation question. Cause and nature coding reflect state reviewer judgment under handbook guidance <span class="citation" data-cites="fns310handbook">(<a href="#ref-fns310handbook" role="doc-biblioref">U.S. Department of Agriculture, Food and Nutrition Service 2023</a>)</span> and carry cross-state coding variation; fiscal 2024 also brought minor revisions to the finding-code sets.
 </figcaption>
 <div aria-describedby="tbl-decompose-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
 <table class="caption-top table">
@@ -472,8 +474,8 @@ <h2 class="anchored" data-anchor-id="sec-decompose">Decomposing measured error:
 </tr>
 <tr class="odd">
 <td>3. Engine replay</td>
-<td>Reconstructed pre-edit facts replayed through the verified engine and compared to issued amounts</td>
-<td>Above-threshold official errors: 21.6% of cases unexplained (upper bound), 78.4% explained as correct arithmetic on wrong facts. Sub-threshold deviations: 8.6% unexplained, 91.4% explained. Blended: 13.1% / 86.9%</td>
+<td>A public solver moves the input named by each case’s first finding until its recomputed benefit reaches the issued amount; the verified engine replays the solver’s inputs</td>
+<td>Not identified. 78.4% of above-threshold official errors and 91.4% of sub-threshold deviations (86.9% blended) reproduce the issued benefit within $5, consistent with a wrong input; the non-reproduced cases mix solver limits, multi-element errors and computation findings</td>
 <td>283 filtered Colorado deviation cases (97 above threshold, 186 below); case shares, not dollar shares</td>
 </tr>
 </tbody>
@@ -481,8 +483,9 @@ <h2 class="anchored" data-anchor-id="sec-decompose">Decomposing measured error:
 </div>
 </figure>
 </div>
-<p>The strictest layer reconstructs each error case’s pre-edit original values using the public reconstruction solver of <span class="citation" data-cites="giannella_snapqc">Giannella and Molin (<a href="#ref-giannella_snapqc" role="doc-biblioref">2026</a>)</span> — which exploits the file’s disclosure-protection perturbation structure to recover original values — adapted to fiscal 2024, then replays those originals through the verified engine. Of 283 deviation cases surviving the solver’s consistency filters (22 of 305 are excluded), 246 reproduce the issued amount at the file’s own $5 editing tolerance: the agency’s arithmetic was correct, applied to wrong facts. The rate differs by threshold status, and the distinction matters for how the headline reads: among the 97 above-threshold official error cases, 76 are explained (78.4%, binomial standard error about 4 points; the 21.6% residual is the computation-side upper bound for <em>official</em> errors), while among the 186 sub-threshold deviations 170 are explained (91.4%). The solver and the engine partition the 283 cases identically — two implementations with distinct codebases and logic reaching the same classification, though both consume the same file, and 33 of the 246 explained cases have deviations of $5 or less, mechanically within the comparison tolerance. The residual class includes solver limitations as well as genuine multi-variable or computational discrepancies. The complete per-case replay output (283 rows), the layer-1 and layer-2 classification outputs, and the reconstruction scripts are committed with this paper.</p>
-<p>The three layers agree on the structure: most measured SNAP payment error among eligible cases is information failure, not calculation failure — consistent with the administrative-burden literature’s account of where compliance costs bind <span class="citation" data-cites="herd2018burden">(<a href="#ref-herd2018burden" role="doc-biblioref">Herd and Moynihan 2018</a>)</span>. Correct-computation guarantees address the smaller share directly — roughly 3 to 10% of error dollars by layers 1–2, and bounded above by the replay’s 21.6% case share among official errors — plus whatever share of information failures better tooling prevents indirectly through verification prompts and documentation-requirement computation at intake; <a href="#sec-adoption" class="quarto-xref">Section&nbsp;7.2</a> prices the direct shares through the formula’s tiers, elections, and delay clause. Ineligibility errors, excluded from the file, are predominantly facts-driven, which would push the information-failure share higher still; that inference, unlike the in-file shares, is not directly computed here. The indirect channel is where policy design enters: rules determine what a caseworker must collect and verify, which is the mechanism the modeling section takes up.</p>
+<p>The third layer adapts the public reconstruction solver of <span class="citation" data-cites="giannella_snapqc">Giannella and Molin (<a href="#ref-giannella_snapqc" role="doc-biblioref">2026</a>)</span> to fiscal 2024 and replays its output through the verified engine. The solver starts from the file’s edited inputs and moves only the input named by the case’s first finding (<code>ELEMENT1</code>): earned or unearned income, rent, the utility allowance, or the medical, dependent care or child-support deduction in $3 steps, or household size by one person. It stops when its own recomputed benefit reaches or passes the issued benefit (<code>RAWBEN</code>), when the input hits zero or a cap, or after 1,000 steps; when the first finding names none of those inputs, nothing moves. Of 283 deviation cases surviving the solver’s consistency filters (22 of 305 are excluded), 246 reproduce the issued amount within the file’s $5 editing tolerance, which is consistent with correct arithmetic applied to wrong facts. In 230 of the 246 the solver moved an input; in the other 16 the issued benefit was already within $5 of <code>FSBEN</code>, so nothing moved and the match restates the parity result. The rate differs by threshold status: among the 97 above-threshold official error cases 76 reproduce (78.4%, binomial standard error about 4 points), and among the 186 sub-threshold deviations 170 do (91.4%). The solver’s own recomputation and the engine agree on the within-$5 classification for all 283 cases, two implementations with distinct codebases reading the same file.</p>
+<p>Neither outcome identifies a computation error. A reproduced case shows that one value of the first-listed input, run through correct arithmetic, yields the issued benefit; it does not show which input the agency had wrong, and a moved input can absorb a computation error. A non-reproduced case can reflect the solver’s limits: it tries one input in one direction, and in 17 of the 37 it moved nothing (14 cases whose first finding names an input it does not adjust, and 3 where it stopped before moving). Of the 26 cases that carry a layer-2 computational finding, 13 reproduce, 10 do not and 3 were not replayed. The complete per-case replay output (283 rows), the layer-1 and layer-2 classification outputs, the reconstruction scripts and <code>audit_claims.py</code>, which regenerates the replay counts into <code>claims_audit.json</code>, are committed with this paper.</p>
+<p>Layers 1 and 2 agree on the structure, and the replay’s reproduction rate fits it: most measured SNAP payment error among eligible cases is information failure, not calculation failure — consistent with the administrative-burden literature’s account of where compliance costs bind <span class="citation" data-cites="herd2018burden">(<a href="#ref-herd2018burden" role="doc-biblioref">Herd and Moynihan 2018</a>)</span>. Correct-computation guarantees address the smaller share directly — roughly 3 to 10% of error dollars by layers 1–2 — plus whatever share of information failures better tooling prevents indirectly through verification prompts and documentation-requirement computation at intake; <a href="#sec-adoption" class="quarto-xref">Section&nbsp;7.2</a> prices the direct shares through the formula’s tiers, elections, and delay clause. Ineligibility errors, excluded from the file, are predominantly facts-driven, which would push the information-failure share higher still; that inference, unlike the in-file shares, is not directly computed here. The indirect channel is where policy design enters: rules determine what a caseworker must collect and verify, which is the mechanism the modeling section takes up.</p>
 </section>
 <section id="sec-model" class="level2">
 <h2 class="anchored" data-anchor-id="sec-model">Modeling the error process</h2>
@@ -539,10 +542,10 @@ <h2 class="anchored" data-anchor-id="sec-simulate">Simulating measured rates und
 <div id="fig-co" class="quarto-float quarto-figure quarto-figure-center anchored">
 <figure class="quarto-float quarto-float-fig figure">
 <div aria-describedby="fig-co-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
-<a href="figures/co_measured_rate.svg" class="lightbox" data-gallery="quarto-lightbox-gallery-1" title="Figure&nbsp;1: Colorado’s simulated measured-rate distribution (100,000 resampled QC-style measurements, official error definition, centered on the official 9.97% rate) against the cost-share tier bands. The state’s official rate sits 0.03 points below the 15%-share boundary; 52% of simulated measurements fall below it and 48% above. Generated by the committed paper/generate_figures.py (seed 11)."><img src="figures/co_measured_rate.svg" class="img-fluid figure-img"></a>
+<a href="figures/co_measured_rate.svg" class="lightbox" data-gallery="quarto-lightbox-gallery-1" title="Figure&nbsp;1: Colorado’s simulated measured-rate distribution (100,000 resampled QC-style measurements, official error definition, centered on the official 9.97% rate) against the cost-share tier bands. The state’s official fiscal 2024 rate sits 0.03 points below the 15%-share boundary, and its fiscal 2025 rate, 10.09%, sits above it; 52% of simulated measurements fall below the boundary and 48% above. Generated by the committed paper/generate_figures.py (seed 11)."><img src="figures/co_measured_rate.svg" class="img-fluid figure-img"></a>
 </div>
 <figcaption class="quarto-float-caption-bottom quarto-float-caption quarto-float-fig" id="fig-co-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
-Figure&nbsp;1: Colorado’s simulated measured-rate distribution (100,000 resampled QC-style measurements, official error definition, centered on the official 9.97% rate) against the cost-share tier bands. The state’s official rate sits 0.03 points below the 15%-share boundary; 52% of simulated measurements fall below it and 48% above. Generated by the committed <code>paper/generate_figures.py</code> (seed 11).
+Figure&nbsp;1: Colorado’s simulated measured-rate distribution (100,000 resampled QC-style measurements, official error definition, centered on the official 9.97% rate) against the cost-share tier bands. The state’s official fiscal 2024 rate sits 0.03 points below the 15%-share boundary, and its fiscal 2025 rate, 10.09%, sits above it; 52% of simulated measurements fall below the boundary and 48% above. Generated by the committed <code>paper/generate_figures.py</code> (seed 11).
 </figcaption>
 </figure>
 </div>
@@ -642,15 +645,15 @@ <h3 class="anchored" data-anchor-id="sec-events">System replacements in the reco
 </section>
 <section id="sec-roadmap" class="level2">
 <h2 class="anchored" data-anchor-id="sec-roadmap">Toward engine-computed counterfactuals</h2>
-<p>The counterfactual gap has begun to close. For one state and one option — Colorado and the standard medical deduction — the pieces now run end to end, and the run itself produced findings. The first is structural: the certified parity chain never consumes Colorado’s standard-medical-deduction rule, because QC editing bakes the deduction into the stored expense fields and the federal computation path reproduces every case exactly without it — a sharper instance of the Minimodel-canon scope point of <a href="#sec-oracle-scope" class="quarto-xref">Section&nbsp;4.1</a>. (A separate raw-facts leg, reconstructing inputs rather than supplying the file’s intermediates, agrees on 845 of 856 cases, its eleven divergences diagnosed to missing public TANF facts and one utility-allowance tier conflict; only the supplied-intermediates certified chain is exact.) A simulation-only composition adapter that binds the state rule into the federal path, validated by exact baseline invariance on all 856 cases, reprices the option’s removal only as bounds, because the public file censors 46 medical-expense records at exactly $165: between −$1.46 and $0 per case-month. The second is a sign disagreement: feeding the engine-recomputed intermediates through the error model’s direct crossing classifier, applied as a level shift, implies a cost-share change of −$1.55M to −$2.20M per year for removing the deduction. That figure is a range across the three censoring variants, not a confidence interval; the paired-bootstrap intervals are wider — the point variant’s is [−$4.1M, +$0.3M] — and span zero in two of the three. The accounting-reverse bound for the same flip is +$7.1M, and the robust form of the disagreement survives the intervals: all three exclude it, with a maximum upper bound of +$0.6M. Neither sign is validated — the artifact itself warns that the fitted models implying lower error without the deduction is not evidence that documentation requirements reduce errors. The argument for retiring the accounting levers from the public simulator is accordingly not that the model is right but that the two mechanisms answer different questions — one is an upper-bound identity scaling observed error dollars by finding category, the other a model-implied association of re-adjudication risk with the flipped feature — and their divergence on a real case shows the substitution was never innocuous. The simulator now serves only the model scenario as a policy lever, level-gated and labeled as an association; the engine-adoption scenarios of <a href="#sec-adoption" class="quarto-xref">Section&nbsp;7.2</a> are accounting constructs by design, kept because there the computing-apparatus classes are the question rather than a proxy for a behavioral response. What remains open is breadth: engine legs for the remaining options (self-employment is not identified from public data, which records net rather than gross income) and the other verified states, feeding the harness raw facts instead of file-derived intermediates (<a href="#sec-oracle-scope" class="quarto-xref">Section&nbsp;4.1</a>), and the cost-sharing formula itself as an encodable target: the sampling formulas of 7 C.F.R. § 275.11(b) and the tier schedule of 7 U.S.C. § 2013(a)(2), with its election and delay clauses, can be encoded and tested at their published boundary values, replacing hand-transcribed constants in simulators like this one.</p>
+<p>The counterfactual gap has begun to close. For one state and one option — Colorado and the standard medical deduction — the pieces now run end to end, and the run itself produced findings. The first is structural: the certified parity chain never consumes Colorado’s standard-medical-deduction rule, because QC editing bakes the deduction into the stored expense fields and the federal computation path reproduces every case exactly without it — a sharper instance of the edited-file scope point of <a href="#sec-oracle-scope" class="quarto-xref">Section&nbsp;4.1</a>. (A separate raw-facts leg, reconstructing inputs rather than supplying the file’s intermediates, agrees on 845 of 856 cases, its eleven divergences diagnosed to missing public TANF facts and one utility-allowance tier conflict; only the supplied-intermediates certified chain is exact.) A simulation-only composition adapter that binds the state rule into the federal path, validated by exact baseline invariance on all 856 cases, reprices the option’s removal only as bounds, because the public file censors 46 medical-expense records at exactly $165: between −$1.46 and $0 per case-month. The second is a sign disagreement: feeding the engine-recomputed intermediates through the error model’s direct crossing classifier, applied as a level shift, implies a cost-share change of −$1.55M to −$2.20M per year for removing the deduction. That figure is a range across the three censoring variants, not a confidence interval; the paired-bootstrap intervals are wider — the point variant’s is [−$4.1M, +$0.3M] — and span zero in two of the three. The accounting-reverse bound for the same flip is +$7.1M, and the robust form of the disagreement survives the intervals: all three exclude it, with a maximum upper bound of +$0.6M. Neither sign is validated — the artifact itself warns that the fitted models implying lower error without the deduction is not evidence that documentation requirements reduce errors. The argument for retiring the accounting levers from the public simulator is accordingly not that the model is right but that the two mechanisms answer different questions — one is an upper-bound identity scaling observed error dollars by finding category, the other a model-implied association of re-adjudication risk with the flipped feature — and their divergence on a real case shows the substitution was never innocuous. The simulator now serves only the model scenario as a policy lever, level-gated and labeled as an association; the engine-adoption scenarios of <a href="#sec-adoption" class="quarto-xref">Section&nbsp;7.2</a> are accounting constructs by design, kept because there the computing-apparatus classes are the question rather than a proxy for a behavioral response. What remains open is breadth: engine legs for the remaining options (self-employment is not identified from public data, which records net rather than gross income) and the other verified states, feeding the harness raw facts instead of file-derived intermediates (<a href="#sec-oracle-scope" class="quarto-xref">Section&nbsp;4.1</a>), and the cost-sharing formula itself as an encodable target: the sampling formulas of 7 C.F.R. § 275.11(b) and the tier schedule of 7 U.S.C. § 2013(a)(2), with its election and delay clauses, can be encoded and tested at their published boundary values, replacing hand-transcribed constants in simulators like this one.</p>
 </section>
 <section id="sec-limitations" class="level2">
 <h2 class="anchored" data-anchor-id="sec-limitations">Limitations</h2>
-<p>The parity result certifies agreement with the edited file’s Minimodel-computed chain from supplied intermediates — not independent derivation from raw facts, and not the reviewer-adjudicated outcome fields (<a href="#sec-oracle-scope" class="quarto-xref">Section&nbsp;4.1</a>); it is also post-correction on the same case set, pending a fiscal 2025 confirmatory run. The public file excludes ineligible-case and full-overissuance error, truncating the error distribution every file-based estimate inherits. The replay decomposition depends on a reconstruction solver whose filters exclude 22 of 305 error cases, and its residual class is an upper bound in case shares, not dollar shares. The error models predict an adjudicated, thresholded label — a property of the measurement system, with a documented manipulation history (<a href="#sec-qc" class="quarto-xref">Section&nbsp;3</a>). Cross-state model transport is imperfect: the model’s raw level underpredicts high-error states, all nine coverage gaps remain one-sided negative after the tail and level-factor fixes — and after the committed location repair, which improves every level but closes only part of the gap — and the deployed simulator accordingly serves the model only for its level-gated standard-medical-deduction scenario, with seven jurisdictions disabled outright; every simulation result in this paper uses the observed-resample engine. That mode resamples i.i.d. from a probability sample whose actual design varies by state, treats the official rate’s federal re-review adjustment as a fixed level with no variance contribution, and holds the error process fixed — no behavioral response of agencies or households to cost sharing, no corrective-feedback channel from audit volume, both live questions the fiscal 2025–27 files will begin to answer under the new incentives. Adoption contrasts are descriptive contrasts of policy bundles; nothing here identifies causal effects of burden policies on error.</p>
+<p>The parity result certifies agreement with the edited file’s Minimodel-computed chain from supplied intermediates — not independent derivation from raw facts, and not the reviewer-adjudicated outcome fields (<a href="#sec-oracle-scope" class="quarto-xref">Section&nbsp;4.1</a>); it is also post-correction on the same case set, pending a fiscal 2025 confirmatory run. The public file excludes ineligible-case and full-overissuance error, truncating the error distribution every file-based estimate inherits. The replay depends on a reconstruction solver whose filters exclude 22 of 305 error cases and which moves one input per case; its shares are case shares, and neither its reproduced nor its non-reproduced class identifies computation errors. The error models predict an adjudicated, thresholded label — a property of the measurement system, with a documented manipulation history (<a href="#sec-qc" class="quarto-xref">Section&nbsp;3</a>). Cross-state model transport is imperfect: the model’s raw level underpredicts high-error states, all nine coverage gaps remain one-sided negative after the tail and level-factor fixes — and after the committed location repair, which improves every level but closes only part of the gap — and the deployed simulator accordingly serves the model only for its level-gated standard-medical-deduction scenario, with seven jurisdictions disabled outright; every simulation result in this paper uses the observed-resample engine. That mode resamples i.i.d. from a probability sample whose actual design varies by state, treats the official rate’s federal re-review adjustment as a fixed level with no variance contribution, and holds the error process fixed — no behavioral response of agencies or households to cost sharing, no corrective-feedback channel from audit volume, both live questions the fiscal 2025–27 files will begin to answer under the new incentives. Adoption contrasts are descriptive contrasts of policy bundles; nothing here identifies causal effects of burden policies on error.</p>
 </section>
 <section id="conclusion" class="level2">
 <h2 class="anchored" data-anchor-id="conclusion">Conclusion</h2>
-<p>A measurement system designed for oversight now sets prices. That makes the QC files three things at once: a case-level verification oracle against the agency’s own computational canon — where the bar can be exact, and holding it exact is what made the residuals informative; the dataset showing that most measured error among eligible cases is information failure rather than calculation failure; and a sampling frame whose noise the new formula converts into near-even-odds tier assignment for boundary states. The modeling bar cannot be exact, and this paper went through six rounds of adversarial review, archived unedited in the repository. The system this points toward is one in which the rules that compute benefits, the requirements they impose on households, and the errors they produce are executable, testable, and priced from the same verified source.</p>
+<p>A measurement system designed for oversight now sets prices. That makes the QC files three things at once: a case-level verification oracle against the benefit computation recorded in the file, where the bar can be exact and holding it exact is what made the residuals informative; the dataset showing that most measured error among eligible cases is information failure rather than calculation failure; and a sampling frame whose noise the new formula converts into near-even-odds tier assignment for boundary states. The modeling bar cannot be exact, and this paper went through six rounds of adversarial review, archived unedited in the repository. The system this points toward is one in which the rules that compute benefits, the requirements they impose on households, and the errors they produce are executable, testable, and priced from the same verified source.</p>
 </section>
 <section id="sec-availability" class="level2">
 <h2 class="anchored" data-anchor-id="sec-availability">Data and code availability</h2>
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diff --git a/paper/FACTS.md b/paper/FACTS.md
index 3fc5e56..9298cd0 100644
--- a/paper/FACTS.md
+++ b/paper/FACTS.md
@@ -31,9 +31,10 @@ Numbers from the superseded pre-audit pipeline (before PR #7) are banned.
 | C1 | Seven states' encoded benefit computations reproduce the QC file's Minimodel chain exactly at zero tolerance for every in-scope case: 6,081 of 6,194 official-universe cases (CO 856/856, NY 847/885, CA 883/883, AZ 922/925, GA 945/945, MD 722/745, TX 906/955; 113 exclusions are enumerated program-structure classes); six asserted values per case | axiom-oracles PRs #244, #268, #269; committed suite reports; per-state suite configs ("in-scope") |
 | C2 | Scope caveat: the comparison bridge supplies several QC-derived intermediates as inputs (QC-calculated medical and child-support deductions, QC utility amount, QC categorical-eligibility status), so parity certifies the downstream benefit arithmetic given those intermediates, not independent derivation of every state-policy path | axiom-oracles bridge (snap_qc_compare.py); engine-leg recon 2026-08-06 |
 | C3 | Reaching parity surfaced and fixed two defects in the encodings themselves: a stale regulatory dollar literal superseded by statute, and a missing whole-dollar rounding step in the benefit computation | rulespec-us PRs (COLA backfill; #826 rounding); axiom-oracles #268 history |
-| C4 | The process also surfaced errata in FNS's own FY2024 technical documentation (variable-description inconsistencies identified while mapping), documented in the lab analysis and report page | paper/snapshot/labs/amterr/ANALYSIS.md; axiom.org/reports/colorado-snap-qc-fy2024 |
+| C4 | WITHDRAWN 2026-10-03: the claim that the process surfaced errata in FNS's FY2024 technical documentation. No source records such errata: neither version of paper/snapshot/labs/amterr/ANALYSIS.md (27e1b09, c4333b3), the axiom.org Colorado report (original and corrected text) nor the axiom-oracles SNAP QC playbook documents one, GitHub issue searches across TheAxiomFoundation and PolicyEngine found none, and round-1 and round-2 referees could not locate the "reported upstream" thread. The manuscript no longer makes the claim (abstract, introduction, @sec-oracle-scope) | searched 2026-10-03; paper/reviews/round-1/referee-redteam.md, round-2/round2-rounddiff.md |
 | C5 | A certification probe (2026-08-06) attested a reproducible toolchain: engine commit de0efdc7 (binary SHA-256 bb8ec236…) × rulespec-us b53ce208 reproduces CO 856/856 benefits and 5,136/5,136 stage cells at zero tolerance; 856 cases in 3.73 s end-to-end (230 cases/s; 279/s with compile amortized) | CERT_REPORT (paper/snapshot/cert/CERT_REPORT.md) |
 | C6 | Extrapolation at certified throughput: 6,081 cases ≈ 26 s; 44,800 × 5 scenarios ≈ 16 minutes single-threaded on a laptop | CERT_REPORT extrapolation table |
+| C7 | What the parity target is: `FSBEN` is a constructed variable ("Final calculated benefit") that Mathematica computes for USDA from each edited case record with the QC Minimodel's benefit formula; the QC Minimodel is "one of FNA's SNAP microsimulation models". The file's other chain values (FSGRINC, FSSTDDED, FSSLTDED, FSNETINC, BENMAX) are constructed too; RAWBEN and AMTERR are reported. Editing Steps 12–13 calculate the benefit and, where it is not within $5 of the issued benefit adjusted for the error amount (Step 13a conditions), try the dependent care deduction, the utility amount and the SMD-demonstration medical deduction, keeping a change only if it produces a match. Colorado: `FSBEN` within $5 of `BENFIX` in 797 of 856 cases (743 exactly equal; same under the May and August 2026 postings, which differ only in weights). Never "the agency's own benefit-calculation software" or "computational canon" | FY2024 QC technical documentation, August 2026 posting: title page and acknowledgments (Mathematica for FNA, contract 12-3198-24-Q-0029), PDF p.15 (Minimodel), pp.36–37 (Steps 12–13), p.65 (Minimodel's FSBEN calculation refers to the codebook entry), p.73 and p.91 (FSBEN constructed; formula); 797/856 recomputed from qc_pub_fy2024.csv (sha256 e871a8e9…), locked in tests/test_retired_claims.py |
 
 ## D. Decomposition of measured error (Colorado FY2024)
 
@@ -42,7 +43,7 @@ Numbers from the superseded pre-audit pipeline (before PR #7) are banned.
 | D1 | Colorado FY2024: 856 sampled cases, 305 with payment errors (STATUS 2/3); $112.6M/yr weighted error dollars on $1.268B issuance (8.88% file-derived; official regression-adjusted rate 9.97% = 7.91 over + 2.06 under) | paper/snapshot/labs/amterr/ANALYSIS.md; FNS FY2024 PER (https://fns-prod.azureedge.us/sites/default/files/resource-files/snap-fy24QC-PER.pdf) |
 | D2 | QC cause coding, SUPERSEDED by K1 (revision 5): the lab tabulation denominated over all weighted deviation dollars ($8.1B national incl. sub-threshold) and gave CO $8.2M/7.3% strict, $11.9M/10.5% broad; national $320M/3.9%, $1.5B/18.4%. The committed artifact denominates over official error dollars and differs (see K1) | paper/snapshot/labs/amterr/native_decomposition.json + ANALYSIS.md (archived) |
 | D3 | Finding-nature classification: pure-math 3.3%, input-with-system-cause 6.6%, mixed 4.2%, other input 86.0% of Colorado weighted deviation dollars. NOW COMMITTED as K3 (revision 5): the artifact reproduces these to the cent under the corrected field semantics (NATUREi carries nature codes; E_FINDGi is impact-only, a lab labeling error that did not affect the numbers) | analysis/cause_shares.json layer 2 (was: paper/snapshot/labs/phase_a_classification.json) |
-| D4 | Replay: reconstructing pre-edit original values (public $3-shift solver, FY2024 adaptation) and replaying through the verified engine explains 246/283 filtered error cases (86.9%) as correct arithmetic on wrong facts; solver and engine partition the 283 cases identically; 37 cases form the computation-side upper bound | paper/snapshot/labs/amterr/amterr_replay_results.json (283 per-case rows) + reconstruct_co_fy2024.R + amterr_replay.py |
+| D4 | Replay: the public solver (FY2024 adaptation of giannella/snap_qc) starts from the file's edited inputs and moves only the ELEMENT1 input ($3 steps; household size by one person for element 150) until its own recomputed benefit reaches or passes RAWBEN, the input hits zero or a cap, or 1,000 steps; it uses no disclosure-perturbation structure. The verified engine on the solver's inputs reproduces RAWBEN within $5 for 246 of 283 filtered error cases (86.9%), consistent with correct arithmetic on a wrong input: 230 after the solver moved an input, 16 with nothing moved (\|RAWBEN − FSBEN\| ≤ $5 already, so the match restates parity). 37 do not reproduce; in 17 nothing moved (14 `no_change` rows whose ELEMENT1 the solver does not adjust, 3 where it stopped before moving). Solver and engine agree on all 283. Neither class identifies computation errors: of 26 cases with a layer-2 computational finding, 13 reproduce, 10 do not, 3 were not replayed. SUPERSEDES the revision-9 wording "explains 246/283 … as correct arithmetic on wrong facts" and "37 cases form the computation-side upper bound" (withdrawn 2026-10-03, matching ANALYSIS.md's revision of the same date) | paper/snapshot/labs/amterr/claims_audit.json case_level.replay (regenerated by audit_claims.py); amterr_replay_results.json (283 rows); reconstruct_co_fy2024.R; amterr_replay.py; ANALYSIS.md layer 3 |
 
 ## E. Error models (corrected pipeline: PRs #7–#8 as refit by #13)
 
@@ -63,7 +64,7 @@ Numbers from the superseded pre-audit pipeline (before PR #7) are banned.
 | # | Fact | Source |
 |---|---|---|
 | F1 | Tier assignment near boundaries is noisy at regulatory sample sizes: P(different tier than the official point rate implies) — ND 50%, WA 48%, CO 49%, KS 47%, NV 46% | paper/snapshot/labs/results_by_state_corrected.json (official error gate, CASE==1) |
-| F2 | Colorado: official 9.97% (0.03pp below the 15% boundary), $1.268B issuance, expected FY2028-rule cost $156.4M/yr, SD $32.8M; the 10%→15% tier step is $63.4M/yr | paper/snapshot/labs/results_by_state_corrected.json |
+| F2 | Colorado: official FY2024 9.97% (0.03pp below the 15% boundary; the FY2025 rate, 10.09%, is above it, J3), $1.268B issuance, expected FY2028-rule cost $156.4M/yr, SD $32.8M; the 10%→15% tier step is $63.4M/yr | paper/snapshot/labs/results_by_state_corrected.json |
 | F3 | Audit volume is a two-sided lever: +500 reviews raises expected cost for just-above-boundary states (CA −$30.3M, PA −$13.1M, TX −$13.1M expectation) and lowers it for just-below states (MO +$3.9M, TN +$3.7M, IN +$3.3M); it nearly always cuts the SD of the bill (CA −$44M) | paper/snapshot/labs/results_by_state_corrected.json |
 | F4 | Category-suppression accounting bound: removing 50% (100%) of observed error dollars in the four named simplification categories corresponds to ≈$609M/yr (≈$1,310M/yr) nationally in expected cost share | paper/snapshot/labs/results_by_state_corrected.json |
 | F5 | These lever numbers are accounting bounds on the observed FY2024 error mix — scenario dials, not causal estimates; the error process is held fixed (no behavioral response, no audit-feedback channel) | mc lab DESIGN; app method notes |
@@ -83,6 +84,15 @@ Numbers from the superseded pre-audit pipeline (before PR #7) are banned.
   something the tool currently serves.
 - No references to private conversations; the reconstruction solver is cited
   as the public software it is.
+- `FSBEN` is a benefit Mathematica computes for USDA from the edited case
+  record (C7). Never "the agency's own computational canon" or "the agency's
+  own benefit-calculation software", and never "a closed model".
+- A replay match is "consistent with" correct arithmetic on a wrong input.
+  Never "explains … as correct arithmetic on wrong facts", and never call
+  the non-reproduced cases a computation-side upper bound (D4).
+- Colorado's 0.03-point margin always carries its year (FY2024) and the
+  FY2025 rate, 10.09%, which is above the 10% line (F2, J3).
+- No claim of errata in FNS's technical documentation (C4 withdrawn).
 
 ## H. System and statute facts added in revision (round-2 verified)
 
@@ -95,7 +105,7 @@ Numbers from the superseded pre-audit pipeline (before PR #7) are banned.
 | H5 | OBBBA cut the federal share of SNAP administrative costs from 50% to 25% beginning FY2027 | OBBBA § 10106; Federal Register 2026-12696 |
 | H6 | OBBBA § 10103 restricts the LIHEAP-triggered heat-and-eat utility allowance to households with an elderly or disabled member | Pub. L. 119-21 § 10103; state implementation letters |
 | H7 | Colorado deviation universe: 305 = 110 above-threshold official errors + 195 sub-threshold deviations; sub-threshold dollars $18.5M of $112.6M (16%); official-gate file-derived rate 7.42% | recomputed from qc_pub_fy2024.csv (official gate); paper/snapshot/labs/amterr/amterr_replay_results.json |
-| H8 | Replay explanation rates by threshold status: 76/97 = 78.4% above threshold; 170/186 = 91.4% below; 33 of 246 explained cases have deviations ≤ $5 (within comparison tolerance mechanically) | paper/snapshot/labs/amterr/amterr_replay_results.json |
+| H8 | Replay reproduction rates by threshold status: 76/97 = 78.4% above threshold; 170/186 = 91.4% below (reproduced within $5; consistent with a wrong input, never "explained"). SUPERSEDES "33 of 246 explained cases have deviations ≤ $5 (within comparison tolerance mechanically)": 33 of the 246 have AMTERR ≤ $5, but the solver moved an input in 17 of them; the matches that follow mechanically from FSBEN parity are the 16 with nothing moved (D4), all among the 33 | analysis/cause_shares.json colorado_replay_reconciliation.slices; claims_audit.json case_level.replay.reproduced_without_move_n; amterr_replay_results.json (amterr field); locked in tests/test_retired_claims.py |
 
 ## I. Engine counterfactuals and the model-primary round (added in revision 3)
 
diff --git a/paper/index.qmd b/paper/index.qmd
index 624294b..608783d 100644
--- a/paper/index.qmd
+++ b/paper/index.qmd
@@ -5,7 +5,7 @@ author:
     affiliations:
       - name: PolicyEngine
     email: max@policyengine.org
-date: 2026-08-16
+date: 2026-10-03
 bibliography: references.bib
 abstract: |
   Beginning in fiscal year 2028, each state's Supplemental Nutrition
@@ -21,12 +21,14 @@ abstract: |
   intermediates, exactly — all 6,081 in-scope cases of the 6,194-case
   official universe (113 documented exclusions), six computation stages
   per case, zero dollar tolerance — a process that surfaced and fixed two
-  defects in the encodings and surfaced errata in the federal technical
-  documentation. Replaying reconstructed pre-edit case facts through the
-  verified engine explains 78.4% of Colorado's above-threshold official
-  error cases — and 91.4% of sub-threshold deviations — as correct
-  arithmetic on wrong facts. Pricing that decomposition through the
-  formula's tiers, elections, and delay clause — removing the error
+  defects in the encodings. For 283 of Colorado's 305 error cases, a
+  public solver moves the input named by each case's first finding, where
+  it can, until its recomputed benefit reaches the issued amount; run on
+  those inputs, the verified engine reproduces the issued benefit within
+  $5 for 78.4% of the above-threshold official error cases and 91.4% of
+  the sub-threshold deviations, which is consistent with correct
+  arithmetic on a wrong input. Pricing a cause-code decomposition through
+  the formula's tiers, elections, and delay clause — removing the error
   dollars the file's cause codes assign to the computing apparatus, an
   accounting scenario rather than a causal estimate — moves summed
   expected fiscal 2028 bills from $7.7B to $7.6B a year under a strict
@@ -104,18 +106,19 @@ encoded as executable rule modules — against the QC file's recorded
 benefit computations in seven states, holding the bar at exact
 reproduction: zero dollar tolerance across six computation stages per
 case (@sec-oracle). The target's precise character matters and is stated
-in @sec-oracle-scope: the file's benefit chain is the FNS QC Minimodel's
-computation on edited, internally consistent case records, so parity
-certifies agreement with the agency's own computational canon, not
-independent adjudication. Reaching it surfaced defects on three sides:
-two in the encodings under test, errata in the federal technical
-documentation, and a residue attributable to state issuance systems.
+in @sec-oracle-scope: the file's benefit chain is a computation Mathematica
+runs for USDA on edited case records with the benefit formula of the FNS
+QC Minimodel, so parity certifies agreement with the benefit computation
+recorded in the file; it does not adjudicate case outcomes. Reaching it surfaced defects on two
+sides: two in the encodings under test, and a residue attributable to
+state issuance systems.
 
 Second, as raw material for decomposing measured error. QC's own cause
-coding, a finding-nature classification, and a replay of reconstructed
-pre-edit case facts through the verified engine give three nested
-estimates of how much of the error rate reflects computation failure
-versus wrong facts correctly processed (@sec-decompose).
+coding and a finding-nature classification give two nested estimates of
+how much of the error rate reflects computation failure versus wrong
+facts correctly processed, and a replay of reconstructed case inputs
+through the verified engine tests whether one wrong input reproduces
+each issued benefit (@sec-decompose).
 
 Third, as training and validation data for a case-level model of the
 error process (@sec-model) and the basis for a distributional simulator
@@ -213,7 +216,7 @@ residents.
 | `RAWBEN` | the allotment the state actually issued |
 | `BENFIX` | the allotment corrected for the reviewer's findings |
 | `AMTERR` | the recorded error amount |
-| `FSBEN` | the formula benefit recomputed by the FNS QC Minimodel from the case's edited inputs |
+| `FSBEN` | the formula benefit Mathematica computes for USDA from the case's edited inputs with the FNS QC Minimodel's benefit formula |
 | `STATUS` | the review disposition (correct, overpayment, underpayment) |
 | `ELEMENT1-9`, `E_FINDG1-9`, `AMOUNT1-9` | up to nine coded error findings: the element in error, the nature of the finding, and its dollar amount |
 | `HWGT` | the case weight |
@@ -296,7 +299,8 @@ testing an independent encoding of the French tax code against the tax
 authority's own published test cases [@merigoux2021compiler].
 Administrative QC data extends that strategy to a benefits program at
 case level: tens of thousands of real cases per year, each carrying
-inputs and the agency's own computed outputs.
+edited inputs and the benefit chain Mathematica computes for USDA from
+the edited record.
 
 I tested encodings of federal SNAP law and seven states' policy manuals
 (Arizona, California, Colorado, Georgia, Maryland, New York, Texas),
@@ -346,27 +350,39 @@ Exactness claims invite overreading, so the scope deserves a precise
 statement — more precise than an earlier draft of this paper gave it.
 
 The compared benefit chain is the one recorded in the edited public-use
-file, whose benefit variable (`FSBEN`) is computed by the FNS QC
-Minimodel — the agency's own benefit-calculation software — and whose
-editing process reconciles case records until internal identities hold,
-adjusting deduction fields in a documented sequence until the calculated
-benefit matches the raw benefit within $5 [@fns2024techdoc]. The
-comparison also supplies several intermediates directly from the file
-rather than deriving them from raw facts: the QC-calculated medical and
+file. Its benefit variable (`FSBEN`) is a constructed variable:
+Mathematica, which edits the file for USDA under contract, computes it
+from each edited case record with the benefit formula of the QC
+Minimodel, one of FNS's SNAP microsimulation models; the file's other
+chain values (gross income, the standard and excess shelter deductions,
+net income, maximum allotment) are constructed variables too
+[@fns2024techdoc]. Where the calculated benefit is more than $5 from the
+issued benefit adjusted for the reviewer's error amount (Step 13a lists
+the exact match conditions), the editing tries a documented sequence of
+changes (setting the dependent care deduction to dependent care costs,
+setting the utility amount in the excess shelter deduction to a standard
+allowance or zero, and zeroing the medical deduction of standard medical
+deduction demonstration participants) and keeps a change only if it
+produces a match [@fns2024techdoc, chap. III, Steps 12–13]. The
+sequence does not always close the gap: in Colorado, `FSBEN` ends within
+$5 of `BENFIX` in 797 of 856 cases. The comparison also supplies several
+intermediates directly from the file rather than deriving them from raw
+facts: the QC-calculated medical and
 child-support deduction amounts, the recorded utility allowance, and the
 recorded categorical-eligibility status; eligibility screens receive
 passing defaults, since every filed case was enrolled.
 
 Parity therefore certifies the following: that the encoded computation
 chain — income aggregation, deduction sequencing, net-income arithmetic,
-allotment lookup, rounding — agrees exactly with the agency's own
-computational canon, over the full diversity of 6,081 real case
+allotment lookup, rounding — agrees exactly with the benefit computation
+recorded in the edited file, over the full diversity of 6,081 real case
 configurations, parameter values, and boundary conditions. It is an
 agreement between two implementations of the same rules, closer in kind
 to the cross-model comparisons this method extends than to independent
 adjudication of case outcomes; what distinguishes it from ordinary
-cross-model comparison is the counterparty (the agency's own canon), the
-case realism, and the zero-tolerance bar. It does not certify independent
+cross-model comparison is the counterparty (a computation Mathematica
+runs for USDA on the edited case records), the case realism, and the
+zero-tolerance bar. It does not certify independent
 derivation of every state eligibility and deduction pathway from raw
 facts. Extending the harness to feed raw facts — recorded medical
 expenses rather than the computed deduction, energy-assistance facts
@@ -381,10 +397,9 @@ runs did not match, and each mismatch class was a finding about
 somebody's rules: two defects in the encodings under test (a regulatory
 dollar literal superseded by statute, and a missing whole-dollar rounding
 step — both fixed in the public repository, with the oracle as the
-detector), errata in the technical documentation's variable descriptions
-(reported upstream), and — once the encodings reproduced the file's
-chain exactly — a residue in *issued* amounts attributable to the
-issuing side, which @sec-decompose takes up. The corrections were global
+detector) and, once the encodings reproduced the file's chain exactly, a
+residue in *issued* amounts attributable to the issuing side, which
+@sec-decompose takes up. The corrections were global
 rule fixes, not case-specific accommodations; still, the exactness result
 is post-correction on the same case set, a reuse the certification
 protocol mitigates but does not eliminate. A confirmatory run on the
@@ -403,8 +418,8 @@ report.
 ## Decomposing measured error: computation versus facts {#sec-decompose}
 
 If a verified engine issued every benefit, how much measured error would
-disappear? The QC file supports three nested answers for Colorado fiscal
-2024, summarized in @tbl-decompose: 305 of 856 sampled cases carry
+disappear? The QC file supports three layers of evidence for Colorado
+fiscal 2024, summarized in @tbl-decompose: 305 of 856 sampled cases carry
 recorded payment deviations (`RAWBEN` ≠ `BENFIX`) — but only 110 of the
 305 are official errors above the $56 threshold; the other 195 are
 sub-threshold deviations the official rate excludes. Weighted, the 305
@@ -420,9 +435,9 @@ construction of the file, that ineligible-case error.
 |---|---|---|---|
 | 1. Cause codes | QC agency-cause codes naming the computing apparatus (programming, arithmetic, mass change), any-presence across a case's nine cause slots, share of official error dollars (HWGT × `AMTERR`) | 7.4% strict ($6.9M/yr); 10.6% ($10.0M/yr) adding policy-misapplied, budgeted-wrong, computer-user | Colorado; nationally 3.5% strict ($232M/yr), 19.3% broad ($1.27B of $6.59B) |
 | 2. Finding natures | `NATURE` codes inherently describing computation (rounding, conversion, wrong standard, benefit miscomputed) plus the arithmetic `ELEMENT`; deduction-amount natures count only when the paired cause is in layer 1's broad computing-apparatus set | 3.3% pure-computation; 6.6% system-caused input; 4.2% mixed; 86.0% other input | Colorado weighted deviation dollars; nationally 4.6% / 11.1% / 6.5% / 77.7% |
-| 3. Engine replay | Reconstructed pre-edit facts replayed through the verified engine and compared to issued amounts | Above-threshold official errors: 21.6% of cases unexplained (upper bound), 78.4% explained as correct arithmetic on wrong facts. Sub-threshold deviations: 8.6% unexplained, 91.4% explained. Blended: 13.1% / 86.9% | 283 filtered Colorado deviation cases (97 above threshold, 186 below); case shares, not dollar shares |
+| 3. Engine replay | A public solver moves the input named by each case's first finding until its recomputed benefit reaches the issued amount; the verified engine replays the solver's inputs | Not identified. 78.4% of above-threshold official errors and 91.4% of sub-threshold deviations (86.9% blended) reproduce the issued benefit within $5, consistent with a wrong input; the non-reproduced cases mix solver limits, multi-element errors and computation findings | 283 filtered Colorado deviation cases (97 above threshold, 186 below); case shares, not dollar shares |
 
-: Three nested decompositions of Colorado fiscal 2024 measured error.
+: Three decompositions of Colorado fiscal 2024 measured error.
 Layers 1–2 quote the committed cause-share artifact
 (`analysis/cause_shares.json`, all 53 jurisdictions, conventions and
 code-to-class maps serialized in the file); layer 3 is a share of
@@ -438,36 +453,49 @@ judgment under handbook guidance [@fns310handbook] and carry
 cross-state coding variation; fiscal 2024 also brought minor revisions
 to the finding-code sets. {#tbl-decompose}
 
-The strictest layer reconstructs each error case's pre-edit original
-values using the public reconstruction solver of @giannella_snapqc —
-which exploits the file's disclosure-protection perturbation structure
-to recover original values — adapted to fiscal 2024, then replays those
-originals through the verified engine. Of 283 deviation cases surviving
-the solver's consistency filters (22 of 305 are excluded), 246 reproduce
-the issued amount at the file's own $5 editing tolerance: the agency's
-arithmetic was correct, applied to wrong facts. The rate differs by
-threshold status, and the distinction matters for how the headline reads:
-among the 97 above-threshold official error cases, 76 are explained
-(78.4%, binomial standard error about 4 points; the 21.6% residual is
-the computation-side upper bound for *official* errors), while among the
-186 sub-threshold deviations 170 are explained (91.4%). The solver and
-the engine partition the 283 cases identically — two implementations
-with distinct codebases and logic reaching the same classification,
-though both consume the same file, and 33 of the 246 explained cases
-have deviations of $5 or less, mechanically within the comparison
-tolerance. The residual class includes solver limitations as well as
-genuine multi-variable or computational discrepancies. The complete
+The third layer adapts the public reconstruction solver of
+@giannella_snapqc to fiscal 2024 and replays its output through the
+verified engine. The solver starts from the file's edited inputs and
+moves only the input named by the case's first finding (`ELEMENT1`):
+earned or unearned income, rent, the utility allowance, or the medical,
+dependent care or child-support deduction in $3 steps, or household size
+by one person. It stops when its own recomputed benefit reaches or passes
+the issued benefit (`RAWBEN`), when the input hits zero or a cap, or
+after 1,000 steps; when the first finding names none of those inputs,
+nothing moves. Of 283 deviation cases surviving the solver's consistency
+filters (22 of 305 are excluded), 246 reproduce the issued amount within
+the file's $5 editing tolerance, which is consistent with correct
+arithmetic applied to wrong facts. In 230 of the 246 the solver moved an
+input; in the other 16 the issued benefit was already within $5 of
+`FSBEN`, so nothing moved and the match restates the parity result. The
+rate differs by threshold status: among the 97 above-threshold official
+error cases 76 reproduce (78.4%, binomial standard error about 4
+points), and among the 186 sub-threshold deviations 170 do (91.4%). The
+solver's own recomputation and the engine agree on the within-$5
+classification for all 283 cases, two implementations with distinct
+codebases reading the same file.
+
+Neither outcome identifies a computation error. A reproduced case shows
+that one value of the first-listed input, run through correct
+arithmetic, yields the issued benefit; it does not show which input the
+agency had wrong, and a moved input can absorb a computation error. A
+non-reproduced case can reflect the solver's limits: it tries one input
+in one direction, and in 17 of the 37 it moved nothing (14 cases whose
+first finding names an input it does not adjust, and 3 where it stopped
+before moving). Of the 26 cases that carry a layer-2 computational
+finding, 13 reproduce, 10 do not and 3 were not replayed. The complete
 per-case replay output (283 rows), the layer-1 and layer-2
-classification outputs, and the reconstruction scripts are committed
-with this paper.
-
-The three layers agree on the structure: most measured SNAP payment
-error among eligible cases is information failure, not calculation
-failure — consistent with the administrative-burden literature's account
-of where compliance costs bind [@herd2018burden]. Correct-computation
-guarantees address the smaller share directly — roughly 3 to 10% of
-error dollars by layers 1–2, and bounded above by the replay's 21.6%
-case share among official errors — plus whatever share of information
+classification outputs, the reconstruction scripts and
+`audit_claims.py`, which regenerates the replay counts into
+`claims_audit.json`, are committed with this paper.
+
+Layers 1 and 2 agree on the structure, and the replay's reproduction
+rate fits it: most measured SNAP payment error among eligible cases is
+information failure, not calculation failure — consistent with the
+administrative-burden literature's account of where compliance costs
+bind [@herd2018burden]. Correct-computation guarantees address the
+smaller share directly — roughly 3 to 10% of error dollars by layers
+1–2 — plus whatever share of information
 failures better tooling
 prevents indirectly through verification prompts and
 documentation-requirement computation at intake; @sec-adoption prices
@@ -709,8 +737,9 @@ boundaries.
 ![Colorado's simulated measured-rate distribution (100,000 resampled
 QC-style measurements, official error definition, centered on the
 official 9.97% rate) against the cost-share tier bands. The state's
-official rate sits 0.03 points below the 15%-share boundary; 52% of
-simulated measurements fall below it and 48% above. Generated by the
+official fiscal 2024 rate sits 0.03 points below the 15%-share boundary,
+and its fiscal 2025 rate, 10.09%, sits above it; 52% of simulated
+measurements fall below the boundary and 48% above. Generated by the
 committed `paper/generate_figures.py` (seed 11).](figures/co_measured_rate.svg){#fig-co}
 
 The formula maps a point estimate to a tier with no
@@ -1020,7 +1049,7 @@ structural: the certified parity chain never consumes Colorado's
 standard-medical-deduction rule, because QC editing bakes the deduction
 into the stored expense fields and the federal computation path
 reproduces every case exactly without it — a sharper instance of the
-Minimodel-canon scope point of @sec-oracle-scope. (A separate
+edited-file scope point of @sec-oracle-scope. (A separate
 raw-facts leg, reconstructing inputs rather than supplying the file's
 intermediates, agrees on 845 of 856 cases, its eleven divergences
 diagnosed to missing public TANF facts and one utility-allowance tier
@@ -1071,10 +1100,11 @@ derivation from raw facts, and not the reviewer-adjudicated outcome
 fields (@sec-oracle-scope); it is also post-correction on the same case
 set, pending a fiscal 2025 confirmatory run. The public file excludes
 ineligible-case and full-overissuance error, truncating the error
-distribution every file-based estimate inherits. The replay
-decomposition depends on a reconstruction solver whose filters exclude
-22 of 305 error cases, and its residual class is an upper bound in case
-shares, not dollar shares. The error models predict an adjudicated,
+distribution every file-based estimate inherits. The replay depends on
+a reconstruction solver whose filters exclude 22 of 305 error cases and
+which moves one input per case; its shares are case shares, and neither
+its reproduced nor its non-reproduced class identifies computation
+errors. The error models predict an adjudicated,
 thresholded label — a property of the measurement system, with a
 documented manipulation history (@sec-qc). Cross-state model transport
 is imperfect: the model's raw level underpredicts high-error states,
@@ -1099,8 +1129,8 @@ burden policies on error.
 
 A measurement system designed for oversight now sets prices. That makes
 the QC files three things at once: a case-level verification oracle
-against the agency's own computational canon — where the bar can be
-exact, and holding it exact is what made the residuals informative; the
+against the benefit computation recorded in the file, where the bar can
+be exact and holding it exact is what made the residuals informative; the
 dataset showing that most measured error among eligible cases is
 information failure rather than calculation failure; and a sampling
 frame whose noise the new formula converts into near-even-odds tier
diff --git a/tests/test_retired_claims.py b/tests/test_retired_claims.py
new file mode 100644
index 0000000..495ebf9
--- /dev/null
+++ b/tests/test_retired_claims.py
@@ -0,0 +1,402 @@
+"""Locks for the October 2026 claim corrections in the working paper.
+
+The manuscript, README, fact catalog and simulator once described FSBEN as
+"the agency's own computational canon", read the error-case replay as
+explaining errors "as correct arithmetic on wrong facts" with a
+"computation-side upper bound", claimed technical-documentation errata no
+source records, and gave Colorado's 0.03-point FY2024 margin without its
+FY2025 rate. These tests tie every corrected figure to a committed artifact
+(``claims_audit.json``, ``cause_shares.json``, the per-case replay rows) and
+keep the retired wording out of the living files. The parity-target figures
+(797 of 856) also need the FY2024 QC file and skip without it.
+"""
+
+from __future__ import annotations
+
+import html
+import importlib.util
+import json
+import re
+import sys
+from pathlib import Path
+
+import pandas as pd
+import pytest
+from hypothesis import given
+from hypothesis import strategies as st
+
+ROOT = Path(__file__).resolve().parents[1]
+LAB = ROOT / "paper/snapshot/labs/amterr"
+MANUSCRIPT = ROOT / "paper/index.qmd"
+RENDERED = ROOT / "app/public/paper/web/index.html"
+README = ROOT / "README.md"
+FACTS = ROOT / "paper/FACTS.md"
+SIMULATOR = ROOT / "app/public/index.html"
+
+AUDIT = json.loads((LAB / "claims_audit.json").read_text(encoding="utf-8"))
+SLICES = json.loads((ROOT / "analysis/cause_shares.json").read_text(encoding="utf-8"))[
+    "colorado_replay_reconciliation"
+]["slices"]
+REPLAY_ROWS = json.loads(
+    (LAB / "amterr_replay_results.json").read_text(encoding="utf-8")
+)
+FY2024_THRESHOLD = 56  # official error threshold, FY2024 (FACTS A6)
+
+# Lowercased, whitespace-collapsed fragments of the retired claims.
+RETIRED = (
+    "agency's own computational canon",
+    "the agency's own canon",
+    "agency's own benefit-calculation software",
+    "agency's own computed outputs",
+    "a closed model",
+    "minimodel-canon",
+    "as correct arithmetic on wrong facts",
+    "explained as correct arithmetic",
+    "the agency's arithmetic was correct",
+    "disclosure-protection perturbation",
+    "computation-side upper bound",
+    "residual class is an upper bound",
+    "bounded above by the replay",
+    "errata in the federal technical documentation",
+    "errata in the technical documentation",
+    "explained cases have deviations",
+)
+
+
+def _normalize(text: str, *, is_html: bool = False) -> str:
+    """Collapse whitespace and typographic apostrophes; for HTML, drop tags."""
+    if is_html:
+        text = html.unescape(re.sub(r"<[^>]+>", " ", text))
+    return " ".join(text.replace("’", "'").split())
+
+
+def _read(path: Path) -> str:
+    return _normalize(path.read_text(encoding="utf-8"), is_html=path.suffix == ".html")
+
+
+def _percent(numerator: int, denominator: int) -> str:
+    return f"{100 * numerator / denominator:.1f}%"
+
+
+# ---------------------------------------------------------------------------
+# retired wording
+
+
+@pytest.mark.parametrize("path", [MANUSCRIPT, RENDERED, README, SIMULATOR])
+def test_retired_wording_is_gone(path):
+    text = _read(path).lower()
+    found = [phrase for phrase in RETIRED if phrase in text]
+    assert not found, f"{path.relative_to(ROOT)} still carries {found}"
+
+
+def test_facts_records_each_withdrawal():
+    """The catalog keeps the retired wording only as a superseded record."""
+    facts = _read(FACTS)
+    assert "| C4 | WITHDRAWN 2026-10-03" in facts
+    assert 'SUPERSEDES the revision-9 wording "explains 246/283' in facts
+    assert 'SUPERSEDES "33 of 246 explained cases' in facts
+
+
+# ---------------------------------------------------------------------------
+# replay figures: manuscript prose against claims_audit.json and the slices
+
+
+def _replay_counts() -> dict[str, int]:
+    replay = AUDIT["case_level"]["replay"]
+    co = AUDIT["by_posting"]["may2026"]["totals"]["CO"]
+    by_posting = AUDIT["by_posting"]["may2026"]["replay"]
+    layer2 = AUDIT["case_level"]["layer2_computational_findings"]["cases"]
+    above = SLICES["official_above_threshold_97"]
+    sub = SLICES["subthreshold_186"]
+    return {
+        "error_cases": co["error_cases"],
+        "n": replay["n"],
+        "reproduced": replay["reproduced_n"],
+        "not_reproduced": replay["not_reproduced_n"],
+        "moved": replay["reproduced_solver_moved_input_n"],
+        "unmoved": replay["reproduced_without_move_n"],
+        "miss_no_change": replay["not_reproduced_solver_no_change_n"],
+        "miss_stopped": replay["not_reproduced_eligible_but_unmoved_n"],
+        "agree": replay["solver_and_engine_agree_n"],
+        "layer2": layer2,
+        "layer2_reproduced": by_posting["reproduced_with_computational_finding"]["n"],
+        "layer2_not_reproduced": by_posting[
+            "not_reproduced_with_computational_finding"
+        ]["n"],
+        "above_n": above["total"]["n"],
+        "above_reproduced": above["outcomes"]["reproduced"]["n"],
+        "sub_n": sub["total"]["n"],
+        "sub_reproduced": sub["outcomes"]["reproduced"]["n"],
+    }
+
+
+def replay_quotes(c: dict[str, int]) -> list[str]:
+    """Whole sentences the manuscript must carry, built from the counts."""
+    above = _percent(c["above_reproduced"], c["above_n"])
+    sub = _percent(c["sub_reproduced"], c["sub_n"])
+    blended = _percent(c["reproduced"], c["n"])
+    unmoved_misses = c["miss_no_change"] + c["miss_stopped"]
+    not_replayed = c["layer2"] - c["layer2_reproduced"] - c["layer2_not_reproduced"]
+    return [
+        # abstract
+        f"For {c['n']} of Colorado's {c['error_cases']} error cases, a public solver",
+        (
+            f"reproduces the issued benefit within $5 for {above} of the "
+            f"above-threshold official error cases and {sub} of the sub-threshold "
+            "deviations, which is consistent with correct arithmetic on a wrong input."
+        ),
+        # decomposition table, layer 3
+        (
+            f"Not identified. {above} of above-threshold official errors and {sub} "
+            f"of sub-threshold deviations ({blended} blended) reproduce the issued "
+            "benefit within $5, consistent with a wrong input"
+        ),
+        # replay paragraph
+        (
+            f"Of {c['n']} deviation cases surviving the solver's consistency filters "
+            f"({c['error_cases'] - c['n']} of {c['error_cases']} are excluded), "
+            f"{c['reproduced']} reproduce the issued amount within the file's $5 "
+            "editing tolerance, which is consistent with correct arithmetic applied "
+            "to wrong facts."
+        ),
+        (
+            f"In {c['moved']} of the {c['reproduced']} the solver moved an input; in "
+            f"the other {c['unmoved']} the issued benefit was already within $5 of "
+            "`FSBEN`, so nothing moved and the match restates the parity result."
+        ),
+        (
+            f"among the {c['above_n']} above-threshold official error cases "
+            f"{c['above_reproduced']} reproduce ({above}, binomial standard error"
+        ),
+        (
+            f"among the {c['sub_n']} sub-threshold deviations {c['sub_reproduced']} "
+            f"do ({sub})."
+        ),
+        (f"agree on the within-$5 classification for all {c['agree']} cases"),
+        (
+            f"in {unmoved_misses} of the {c['not_reproduced']} it moved nothing "
+            f"({c['miss_no_change']} cases whose first finding names an input it "
+            f"does not adjust, and {c['miss_stopped']} where it stopped before moving)"
+        ),
+        (
+            f"Of the {c['layer2']} cases that carry a layer-2 computational finding, "
+            f"{c['layer2_reproduced']} reproduce, {c['layer2_not_reproduced']} do not "
+            f"and {not_replayed} were not replayed."
+        ),
+    ]
+
+
+def test_manuscript_quotes_the_replay_audit():
+    text = _read(MANUSCRIPT)
+    missing = [quote for quote in replay_quotes(_replay_counts()) if quote not in text]
+    assert not missing, missing
+
+
+def test_replay_counts_partition():
+    """The identities the replay paragraph's arithmetic relies on."""
+    c = _replay_counts()
+    assert c["reproduced"] + c["not_reproduced"] == c["n"] == c["agree"]
+    assert c["moved"] + c["unmoved"] == c["reproduced"]
+    assert c["above_n"] + c["sub_n"] == c["n"]
+    assert c["above_reproduced"] + c["sub_reproduced"] == c["reproduced"]
+    assert c["layer2_reproduced"] + c["layer2_not_reproduced"] <= c["layer2"]
+    assert c["n"] <= c["error_cases"]
+    # "nothing moved and the match restates the parity result" needs every
+    # unmoved match to sit within $5 of FSBEN already.
+    replay = AUDIT["case_level"]["replay"]
+    assert replay["reproduced_without_move_rawben_within_5_of_fsben_n"] == c["unmoved"]
+
+
+def test_threshold_split_differential():
+    """cause_shares.json's slices against the per-case replay rows."""
+    above = [r for r in REPLAY_ROWS if r["amterr"] > FY2024_THRESHOLD]
+    below = [r for r in REPLAY_ROWS if r["amterr"] <= FY2024_THRESHOLD]
+    c = _replay_counts()
+    assert (len(above), sum(r["within5"] for r in above)) == (
+        c["above_n"],
+        c["above_reproduced"],
+    )
+    assert (len(below), sum(r["within5"] for r in below)) == (
+        c["sub_n"],
+        c["sub_reproduced"],
+    )
+
+
+def test_unmoved_matches_restate_parity():
+    """The 16 unmoved matches replay the file's inputs, so the engine gives FSBEN."""
+    unchanged = set(AUDIT["case_level"]["replay_inputs_unchanged_keys"])
+    rows = [
+        r
+        for r in REPLAY_ROWS
+        if f"{r['yrmonth']}-{r['hhldno']}" in unchanged and r["within5"]
+    ]
+    assert len(rows) == _replay_counts()["unmoved"]
+    for r in rows:
+        assert r["engine_on_original"] == r["fsben"]
+        assert abs(r["rawben"] - r["fsben"]) <= 5
+
+
+def test_superseded_33_reconciles_with_the_16():
+    """FACTS H8: 33 reproduced cases have AMTERR <= $5; 16 of them moved nothing."""
+    unchanged = set(AUDIT["case_level"]["replay_inputs_unchanged_keys"])
+    small = [r for r in REPLAY_ROWS if r["within5"] and r["amterr"] <= 5]
+    unmoved = [r for r in small if f"{r['yrmonth']}-{r['hhldno']}" in unchanged]
+    assert (len(small), len(unmoved)) == (33, 16)
+    assert len(unmoved) == _replay_counts()["unmoved"]
+    facts = _read(FACTS)
+    assert f"{len(small)} of the 246 have AMTERR ≤ $5" in facts
+    assert f"the solver moved an input in {len(small) - len(unmoved)} of them" in facts
+
+
+# ---------------------------------------------------------------------------
+# Colorado's margin always carries the FY2025 rate
+
+
+def test_colorado_margin_carries_both_years():
+    rates = AUDIT["inputs"]["official_rates_percent"]
+    fy24, fy25 = rates["fy2024"]["CO"], rates["fy2025"]["CO"]
+    margin = round(10 - fy24, 2)
+    # The sentences say "below" for FY2024 and "above" for FY2025.
+    assert fy24 < 10 <= fy25
+    assert (
+        f"official fiscal 2024 rate sits {margin:.2f} points below the 15%-share "
+        f"boundary, and its fiscal 2025 rate, {fy25:.2f}%, sits above it"
+    ) in _read(MANUSCRIPT)
+    assert (
+        f"official FY 2024 rate {fy24:.2f}%, {margin:.2f} points below the 15% "
+        f"boundary, with a ±0.9-point sampling SD; its FY 2025 rate, {fy25:.2f}%, "
+        "is above the boundary"
+    ) in _read(README)
+    assert (
+        f"official FY2024 {fy24:.2f}% ({margin:.2f}pp below the 15% boundary; the "
+        f"FY2025 rate, {fy25:.2f}%, is above it, J3)"
+    ) in _read(FACTS)
+
+
+def test_every_margin_mention_names_fy2025():
+    for path in (MANUSCRIPT, README):
+        text = _read(path)
+        for match in re.finditer(r"0\.03 points", text):
+            window = text[match.start() : match.start() + 200]
+            assert "10.09%" in window, (path.name, window)
+
+
+# ---------------------------------------------------------------------------
+# the parity target: FSBEN's description and the 797-of-856 figure
+
+
+def test_simulator_describes_fsben_as_mathematicas_computation():
+    text = _read(SIMULATOR)
+    assert (
+        "Mathematica computes the QC file's formula benefit (FSBEN) for USDA from "
+        "each edited case record with the benefit formula of the QC Minimodel"
+    ) in text
+
+
+def _load_audit_module():
+    spec = importlib.util.spec_from_file_location(
+        "amterr_audit_claims_for_retired_claims", LAB / "audit_claims.py"
+    )
+    module = importlib.util.module_from_spec(spec)
+    sys.modules[spec.name] = module
+    spec.loader.exec_module(module)
+    return module
+
+
+audit = _load_audit_module()
+
+
+def _available_postings() -> list[str]:
+    out = []
+    for label, spec in audit.POSTINGS.items():
+        path = audit.posting_path(label)
+        if path.exists() and audit.sha256(path) == spec["csv_sha256"]:
+            out.append(label)
+    return out
+
+
+@pytest.mark.skipif(not _available_postings(), reason="FY2024 QC posting absent")
+@pytest.mark.parametrize("posting", _available_postings() or ["none"])
+def test_fsben_within_five_of_benfix_in_colorado(posting):
+    frame = pd.read_csv(
+        audit.posting_path(posting),
+        usecols=["STATE", "FSBEN", "BENFIX"],
+        low_memory=False,
+    )
+    co = frame[frame["STATE"] == 8]
+    within = int(((co["FSBEN"] - co["BENFIX"]).abs() <= 5).sum())
+    exact = int((co["FSBEN"] == co["BENFIX"]).sum())
+    assert (within, len(co), exact) == (797, 856, 743)
+    assert (
+        f"`FSBEN` ends within $5 of `BENFIX` in {within} of {len(co)} cases"
+    ) in _read(MANUSCRIPT)
+    facts = _read(FACTS)
+    assert (
+        f"within $5 of `BENFIX` in {within} of {len(co)} cases ({exact} exactly"
+        in facts
+    )
+
+
+# ---------------------------------------------------------------------------
+# property tests on the lock itself
+
+
+@st.composite
+def replay_counts(draw):
+    above_n = draw(st.integers(1, 400))
+    sub_n = draw(st.integers(1, 400))
+    above_reproduced = draw(st.integers(0, above_n))
+    sub_reproduced = draw(st.integers(0, sub_n))
+    n = above_n + sub_n
+    reproduced = above_reproduced + sub_reproduced
+    not_reproduced = n - reproduced
+    unmoved = draw(st.integers(0, reproduced))
+    miss_no_change = draw(st.integers(0, not_reproduced))
+    miss_stopped = draw(st.integers(0, not_reproduced - miss_no_change))
+    layer2_reproduced = draw(st.integers(0, 50))
+    layer2_not_reproduced = draw(st.integers(0, 50))
+    layer2 = layer2_reproduced + layer2_not_reproduced + draw(st.integers(0, 20))
+    return {
+        "error_cases": n + draw(st.integers(0, 100)),
+        "n": n,
+        "reproduced": reproduced,
+        "not_reproduced": not_reproduced,
+        "moved": reproduced - unmoved,
+        "unmoved": unmoved,
+        "miss_no_change": miss_no_change,
+        "miss_stopped": miss_stopped,
+        "agree": n,
+        "layer2": layer2,
+        "layer2_reproduced": layer2_reproduced,
+        "layer2_not_reproduced": layer2_not_reproduced,
+        "above_n": above_n,
+        "above_reproduced": above_reproduced,
+        "sub_n": sub_n,
+        "sub_reproduced": sub_reproduced,
+    }
+
+
+@given(replay_counts(), st.data())
+def test_lock_detects_any_changed_count(counts, data):
+    """Every count is load-bearing: changing one changes the required prose."""
+    key = data.draw(st.sampled_from(sorted(counts)))
+    changed = dict(counts, **{key: counts[key] + 1})
+    assert replay_quotes(changed) != replay_quotes(counts)
+
+
+@given(replay_counts())
+def test_lock_is_deterministic_and_counts_render_verbatim(counts):
+    quotes = replay_quotes(counts)
+    assert quotes == replay_quotes(counts)
+    joined = " ".join(quotes)
+    for key in ("n", "reproduced", "moved", "unmoved", "above_n", "sub_n"):
+        assert re.search(rf"(?<![\d.]){counts[key]}(?![\d.])", joined), key
+
+
+@given(st.integers(1, 10_000), st.data())
+def test_complementary_shares_sum_to_100(denominator, data):
+    numerator = data.draw(st.integers(0, denominator))
+    total = float(_percent(numerator, denominator)[:-1]) + float(
+        _percent(denominator - numerator, denominator)[:-1]
+    )
+    assert abs(total - 100.0) <= 0.1 + 1e-9

From 8de4b13516a599f233b5f488b932fc0a385e2660 Mon Sep 17 00:00:00 2001
From: Max Ghenis <mghenis@gmail.com>
Date: Sat, 3 Oct 2026 22:45:19 -0400
Subject: [PATCH 2/9] Address review round 1 on the retired-claim corrections

- Disclosure: parity replays "against the benefit computation recorded in
  the QC file", replacing "the government's own recorded values" (a missed
  paraphrase of the retired claim).
- Intro: the issuance residue is "a residue in issued amounts", no longer
  a defect of state systems; "two nested estimates" drops "nested".
- The replay no longer supports the information-failure conclusion;
  layers 1 and 2 carry it.
- Replay paragraph: "a wrong input" (the solver moves one input), and the
  utility-amount snap after the $3 steps is described.
- Layer-3 table cell: of the 37 misses, 17 moved nothing and 10 carry a
  computational finding (8 are both), from claims_audit.json.
- "305 cases with a recorded payment deviation", matching section 5;
  limitations say "deviation cases"; "Minimodel-computed chain" becomes
  "computed benefit chain".
- README and FACTS F2 name the 10% rate where the 15% share begins.
- Wrapper says six rounds of adversarial review, as the paper does.
- Tests: retire "agency's own" and "government's own", check the PDF
  (pdftotext), cover the "0.03pp" form in FACTS, and lock the 8-case
  overlap. Re-rendered app/public/paper/web.

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
---
 README.md                       |   4 +--
 app/public/paper/index.html     |   2 +-
 app/public/paper/web/index.html |  20 +++++------
 app/public/paper/web/index.pdf  | Bin 211731 -> 212053 bytes
 paper/FACTS.md                  |   4 +--
 paper/index.qmd                 |  32 +++++++++--------
 tests/test_retired_claims.py    |  59 ++++++++++++++++++++++++++------
 7 files changed, 81 insertions(+), 40 deletions(-)

diff --git a/README.md b/README.md
index 4f9b407..caa12bc 100644
--- a/README.md
+++ b/README.md
@@ -29,8 +29,8 @@ noise alone materially affects tier assignment.
 
 - **Tier assignment is noisy.** Several states' tiers are near coin flips
   under QC sampling variation (Colorado: official FY 2024 rate 9.97%, 0.03
-  points below the 15% boundary, with a ±0.9-point sampling SD; its FY 2025
-  rate, 10.09%, is above the boundary).
+  points below the 10% rate where the 15% cost share begins, with a
+  ±0.9-point sampling SD; its FY 2025 rate, 10.09%, is above it).
 - **Simplification options carry large expected values** where they can move
   a state across a boundary: about $609M/yr in combined expected state
   cost-share reduction nationally at 50% category-suppression effectiveness
diff --git a/app/public/paper/index.html b/app/public/paper/index.html
index 12526af..506aae1 100644
--- a/app/public/paper/index.html
+++ b/app/public/paper/index.html
@@ -81,7 +81,7 @@ <h1>Administrative quality control data as an oracle</h1>
       error-case replay, adds Colorado's fiscal 2025 rate beside its fiscal
       2024 margin, and withdraws a claim of errata in the federal technical
       documentation that no source records; the fact catalog lists each
-      change. The manuscript carries five rounds of adversarial review,
+      change. The manuscript carries six rounds of adversarial review,
       archived unedited, and adds the engine-adoption accounting bounds, the
       official-vs-file wedge decomposition, and the pre-registered
       model-repair experiments.
diff --git a/app/public/paper/web/index.html b/app/public/paper/web/index.html
index f9c322e..13868cd 100644
--- a/app/public/paper/web/index.html
+++ b/app/public/paper/web/index.html
@@ -106,8 +106,8 @@
 intermediates, exactly — all 6,081 in-scope cases of the 6,194-case
 official universe (113 documented exclusions), six computation stages
 per case, zero dollar tolerance — a process that surfaced and fixed two
-defects in the encodings. For 283 of Colorado's 305 error cases, a
-public solver moves the input named by each case's first finding, where
+defects in the encodings. For 283 of the 305 Colorado cases with a
+recorded payment deviation, a public solver moves the input named by each case's first finding, where
 it can, until its recomputed benefit reaches the issued amount; run on
 those inputs, the verified engine reproduces the issued benefit within
 $5 for 78.4% of the above-threshold official error cases and 91.4% of
@@ -218,7 +218,7 @@ <h1 class="title">Administrative quality control data as an oracle: verifying en
     <div>
       <div class="abstract">
         <div class="block-title">Abstract</div>
-        <p>Beginning in fiscal year 2028, each state’s Supplemental Nutrition Assistance Program (SNAP) payment error rate sets its share of benefit costs, in steps of five percentage points at rates of 6, 8, and 10 percent. A measurement system built for oversight now moves large sums, which raises two questions this paper treats together: whether the rules that produce benefit amounts can be executed correctly, and how the errors that remain should be measured and simulated. I use the USDA Quality Control (QC) public-use files as a verification oracle for independently encoded SNAP rules: across seven states, the encodings reproduce the file’s benefit-computation chain, from file-recorded intermediates, exactly — all 6,081 in-scope cases of the 6,194-case official universe (113 documented exclusions), six computation stages per case, zero dollar tolerance — a process that surfaced and fixed two defects in the encodings. For 283 of Colorado’s 305 error cases, a public solver moves the input named by each case’s first finding, where it can, until its recomputed benefit reaches the issued amount; run on those inputs, the verified engine reproduces the issued benefit within $5 for 78.4% of the above-threshold official error cases and 91.4% of the sub-threshold deviations, which is consistent with correct arithmetic on a wrong input. Pricing a cause-code decomposition through the formula’s tiers, elections, and delay clause — removing the error dollars the file’s cause codes assign to the computing apparatus, an accounting scenario rather than a causal estimate — moves summed expected fiscal 2028 bills from $7.7B to $7.6B a year under a strict class convention and $6.9B under a broad one, and not monotonically: a state just under the statute’s delay threshold sees its near-term expected bill rise as its measured rate falls. I also fit and validate a distributional model of case-level payment deviations, reporting its validation failures alongside its results, and serve one policy counterfactual from it in the deployed simulator: a standard medical deduction scenario computed on engine-recomputed case parameters, whose model-implied Colorado effect opposes the naive accounting bound in sign. The central finding, from an open Monte Carlo simulator resampling each state’s own QC cases, is about measurement: at regulatory sample sizes, several states face nearly even odds — 46 to 50 percent — of being assigned a different cost tier than their official point rate implies. The fiscal 2025 rates, published in June 2026, land where that mechanism says realized years should: 18 of 53 jurisdictions changed cost-sharing tiers in one year, and only seven of those changes exceed the 95% band that two years of sampling noise — estimated from fiscal 2024 microdata alone — implies.</p>
+        <p>Beginning in fiscal year 2028, each state’s Supplemental Nutrition Assistance Program (SNAP) payment error rate sets its share of benefit costs, in steps of five percentage points at rates of 6, 8, and 10 percent. A measurement system built for oversight now moves large sums, which raises two questions this paper treats together: whether the rules that produce benefit amounts can be executed correctly, and how the errors that remain should be measured and simulated. I use the USDA Quality Control (QC) public-use files as a verification oracle for independently encoded SNAP rules: across seven states, the encodings reproduce the file’s benefit-computation chain, from file-recorded intermediates, exactly — all 6,081 in-scope cases of the 6,194-case official universe (113 documented exclusions), six computation stages per case, zero dollar tolerance — a process that surfaced and fixed two defects in the encodings. For 283 of the 305 Colorado cases with a recorded payment deviation, a public solver moves the input named by each case’s first finding, where it can, until its recomputed benefit reaches the issued amount; run on those inputs, the verified engine reproduces the issued benefit within $5 for 78.4% of the above-threshold official error cases and 91.4% of the sub-threshold deviations, which is consistent with correct arithmetic on a wrong input. Pricing a cause-code decomposition through the formula’s tiers, elections, and delay clause — removing the error dollars the file’s cause codes assign to the computing apparatus, an accounting scenario rather than a causal estimate — moves summed expected fiscal 2028 bills from $7.7B to $7.6B a year under a strict class convention and $6.9B under a broad one, and not monotonically: a state just under the statute’s delay threshold sees its near-term expected bill rise as its measured rate falls. I also fit and validate a distributional model of case-level payment deviations, reporting its validation failures alongside its results, and serve one policy counterfactual from it in the deployed simulator: a standard medical deduction scenario computed on engine-recomputed case parameters, whose model-implied Colorado effect opposes the naive accounting bound in sign. The central finding, from an open Monte Carlo simulator resampling each state’s own QC cases, is about measurement: at regulatory sample sizes, several states face nearly even odds — 46 to 50 percent — of being assigned a different cost tier than their official point rate implies. The fiscal 2025 rates, published in June 2026, land where that mechanism says realized years should: 18 of 53 jurisdictions changed cost-sharing tiers in one year, and only seven of those changes exceed the 95% band that two years of sampling noise — estimated from fiscal 2024 microdata alone — implies.</p>
       </div>
     </div>
 
@@ -270,8 +270,8 @@ <h2 class="anchored" data-anchor-id="introduction">Introduction</h2>
 <p>Under 7 U.S.C. § 2013(a)(2), as amended by the One Big Beautiful Bill Act (Pub. L. No.&nbsp;119-21, 2025; OBBBA), a state whose SNAP payment error rate reaches 6% will pay 5% of benefit costs beginning in fiscal year 2028; at 8%, the share is 10%; at 10%, it is 15%. The rate that determines the FY2028 share is the state’s fiscal 2025 or 2026 rate, at its election; later years use the third preceding year. Implementation is delayed for the highest-error states: a state whose fiscal 2025 rate times 1.5 reaches 20% — that is, a rate above 13.33% (13.34% at the published two-decimal precision) — starts in FY2029 instead, and a state whose fiscal 2026 rate crosses the same test starts in FY2030.<a href="#fn1" class="footnote-ref" id="fnref1" role="doc-noteref"><sup>1</sup></a> For a mid-sized state, one tier is worth tens of millions of dollars a year; for the largest states, more than a hundred million.</p>
 <p>The error rate is estimated by the Quality Control system: each state re-reviews a probability sample of its own active cases — by formula in 7 C.F.R. § 275.11, between 300 and 2,400 annually, or 300 to 1,020 under an optional reduced schedule that every state elected in fiscal 2024 <span class="citation" data-cites="fns2024techdoc">(<a href="#ref-fns2024techdoc" role="doc-biblioref">U.S. Department of Agriculture, Food and Nutrition Service 2025b</a>)</span> — and federal reviewers re-review a subsample, from which the Food and Nutrition Service (FNS; renamed the Food and Nutrition Administration in 2026 — the fiscal 2025 rate publication carries the new letterhead <span class="citation" data-cites="fna2025per">(<a href="#ref-fna2025per" role="doc-biblioref">U.S. Department of Agriculture, Food and Nutrition Administration 2026</a>)</span>, and this paper keeps FNS, the name on every document the fiscal 2024 data cite) estimates official state rates with adjustments including federal re-review integration. The public-use files that result record, for roughly 45,000 cases a year, what the agency issued and what the review concluded the case should have received, with coded findings for every discrepancy.</p>
 <p>This paper treats those files as a research instrument with three uses, the first in the software-testing sense of an <em>oracle</em>: an external source of ground-truth outputs for given inputs.</p>
-<p>First, as a verification target for rules as code. I test independently encoded SNAP rules — statute, regulations, and state policy manuals encoded as executable rule modules — against the QC file’s recorded benefit computations in seven states, holding the bar at exact reproduction: zero dollar tolerance across six computation stages per case (<a href="#sec-oracle" class="quarto-xref">Section&nbsp;4</a>). The target’s precise character matters and is stated in <a href="#sec-oracle-scope" class="quarto-xref">Section&nbsp;4.1</a>: the file’s benefit chain is a computation Mathematica runs for USDA on edited case records with the benefit formula of the FNS QC Minimodel, so parity certifies agreement with the benefit computation recorded in the file; it does not adjudicate case outcomes. Reaching it surfaced defects on two sides: two in the encodings under test, and a residue attributable to state issuance systems.</p>
-<p>Second, as raw material for decomposing measured error. QC’s own cause coding and a finding-nature classification give two nested estimates of how much of the error rate reflects computation failure versus wrong facts correctly processed, and a replay of reconstructed case inputs through the verified engine tests whether one wrong input reproduces each issued benefit (<a href="#sec-decompose" class="quarto-xref">Section&nbsp;5</a>).</p>
+<p>First, as a verification target for rules as code. I test independently encoded SNAP rules — statute, regulations, and state policy manuals encoded as executable rule modules — against the QC file’s recorded benefit computations in seven states, holding the bar at exact reproduction: zero dollar tolerance across six computation stages per case (<a href="#sec-oracle" class="quarto-xref">Section&nbsp;4</a>). The target’s precise character matters and is stated in <a href="#sec-oracle-scope" class="quarto-xref">Section&nbsp;4.1</a>: the file’s benefit chain is a computation Mathematica runs for USDA on edited case records with the benefit formula of the FNS QC Minimodel, so parity certifies agreement with the benefit computation recorded in the file; it does not adjudicate case outcomes. Reaching it surfaced two defects in the encodings under test and left a residue in issued amounts, which <a href="#sec-decompose" class="quarto-xref">Section&nbsp;5</a> decomposes.</p>
+<p>Second, as raw material for decomposing measured error. QC’s own cause coding and a finding-nature classification give two estimates of how much of the error rate reflects computation failure versus wrong facts correctly processed, and a replay of reconstructed case inputs through the verified engine tests whether one wrong input reproduces each issued benefit (<a href="#sec-decompose" class="quarto-xref">Section&nbsp;5</a>).</p>
 <p>Third, as training and validation data for a case-level model of the error process (<a href="#sec-model" class="quarto-xref">Section&nbsp;6</a>) and the basis for a distributional simulator of measured rates under the cost-sharing formula (<a href="#sec-simulate" class="quarto-xref">Section&nbsp;7</a>), public at policyengine.org/us/snap-payment-error-simulator. The simulator’s central output is a measurement result: near tier boundaries, sampling noise alone gives several states nearly even odds of a different cost tier than their point rate implies.</p>
 <p>The analysis pipeline, simulator, and this manuscript live in one public repository; a fact catalog maps every quantitative claim to a committed artifact (<a href="#sec-availability" class="quarto-xref">Section&nbsp;11</a>). The pipeline is deterministic — independent clean-room reruns reproduce every analysis artifact byte-for-byte, and the headline simulation artifact was independently reproduced from the raw file to four decimals — and the manuscript has been through six rounds of adversarial review, archived unedited in the repository.</p>
 </section>
@@ -475,7 +475,7 @@ <h2 class="anchored" data-anchor-id="sec-decompose">Decomposing measured error:
 <tr class="odd">
 <td>3. Engine replay</td>
 <td>A public solver moves the input named by each case’s first finding until its recomputed benefit reaches the issued amount; the verified engine replays the solver’s inputs</td>
-<td>Not identified. 78.4% of above-threshold official errors and 91.4% of sub-threshold deviations (86.9% blended) reproduce the issued benefit within $5, consistent with a wrong input; the non-reproduced cases mix solver limits, multi-element errors and computation findings</td>
+<td>Not identified. 78.4% of above-threshold official errors and 91.4% of sub-threshold deviations (86.9% blended) reproduce the issued benefit within $5, consistent with a wrong input; of the 37 that do not, the solver moved nothing in 17 and 10 carry a computational finding (8 are both)</td>
 <td>283 filtered Colorado deviation cases (97 above threshold, 186 below); case shares, not dollar shares</td>
 </tr>
 </tbody>
@@ -483,9 +483,9 @@ <h2 class="anchored" data-anchor-id="sec-decompose">Decomposing measured error:
 </div>
 </figure>
 </div>
-<p>The third layer adapts the public reconstruction solver of <span class="citation" data-cites="giannella_snapqc">Giannella and Molin (<a href="#ref-giannella_snapqc" role="doc-biblioref">2026</a>)</span> to fiscal 2024 and replays its output through the verified engine. The solver starts from the file’s edited inputs and moves only the input named by the case’s first finding (<code>ELEMENT1</code>): earned or unearned income, rent, the utility allowance, or the medical, dependent care or child-support deduction in $3 steps, or household size by one person. It stops when its own recomputed benefit reaches or passes the issued benefit (<code>RAWBEN</code>), when the input hits zero or a cap, or after 1,000 steps; when the first finding names none of those inputs, nothing moves. Of 283 deviation cases surviving the solver’s consistency filters (22 of 305 are excluded), 246 reproduce the issued amount within the file’s $5 editing tolerance, which is consistent with correct arithmetic applied to wrong facts. In 230 of the 246 the solver moved an input; in the other 16 the issued benefit was already within $5 of <code>FSBEN</code>, so nothing moved and the match restates the parity result. The rate differs by threshold status: among the 97 above-threshold official error cases 76 reproduce (78.4%, binomial standard error about 4 points), and among the 186 sub-threshold deviations 170 do (91.4%). The solver’s own recomputation and the engine agree on the within-$5 classification for all 283 cases, two implementations with distinct codebases reading the same file.</p>
+<p>The third layer adapts the public reconstruction solver of <span class="citation" data-cites="giannella_snapqc">Giannella and Molin (<a href="#ref-giannella_snapqc" role="doc-biblioref">2026</a>)</span> to fiscal 2024 and replays its output through the verified engine. The solver starts from the file’s edited inputs and moves only the input named by the case’s first finding (<code>ELEMENT1</code>): earned or unearned income, rent, the utility allowance, or the medical, dependent care or child-support deduction in $3 steps, or household size by one person. It stops when its own recomputed benefit reaches or passes the issued benefit (<code>RAWBEN</code>), when the input hits zero or a cap, or after 1,000 steps; a moved utility amount is then reset to the nearest value above or below the file’s that more than five filtered cases in the same state and calendar year use. When the first finding names none of those inputs, nothing moves. Of 283 deviation cases surviving the solver’s consistency filters (22 of 305 are excluded), 246 reproduce the issued amount within the file’s $5 editing tolerance, which is consistent with correct arithmetic applied to a wrong input. In 230 of the 246 the solver moved an input; in the other 16 the issued benefit was already within $5 of <code>FSBEN</code>, so nothing moved and the match restates the parity result. The rate differs by threshold status: among the 97 above-threshold official error cases 76 reproduce (78.4%, binomial standard error about 4 points), and among the 186 sub-threshold deviations 170 do (91.4%). The solver’s own recomputation and the engine agree on the within-$5 classification for all 283 cases, two implementations with distinct codebases reading the same file.</p>
 <p>Neither outcome identifies a computation error. A reproduced case shows that one value of the first-listed input, run through correct arithmetic, yields the issued benefit; it does not show which input the agency had wrong, and a moved input can absorb a computation error. A non-reproduced case can reflect the solver’s limits: it tries one input in one direction, and in 17 of the 37 it moved nothing (14 cases whose first finding names an input it does not adjust, and 3 where it stopped before moving). Of the 26 cases that carry a layer-2 computational finding, 13 reproduce, 10 do not and 3 were not replayed. The complete per-case replay output (283 rows), the layer-1 and layer-2 classification outputs, the reconstruction scripts and <code>audit_claims.py</code>, which regenerates the replay counts into <code>claims_audit.json</code>, are committed with this paper.</p>
-<p>Layers 1 and 2 agree on the structure, and the replay’s reproduction rate fits it: most measured SNAP payment error among eligible cases is information failure, not calculation failure — consistent with the administrative-burden literature’s account of where compliance costs bind <span class="citation" data-cites="herd2018burden">(<a href="#ref-herd2018burden" role="doc-biblioref">Herd and Moynihan 2018</a>)</span>. Correct-computation guarantees address the smaller share directly — roughly 3 to 10% of error dollars by layers 1–2 — plus whatever share of information failures better tooling prevents indirectly through verification prompts and documentation-requirement computation at intake; <a href="#sec-adoption" class="quarto-xref">Section&nbsp;7.2</a> prices the direct shares through the formula’s tiers, elections, and delay clause. Ineligibility errors, excluded from the file, are predominantly facts-driven, which would push the information-failure share higher still; that inference, unlike the in-file shares, is not directly computed here. The indirect channel is where policy design enters: rules determine what a caseworker must collect and verify, which is the mechanism the modeling section takes up.</p>
+<p>Layers 1 and 2 agree on the structure: most measured SNAP payment error among eligible cases is information failure, not calculation failure — consistent with the administrative-burden literature’s account of where compliance costs bind <span class="citation" data-cites="herd2018burden">(<a href="#ref-herd2018burden" role="doc-biblioref">Herd and Moynihan 2018</a>)</span>. Correct-computation guarantees address the smaller share directly — roughly 3 to 10% of error dollars by layers 1–2 — plus whatever share of information failures better tooling prevents indirectly through verification prompts and documentation-requirement computation at intake; <a href="#sec-adoption" class="quarto-xref">Section&nbsp;7.2</a> prices the direct shares through the formula’s tiers, elections, and delay clause. Ineligibility errors, excluded from the file, are predominantly facts-driven, which would push the information-failure share higher still; that inference, unlike the in-file shares, is not directly computed here. The indirect channel is where policy design enters: rules determine what a caseworker must collect and verify, which is the mechanism the modeling section takes up.</p>
 </section>
 <section id="sec-model" class="level2">
 <h2 class="anchored" data-anchor-id="sec-model">Modeling the error process</h2>
@@ -649,7 +649,7 @@ <h2 class="anchored" data-anchor-id="sec-roadmap">Toward engine-computed counter
 </section>
 <section id="sec-limitations" class="level2">
 <h2 class="anchored" data-anchor-id="sec-limitations">Limitations</h2>
-<p>The parity result certifies agreement with the edited file’s Minimodel-computed chain from supplied intermediates — not independent derivation from raw facts, and not the reviewer-adjudicated outcome fields (<a href="#sec-oracle-scope" class="quarto-xref">Section&nbsp;4.1</a>); it is also post-correction on the same case set, pending a fiscal 2025 confirmatory run. The public file excludes ineligible-case and full-overissuance error, truncating the error distribution every file-based estimate inherits. The replay depends on a reconstruction solver whose filters exclude 22 of 305 error cases and which moves one input per case; its shares are case shares, and neither its reproduced nor its non-reproduced class identifies computation errors. The error models predict an adjudicated, thresholded label — a property of the measurement system, with a documented manipulation history (<a href="#sec-qc" class="quarto-xref">Section&nbsp;3</a>). Cross-state model transport is imperfect: the model’s raw level underpredicts high-error states, all nine coverage gaps remain one-sided negative after the tail and level-factor fixes — and after the committed location repair, which improves every level but closes only part of the gap — and the deployed simulator accordingly serves the model only for its level-gated standard-medical-deduction scenario, with seven jurisdictions disabled outright; every simulation result in this paper uses the observed-resample engine. That mode resamples i.i.d. from a probability sample whose actual design varies by state, treats the official rate’s federal re-review adjustment as a fixed level with no variance contribution, and holds the error process fixed — no behavioral response of agencies or households to cost sharing, no corrective-feedback channel from audit volume, both live questions the fiscal 2025–27 files will begin to answer under the new incentives. Adoption contrasts are descriptive contrasts of policy bundles; nothing here identifies causal effects of burden policies on error.</p>
+<p>The parity result certifies agreement with the edited file’s computed benefit chain from supplied intermediates — not independent derivation from raw facts, and not the reviewer-adjudicated outcome fields (<a href="#sec-oracle-scope" class="quarto-xref">Section&nbsp;4.1</a>); it is also post-correction on the same case set, pending a fiscal 2025 confirmatory run. The public file excludes ineligible-case and full-overissuance error, truncating the error distribution every file-based estimate inherits. The replay depends on a reconstruction solver whose filters exclude 22 of 305 deviation cases and which moves one input per case; its shares are case shares, and neither its reproduced nor its non-reproduced class identifies computation errors. The error models predict an adjudicated, thresholded label — a property of the measurement system, with a documented manipulation history (<a href="#sec-qc" class="quarto-xref">Section&nbsp;3</a>). Cross-state model transport is imperfect: the model’s raw level underpredicts high-error states, all nine coverage gaps remain one-sided negative after the tail and level-factor fixes — and after the committed location repair, which improves every level but closes only part of the gap — and the deployed simulator accordingly serves the model only for its level-gated standard-medical-deduction scenario, with seven jurisdictions disabled outright; every simulation result in this paper uses the observed-resample engine. That mode resamples i.i.d. from a probability sample whose actual design varies by state, treats the official rate’s federal re-review adjustment as a fixed level with no variance contribution, and holds the error process fixed — no behavioral response of agencies or households to cost sharing, no corrective-feedback channel from audit volume, both live questions the fiscal 2025–27 files will begin to answer under the new incentives. Adoption contrasts are descriptive contrasts of policy bundles; nothing here identifies causal effects of burden policies on error.</p>
 </section>
 <section id="conclusion" class="level2">
 <h2 class="anchored" data-anchor-id="conclusion">Conclusion</h2>
@@ -661,7 +661,7 @@ <h2 class="anchored" data-anchor-id="sec-availability">Data and code availabilit
 </section>
 <section id="disclosure" class="level2">
 <h2 class="anchored" data-anchor-id="disclosure">Disclosure</h2>
-<p>The author leads PolicyEngine, which builds the simulator described here, and The Axiom Foundation, which builds the rulespec-us encodings and the axiom-oracles verification harness whose parity this paper reports — the paper is an evaluation of systems its author is responsible for. The mitigations are structural: every parity claim replays against the government’s own recorded values at zero tolerance from committed, hash-pinned artifacts any reader can rerun, and the manuscript’s six rounds of adversarial review are archived unedited in the repository.</p>
+<p>The author leads PolicyEngine, which builds the simulator described here, and The Axiom Foundation, which builds the rulespec-us encodings and the axiom-oracles verification harness whose parity this paper reports — the paper is an evaluation of systems its author is responsible for. The mitigations are structural: every parity claim replays against the benefit computation recorded in the QC file at zero tolerance from committed, hash-pinned artifacts any reader can rerun, and the manuscript’s six rounds of adversarial review are archived unedited in the repository.</p>
 </section>
 <section id="acknowledgments" class="level2">
 <h2 class="anchored" data-anchor-id="acknowledgments">Acknowledgments</h2>
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diff --git a/paper/FACTS.md b/paper/FACTS.md
index 9298cd0..975bc8d 100644
--- a/paper/FACTS.md
+++ b/paper/FACTS.md
@@ -43,7 +43,7 @@ Numbers from the superseded pre-audit pipeline (before PR #7) are banned.
 | D1 | Colorado FY2024: 856 sampled cases, 305 with payment errors (STATUS 2/3); $112.6M/yr weighted error dollars on $1.268B issuance (8.88% file-derived; official regression-adjusted rate 9.97% = 7.91 over + 2.06 under) | paper/snapshot/labs/amterr/ANALYSIS.md; FNS FY2024 PER (https://fns-prod.azureedge.us/sites/default/files/resource-files/snap-fy24QC-PER.pdf) |
 | D2 | QC cause coding, SUPERSEDED by K1 (revision 5): the lab tabulation denominated over all weighted deviation dollars ($8.1B national incl. sub-threshold) and gave CO $8.2M/7.3% strict, $11.9M/10.5% broad; national $320M/3.9%, $1.5B/18.4%. The committed artifact denominates over official error dollars and differs (see K1) | paper/snapshot/labs/amterr/native_decomposition.json + ANALYSIS.md (archived) |
 | D3 | Finding-nature classification: pure-math 3.3%, input-with-system-cause 6.6%, mixed 4.2%, other input 86.0% of Colorado weighted deviation dollars. NOW COMMITTED as K3 (revision 5): the artifact reproduces these to the cent under the corrected field semantics (NATUREi carries nature codes; E_FINDGi is impact-only, a lab labeling error that did not affect the numbers) | analysis/cause_shares.json layer 2 (was: paper/snapshot/labs/phase_a_classification.json) |
-| D4 | Replay: the public solver (FY2024 adaptation of giannella/snap_qc) starts from the file's edited inputs and moves only the ELEMENT1 input ($3 steps; household size by one person for element 150) until its own recomputed benefit reaches or passes RAWBEN, the input hits zero or a cap, or 1,000 steps; it uses no disclosure-perturbation structure. The verified engine on the solver's inputs reproduces RAWBEN within $5 for 246 of 283 filtered error cases (86.9%), consistent with correct arithmetic on a wrong input: 230 after the solver moved an input, 16 with nothing moved (\|RAWBEN − FSBEN\| ≤ $5 already, so the match restates parity). 37 do not reproduce; in 17 nothing moved (14 `no_change` rows whose ELEMENT1 the solver does not adjust, 3 where it stopped before moving). Solver and engine agree on all 283. Neither class identifies computation errors: of 26 cases with a layer-2 computational finding, 13 reproduce, 10 do not, 3 were not replayed. SUPERSEDES the revision-9 wording "explains 246/283 … as correct arithmetic on wrong facts" and "37 cases form the computation-side upper bound" (withdrawn 2026-10-03, matching ANALYSIS.md's revision of the same date) | paper/snapshot/labs/amterr/claims_audit.json case_level.replay (regenerated by audit_claims.py); amterr_replay_results.json (283 rows); reconstruct_co_fy2024.R; amterr_replay.py; ANALYSIS.md layer 3 |
+| D4 | Replay: the public solver (FY2024 adaptation of giannella/snap_qc) starts from the file's edited inputs and moves only the ELEMENT1 input ($3 steps; household size by one person for element 150) until its own recomputed benefit reaches or passes RAWBEN, the input hits zero or a cap, or 1,000 steps; a moved utility amount is then reset to the nearest value above or below the file's UTIL that more than five filtered cases in the state and calendar year use. It uses no disclosure-perturbation structure. The verified engine on the solver's inputs reproduces RAWBEN within $5 for 246 of 283 filtered error cases (86.9%), consistent with correct arithmetic on a wrong input: 230 after the solver moved an input, 16 with nothing moved (\|RAWBEN − FSBEN\| ≤ $5 already, so the match restates parity). 37 do not reproduce; in 17 nothing moved (14 `no_change` rows whose ELEMENT1 the solver does not adjust, 3 where it stopped before moving), and 10 carry a layer-2 computational finding (8 of them among the 17). Solver and engine agree on all 283. Neither class identifies computation errors: of 26 cases with a layer-2 computational finding, 13 reproduce, 10 do not, 3 were not replayed. SUPERSEDES the revision-9 wording "explains 246/283 … as correct arithmetic on wrong facts" and "37 cases form the computation-side upper bound" (withdrawn 2026-10-03, matching ANALYSIS.md's revision of the same date) | paper/snapshot/labs/amterr/claims_audit.json case_level.replay (regenerated by audit_claims.py); amterr_replay_results.json (283 rows); reconstruct_co_fy2024.R; amterr_replay.py; ANALYSIS.md layer 3 |
 
 ## E. Error models (corrected pipeline: PRs #7–#8 as refit by #13)
 
@@ -64,7 +64,7 @@ Numbers from the superseded pre-audit pipeline (before PR #7) are banned.
 | # | Fact | Source |
 |---|---|---|
 | F1 | Tier assignment near boundaries is noisy at regulatory sample sizes: P(different tier than the official point rate implies) — ND 50%, WA 48%, CO 49%, KS 47%, NV 46% | paper/snapshot/labs/results_by_state_corrected.json (official error gate, CASE==1) |
-| F2 | Colorado: official FY2024 9.97% (0.03pp below the 15% boundary; the FY2025 rate, 10.09%, is above it, J3), $1.268B issuance, expected FY2028-rule cost $156.4M/yr, SD $32.8M; the 10%→15% tier step is $63.4M/yr | paper/snapshot/labs/results_by_state_corrected.json |
+| F2 | Colorado: official FY2024 9.97% (0.03pp below the 10% rate where the 15% share begins; the FY2025 rate, 10.09%, is above it, J3), $1.268B issuance, expected FY2028-rule cost $156.4M/yr, SD $32.8M; the 10%→15% tier step is $63.4M/yr | paper/snapshot/labs/results_by_state_corrected.json |
 | F3 | Audit volume is a two-sided lever: +500 reviews raises expected cost for just-above-boundary states (CA −$30.3M, PA −$13.1M, TX −$13.1M expectation) and lowers it for just-below states (MO +$3.9M, TN +$3.7M, IN +$3.3M); it nearly always cuts the SD of the bill (CA −$44M) | paper/snapshot/labs/results_by_state_corrected.json |
 | F4 | Category-suppression accounting bound: removing 50% (100%) of observed error dollars in the four named simplification categories corresponds to ≈$609M/yr (≈$1,310M/yr) nationally in expected cost share | paper/snapshot/labs/results_by_state_corrected.json |
 | F5 | These lever numbers are accounting bounds on the observed FY2024 error mix — scenario dials, not causal estimates; the error process is held fixed (no behavioral response, no audit-feedback channel) | mc lab DESIGN; app method notes |
diff --git a/paper/index.qmd b/paper/index.qmd
index 608783d..b0f2f8f 100644
--- a/paper/index.qmd
+++ b/paper/index.qmd
@@ -21,8 +21,8 @@ abstract: |
   intermediates, exactly — all 6,081 in-scope cases of the 6,194-case
   official universe (113 documented exclusions), six computation stages
   per case, zero dollar tolerance — a process that surfaced and fixed two
-  defects in the encodings. For 283 of Colorado's 305 error cases, a
-  public solver moves the input named by each case's first finding, where
+  defects in the encodings. For 283 of the 305 Colorado cases with a
+  recorded payment deviation, a public solver moves the input named by each case's first finding, where
   it can, until its recomputed benefit reaches the issued amount; run on
   those inputs, the verified engine reproduces the issued benefit within
   $5 for 78.4% of the above-threshold official error cases and 91.4% of
@@ -109,12 +109,12 @@ case (@sec-oracle). The target's precise character matters and is stated
 in @sec-oracle-scope: the file's benefit chain is a computation Mathematica
 runs for USDA on edited case records with the benefit formula of the FNS
 QC Minimodel, so parity certifies agreement with the benefit computation
-recorded in the file; it does not adjudicate case outcomes. Reaching it surfaced defects on two
-sides: two in the encodings under test, and a residue attributable to
-state issuance systems.
+recorded in the file; it does not adjudicate case outcomes. Reaching it surfaced two defects in
+the encodings under test and left a residue in issued amounts, which
+@sec-decompose decomposes.
 
 Second, as raw material for decomposing measured error. QC's own cause
-coding and a finding-nature classification give two nested estimates of
+coding and a finding-nature classification give two estimates of
 how much of the error rate reflects computation failure versus wrong
 facts correctly processed, and a replay of reconstructed case inputs
 through the verified engine tests whether one wrong input reproduces
@@ -435,7 +435,7 @@ construction of the file, that ineligible-case error.
 |---|---|---|---|
 | 1. Cause codes | QC agency-cause codes naming the computing apparatus (programming, arithmetic, mass change), any-presence across a case's nine cause slots, share of official error dollars (HWGT × `AMTERR`) | 7.4% strict ($6.9M/yr); 10.6% ($10.0M/yr) adding policy-misapplied, budgeted-wrong, computer-user | Colorado; nationally 3.5% strict ($232M/yr), 19.3% broad ($1.27B of $6.59B) |
 | 2. Finding natures | `NATURE` codes inherently describing computation (rounding, conversion, wrong standard, benefit miscomputed) plus the arithmetic `ELEMENT`; deduction-amount natures count only when the paired cause is in layer 1's broad computing-apparatus set | 3.3% pure-computation; 6.6% system-caused input; 4.2% mixed; 86.0% other input | Colorado weighted deviation dollars; nationally 4.6% / 11.1% / 6.5% / 77.7% |
-| 3. Engine replay | A public solver moves the input named by each case's first finding until its recomputed benefit reaches the issued amount; the verified engine replays the solver's inputs | Not identified. 78.4% of above-threshold official errors and 91.4% of sub-threshold deviations (86.9% blended) reproduce the issued benefit within $5, consistent with a wrong input; the non-reproduced cases mix solver limits, multi-element errors and computation findings | 283 filtered Colorado deviation cases (97 above threshold, 186 below); case shares, not dollar shares |
+| 3. Engine replay | A public solver moves the input named by each case's first finding until its recomputed benefit reaches the issued amount; the verified engine replays the solver's inputs | Not identified. 78.4% of above-threshold official errors and 91.4% of sub-threshold deviations (86.9% blended) reproduce the issued benefit within $5, consistent with a wrong input; of the 37 that do not, the solver moved nothing in 17 and 10 carry a computational finding (8 are both) | 283 filtered Colorado deviation cases (97 above threshold, 186 below); case shares, not dollar shares |
 
 : Three decompositions of Colorado fiscal 2024 measured error.
 Layers 1–2 quote the committed cause-share artifact
@@ -461,11 +461,13 @@ earned or unearned income, rent, the utility allowance, or the medical,
 dependent care or child-support deduction in $3 steps, or household size
 by one person. It stops when its own recomputed benefit reaches or passes
 the issued benefit (`RAWBEN`), when the input hits zero or a cap, or
-after 1,000 steps; when the first finding names none of those inputs,
-nothing moves. Of 283 deviation cases surviving the solver's consistency
+after 1,000 steps; a moved utility amount is then reset to the nearest
+value above or below the file's that more than five filtered cases in
+the same state and calendar year use. When the first finding names none
+of those inputs, nothing moves. Of 283 deviation cases surviving the solver's consistency
 filters (22 of 305 are excluded), 246 reproduce the issued amount within
 the file's $5 editing tolerance, which is consistent with correct
-arithmetic applied to wrong facts. In 230 of the 246 the solver moved an
+arithmetic applied to a wrong input. In 230 of the 246 the solver moved an
 input; in the other 16 the issued benefit was already within $5 of
 `FSBEN`, so nothing moved and the match restates the parity result. The
 rate differs by threshold status: among the 97 above-threshold official
@@ -489,8 +491,7 @@ classification outputs, the reconstruction scripts and
 `audit_claims.py`, which regenerates the replay counts into
 `claims_audit.json`, are committed with this paper.
 
-Layers 1 and 2 agree on the structure, and the replay's reproduction
-rate fits it: most measured SNAP payment error among eligible cases is
+Layers 1 and 2 agree on the structure: most measured SNAP payment error among eligible cases is
 information failure, not calculation failure — consistent with the
 administrative-burden literature's account of where compliance costs
 bind [@herd2018burden]. Correct-computation guarantees address the
@@ -1095,13 +1096,13 @@ one.
 ## Limitations {#sec-limitations}
 
 The parity result certifies agreement with the edited file's
-Minimodel-computed chain from supplied intermediates — not independent
+computed benefit chain from supplied intermediates — not independent
 derivation from raw facts, and not the reviewer-adjudicated outcome
 fields (@sec-oracle-scope); it is also post-correction on the same case
 set, pending a fiscal 2025 confirmatory run. The public file excludes
 ineligible-case and full-overissuance error, truncating the error
 distribution every file-based estimate inherits. The replay depends on
-a reconstruction solver whose filters exclude 22 of 305 error cases and
+a reconstruction solver whose filters exclude 22 of 305 deviation cases and
 which moves one input per case; its shares are case shares, and neither
 its reproduced nor its non-reproduced class identifies computation
 errors. The error models predict an adjudicated,
@@ -1170,7 +1171,8 @@ here, and The Axiom Foundation, which builds the rulespec-us encodings
 and the axiom-oracles verification harness whose parity this paper
 reports — the paper is an evaluation of systems its author is
 responsible for. The mitigations are structural: every parity claim
-replays against the government's own recorded values at zero tolerance
+replays against the benefit computation recorded in the QC file at zero
+tolerance
 from committed, hash-pinned artifacts any reader can rerun, and the
 manuscript's six rounds of adversarial review are archived unedited in
 the repository.
diff --git a/tests/test_retired_claims.py b/tests/test_retired_claims.py
index 495ebf9..ec1c2a1 100644
--- a/tests/test_retired_claims.py
+++ b/tests/test_retired_claims.py
@@ -17,6 +17,8 @@
 import importlib.util
 import json
 import re
+import shutil
+import subprocess
 import sys
 from pathlib import Path
 
@@ -29,6 +31,7 @@
 LAB = ROOT / "paper/snapshot/labs/amterr"
 MANUSCRIPT = ROOT / "paper/index.qmd"
 RENDERED = ROOT / "app/public/paper/web/index.html"
+RENDERED_PDF = ROOT / "app/public/paper/web/index.pdf"
 README = ROOT / "README.md"
 FACTS = ROOT / "paper/FACTS.md"
 SIMULATOR = ROOT / "app/public/index.html"
@@ -44,6 +47,8 @@
 
 # Lowercased, whitespace-collapsed fragments of the retired claims.
 RETIRED = (
+    "agency's own",
+    "government's own",
     "agency's own computational canon",
     "the agency's own canon",
     "agency's own benefit-calculation software",
@@ -89,6 +94,20 @@ def test_retired_wording_is_gone(path):
     assert not found, f"{path.relative_to(ROOT)} still carries {found}"
 
 
+@pytest.mark.skipif(shutil.which("pdftotext") is None, reason="pdftotext absent")
+def test_retired_wording_is_gone_from_the_pdf():
+    text = subprocess.run(
+        ["pdftotext", str(RENDERED_PDF), "-"],
+        capture_output=True,
+        text=True,
+        check=True,
+    ).stdout
+    # pdftotext breaks hyphenated words across lines; rejoin them first.
+    text = _normalize(re.sub(r"-\n", "-", text)).lower()
+    found = [phrase for phrase in RETIRED if phrase in text]
+    assert not found, f"index.pdf still carries {found}"
+
+
 def test_facts_records_each_withdrawal():
     """The catalog keeps the retired wording only as a superseded record."""
     facts = _read(FACTS)
@@ -101,6 +120,15 @@ def test_facts_records_each_withdrawal():
 # replay figures: manuscript prose against claims_audit.json and the slices
 
 
+def _unmoved_computational_misses() -> int:
+    """Non-reproduced cases that both moved nothing and carry a computational finding."""
+    cases = AUDIT["case_level"]
+    unmoved = {
+        c["key"] for c in cases["not_reproduced_cases"] if not c["solver_moved_input"]
+    }
+    return len(unmoved & set(cases["not_reproduced_with_computational_finding_keys"]))
+
+
 def _replay_counts() -> dict[str, int]:
     replay = AUDIT["case_level"]["replay"]
     co = AUDIT["by_posting"]["may2026"]["totals"]["CO"]
@@ -123,6 +151,7 @@ def _replay_counts() -> dict[str, int]:
         "layer2_not_reproduced": by_posting[
             "not_reproduced_with_computational_finding"
         ]["n"],
+        "miss_overlap": _unmoved_computational_misses(),
         "above_n": above["total"]["n"],
         "above_reproduced": above["outcomes"]["reproduced"]["n"],
         "sub_n": sub["total"]["n"],
@@ -139,7 +168,10 @@ def replay_quotes(c: dict[str, int]) -> list[str]:
     not_replayed = c["layer2"] - c["layer2_reproduced"] - c["layer2_not_reproduced"]
     return [
         # abstract
-        f"For {c['n']} of Colorado's {c['error_cases']} error cases, a public solver",
+        (
+            f"For {c['n']} of the {c['error_cases']} Colorado cases with a recorded "
+            "payment deviation, a public solver"
+        ),
         (
             f"reproduces the issued benefit within $5 for {above} of the "
             f"above-threshold official error cases and {sub} of the sub-threshold "
@@ -149,7 +181,10 @@ def replay_quotes(c: dict[str, int]) -> list[str]:
         (
             f"Not identified. {above} of above-threshold official errors and {sub} "
             f"of sub-threshold deviations ({blended} blended) reproduce the issued "
-            "benefit within $5, consistent with a wrong input"
+            "benefit within $5, consistent with a wrong input; of the "
+            f"{c['not_reproduced']} that do not, the solver moved nothing in "
+            f"{unmoved_misses} and {c['layer2_not_reproduced']} carry a computational "
+            f"finding ({c['miss_overlap']} are both)"
         ),
         # replay paragraph
         (
@@ -157,7 +192,7 @@ def replay_quotes(c: dict[str, int]) -> list[str]:
             f"({c['error_cases'] - c['n']} of {c['error_cases']} are excluded), "
             f"{c['reproduced']} reproduce the issued amount within the file's $5 "
             "editing tolerance, which is consistent with correct arithmetic applied "
-            "to wrong facts."
+            "to a wrong input."
         ),
         (
             f"In {c['moved']} of the {c['reproduced']} the solver moved an input; in "
@@ -263,20 +298,20 @@ def test_colorado_margin_carries_both_years():
         f"boundary, and its fiscal 2025 rate, {fy25:.2f}%, sits above it"
     ) in _read(MANUSCRIPT)
     assert (
-        f"official FY 2024 rate {fy24:.2f}%, {margin:.2f} points below the 15% "
-        f"boundary, with a ±0.9-point sampling SD; its FY 2025 rate, {fy25:.2f}%, "
-        "is above the boundary"
+        f"official FY 2024 rate {fy24:.2f}%, {margin:.2f} points below the 10% "
+        "rate where the 15% cost share begins, with a ±0.9-point sampling SD; its "
+        f"FY 2025 rate, {fy25:.2f}%, is above it"
     ) in _read(README)
     assert (
-        f"official FY2024 {fy24:.2f}% ({margin:.2f}pp below the 15% boundary; the "
-        f"FY2025 rate, {fy25:.2f}%, is above it, J3)"
+        f"official FY2024 {fy24:.2f}% ({margin:.2f}pp below the 10% rate where the "
+        f"15% share begins; the FY2025 rate, {fy25:.2f}%, is above it, J3)"
     ) in _read(FACTS)
 
 
 def test_every_margin_mention_names_fy2025():
-    for path in (MANUSCRIPT, README):
+    for path in (MANUSCRIPT, README, FACTS):
         text = _read(path)
-        for match in re.finditer(r"0\.03 points", text):
+        for match in re.finditer(r"(?<![\d.])0\.03(?!\d)[- ]?(points?|pp)", text):
             window = text[match.start() : match.start() + 200]
             assert "10.09%" in window, (path.name, window)
 
@@ -356,6 +391,9 @@ def replay_counts(draw):
     layer2_reproduced = draw(st.integers(0, 50))
     layer2_not_reproduced = draw(st.integers(0, 50))
     layer2 = layer2_reproduced + layer2_not_reproduced + draw(st.integers(0, 20))
+    miss_overlap = draw(
+        st.integers(0, min(miss_no_change + miss_stopped, layer2_not_reproduced))
+    )
     return {
         "error_cases": n + draw(st.integers(0, 100)),
         "n": n,
@@ -369,6 +407,7 @@ def replay_counts(draw):
         "layer2": layer2,
         "layer2_reproduced": layer2_reproduced,
         "layer2_not_reproduced": layer2_not_reproduced,
+        "miss_overlap": miss_overlap,
         "above_n": above_n,
         "above_reproduced": above_reproduced,
         "sub_n": sub_n,

From b8a35e7d6a8fd87323f10c2d7ec5ec64a1a2fed3 Mon Sep 17 00:00:00 2001
From: Max Ghenis <mghenis@gmail.com>
Date: Sun, 4 Oct 2026 07:14:32 -0400
Subject: [PATCH 3/9] Address review round 2 on the retired-claim corrections

- Simulator: the engine panel's FSBEN line now says Mathematica computes
  it for USDA with the QC Minimodel's benefit formula ("the Minimodel's
  full-formula recomputation" retired); the panel heading and hint name
  the QC file's benefit chain. ASSET_V 20261004a, since app.js changed.
  The engine-comparison generator and its report carry the same wording
  (engine_data.json regenerates byte-identically, sha256 2517d26e).
- Solver description: household size moves only for element 150 with
  natures 7, 12, 14 or 16; the stop rules include a $0 benefit; every
  case labeled a utility error is snapped (to zero if nothing lies
  below), which moves 2 amounts the steps left unchanged. The 14 misses
  are cases whose first finding falls outside the solver's codes.
- Oracle scope: the issuance gap is no longer called a finding about
  somebody's rules.
- Intro: "computation failure versus information failure".
- FACTS: D4 mirrors the solver description; H8 drops its "X, never Y";
  C7 cites PDF p.71 for FSGRINC and FSNETINC.
- references.bib: drops "pre-edit" from the solver note.
- Tests: Hypothesis deadline=None (deadline flakes under load); scan
  app.js and retire the Minimodel recomputation phrasings.

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
---
 analysis/ENGINE_COMPARISON.md   |   2 +-
 analysis/engine_comparison.py   |  12 +++++-----
 app/public/app.js               |   8 +++----
 app/public/index.html           |   6 ++---
 app/public/paper/web/index.html |   8 +++----
 app/public/paper/web/index.pdf  | Bin 212053 -> 212256 bytes
 paper/FACTS.md                  |   6 ++---
 paper/index.qmd                 |  40 ++++++++++++++++++--------------
 paper/references.bib            |   2 +-
 tests/test_retired_claims.py    |  17 ++++++++++----
 10 files changed, 57 insertions(+), 44 deletions(-)

diff --git a/analysis/ENGINE_COMPARISON.md b/analysis/ENGINE_COMPARISON.md
index 94ce6b6..4f309b6 100644
--- a/analysis/ENGINE_COMPARISON.md
+++ b/analysis/ENGINE_COMPARISON.md
@@ -20,7 +20,7 @@ Stage concepts are pinned per state in the suite reports (Colorado exposes `snap
 
 ## Formula-benefit divergence catalog
 
-`|RAWBEN - BENFIX|` equals the recorded error amount for every official-universe case in the seven states (asserted at build), and nationally for 99.997% of weighted FY2024 cases (n=44,800). The Minimodel's formula benefit taken as the deviation anchor — `|RAWBEN - FSBEN|` — matches the recorded error for only 83.64% weighted, because the formula recomputation conflates recorded allotment adjustments with error. The catalog partitions each state's `FSBEN != BENFIX` cases three ways:
+`|RAWBEN - BENFIX|` equals the recorded error amount for every official-universe case in the seven states (asserted at build), and nationally for 99.997% of weighted FY2024 cases (n=44,800). The formula benefit (FSBEN) taken as the deviation anchor — `|RAWBEN - FSBEN|` — matches the recorded error for only 83.64% weighted, because the formula recomputation conflates recorded allotment adjustments with error. The catalog partitions each state's `FSBEN != BENFIX` cases three ways:
 
 1. **Coded allotment adjustment** — `ALLADJ` records a prorated (2) or other (3) adjustment; the recorded allotment is not the full-month formula amount by design.
 2. **Error-correction arithmetic** — no coded adjustment and `AMTERR > 0`: the reviewer's corrected allotment differs from the full recomputation (about two-thirds within $2, rounding-scale).
diff --git a/analysis/engine_comparison.py b/analysis/engine_comparison.py
index da2f391..141d34c 100644
--- a/analysis/engine_comparison.py
+++ b/analysis/engine_comparison.py
@@ -9,8 +9,8 @@
   each state's FY2024 QC public-use reviews through the Axiom RuleSpec
   SNAP composition and compare six recorded stages — gross income,
   standard deduction, excess-shelter deduction, net income, maximum
-  allotment, and the benefit — at zero tolerance against the file's own
-  Minimodel-recomputed values (``FSBEN`` and companions).  This module
+  allotment, and the benefit — at zero tolerance against the file's
+  computed values (``FSBEN`` and companions).  This module
   re-derives the per-state parity counts from those reports and asserts
   the exactness the app displays (every compared case matches at every
   stage; excluded cases are enumerated program-structure classes).
@@ -18,8 +18,8 @@
 * **The formula-benefit divergence catalog** — the public file records
   three benefit anchors per case: ``RAWBEN`` (the allotment the state
   issued), ``BENFIX`` (the allotment corrected for the reviewer's
-  findings), and ``FSBEN`` (the Minimodel's full-formula recomputation
-  from the edited inputs).  ``|RAWBEN - BENFIX|`` equals the recorded
+  findings), and ``FSBEN`` (the formula benefit Mathematica computes for
+  USDA from the edited inputs).  ``|RAWBEN - BENFIX|`` equals the recorded
   error amount ``AMTERR`` for every official-universe case in the seven
   states; ``FSBEN`` differs from ``BENFIX`` for a catalogued minority,
   so a formula recomputation alone conflates recorded allotment
@@ -495,8 +495,8 @@ def render_report(artifact: dict[str, Any]) -> str:
                 "official-universe case in the seven states (asserted at "
                 "build), and nationally for "
                 f"{100 * national['concordance_weighted']['benfix']:.3f}% of "
-                f"weighted FY2024 cases (n={national['n']:,}). The Minimodel's "
-                "formula benefit taken as the deviation anchor — "
+                f"weighted FY2024 cases (n={national['n']:,}). The formula "
+                "benefit (FSBEN) taken as the deviation anchor — "
                 "`|RAWBEN - FSBEN|` — matches the recorded error for only "
                 f"{100 * national['concordance_weighted']['fsben']:.2f}% "
                 "weighted, because the formula recomputation conflates "
diff --git a/app/public/app.js b/app/public/app.js
index 70006e5..b825cc6 100644
--- a/app/public/app.js
+++ b/app/public/app.js
@@ -7,7 +7,7 @@ const TIERS = [[6, 0], [8, 5], [10, 10], [Infinity, 15]];
 const TIER_LABELS = { 0: "0% share", 5: "5% share", 10: "10% share", 15: "15% share" };
 const TIER_VARS = { 0: "--tier-0", 5: "--tier-5", 10: "--tier-10", 15: "--tier-15" };
 const DRAWS = 4000;
-const ASSET_V = "20261003a"; // bump with index.html's app.js?v= on every deploy that changes any asset
+const ASSET_V = "20261004a"; // bump with index.html's app.js?v= on every deploy that changes any asset
 const SCEN_SCHEMA = "snap_qc_sim.model_scenarios.v1";
 const ENGINE_SCHEMA = "snap_qc_sim.engine_comparison.v1";
 const INTERVENTIONS_SCHEMA = "snap_qc_sim.interventions.v1";
@@ -813,7 +813,7 @@ function renderFy2027(code) {
 // ---- Axiom rules-engine comparison mode ----------------------------------
 // The engine mode surfaces precomputed verification artifacts: the committed
 // axiom-oracles suite reports (stage-by-stage parity of the Axiom RuleSpec
-// computation against the QC file's Minimodel-recomputed benefit chain) and
+// computation against the QC file's computed benefit chain) and
 // the formula-benefit divergence catalog built by analysis/engine_comparison.py.
 // It does not change the Monte Carlo simulation — engine runs are precomputed,
 // never recomputed in the browser.
@@ -866,8 +866,8 @@ function enginePanelBody(code) {
   const conflates =
     `<h3>What the formula benefit conflates with error</h3>` +
     `<p>The file records three benefit anchors per case: the allotment the state issued (RAWBEN), ` +
-    `the allotment corrected for the reviewer's findings (BENFIX), and the Minimodel's full-formula ` +
-    `recomputation (FSBEN). |RAWBEN − BENFIX| equals the recorded error amount for every ` +
+    `the allotment corrected for the reviewer's findings (BENFIX), and the formula benefit Mathematica ` +
+    `computes for USDA from the edited record with the QC Minimodel's benefit formula (FSBEN). |RAWBEN − BENFIX| equals the recorded error amount for every ` +
     `official-universe case in the seven verified states — nationally ${pct(nat.concordance_weighted.benfix, 3)} of ` +
     `weighted FY 2024 cases. Taking the formula benefit as the error anchor instead — |RAWBEN − FSBEN| — matches ` +
     `the recorded error for only ${pct(nat.concordance_weighted.fsben, 1)} weighted: a formula recomputation alone ` +
diff --git a/app/public/index.html b/app/public/index.html
index 535225f..eef4396 100644
--- a/app/public/index.html
+++ b/app/public/index.html
@@ -118,7 +118,7 @@ <h2>Targeted review scenarios</h2>
     <div class="control levers">
       <span class="label">Benefit recomputation</span>
       <label><input type="checkbox" id="engine-mode"> <span>Axiom rules engine — verification view</span></label>
-      <div class="hint">Shows how the open, certified Axiom RuleSpec computation compares against the QC file's FNA Minimodel benefit chain, stage by stage, with the formula-benefit divergence catalog. Precomputed artifacts; the simulation is unchanged.</div>
+      <div class="hint">Shows how the open, certified Axiom RuleSpec computation compares against the QC file's computed benefit chain, stage by stage, with the formula-benefit divergence catalog. Precomputed artifacts; the simulation is unchanged.</div>
     </div>
     <div class="control levers">
       <span class="label">Uncertainty</span>
@@ -129,7 +129,7 @@ <h2>Targeted review scenarios</h2>
   <p class="hint mechanism" id="mechanism">Sampling engine: resamples the state's observed FY 2024 QC error dollars, centered on the official rate.</p>
 
   <section class="chart-card engine-panel" id="engine-panel" hidden>
-    <h2>Axiom rules engine vs the FNA QC Minimodel</h2>
+    <h2>Axiom rules engine vs the QC file's benefit chain</h2>
     <p class="sub">Mathematica computes the QC file's formula benefit (FSBEN) for USDA from each edited case record with the benefit formula of the QC Minimodel, one of the SNAP microsimulation models of the Food and Nutrition Administration (formerly FNS). The Axiom rules engine replays the same reviews through open, per-provision-cited RuleSpec encodings — compared stage by stage below against the file's own recorded values.</p>
     <div id="engine-body"></div>
   </section>
@@ -236,6 +236,6 @@ <h2>All states at a glance</h2>
   </nav>
 </div></footer>
 
-<script type="module" src="app.js?v=20261003a"></script>
+<script type="module" src="app.js?v=20261004a"></script>
 </body>
 </html>
diff --git a/app/public/paper/web/index.html b/app/public/paper/web/index.html
index 13868cd..4c0c4fc 100644
--- a/app/public/paper/web/index.html
+++ b/app/public/paper/web/index.html
@@ -271,7 +271,7 @@ <h2 class="anchored" data-anchor-id="introduction">Introduction</h2>
 <p>The error rate is estimated by the Quality Control system: each state re-reviews a probability sample of its own active cases — by formula in 7 C.F.R. § 275.11, between 300 and 2,400 annually, or 300 to 1,020 under an optional reduced schedule that every state elected in fiscal 2024 <span class="citation" data-cites="fns2024techdoc">(<a href="#ref-fns2024techdoc" role="doc-biblioref">U.S. Department of Agriculture, Food and Nutrition Service 2025b</a>)</span> — and federal reviewers re-review a subsample, from which the Food and Nutrition Service (FNS; renamed the Food and Nutrition Administration in 2026 — the fiscal 2025 rate publication carries the new letterhead <span class="citation" data-cites="fna2025per">(<a href="#ref-fna2025per" role="doc-biblioref">U.S. Department of Agriculture, Food and Nutrition Administration 2026</a>)</span>, and this paper keeps FNS, the name on every document the fiscal 2024 data cite) estimates official state rates with adjustments including federal re-review integration. The public-use files that result record, for roughly 45,000 cases a year, what the agency issued and what the review concluded the case should have received, with coded findings for every discrepancy.</p>
 <p>This paper treats those files as a research instrument with three uses, the first in the software-testing sense of an <em>oracle</em>: an external source of ground-truth outputs for given inputs.</p>
 <p>First, as a verification target for rules as code. I test independently encoded SNAP rules — statute, regulations, and state policy manuals encoded as executable rule modules — against the QC file’s recorded benefit computations in seven states, holding the bar at exact reproduction: zero dollar tolerance across six computation stages per case (<a href="#sec-oracle" class="quarto-xref">Section&nbsp;4</a>). The target’s precise character matters and is stated in <a href="#sec-oracle-scope" class="quarto-xref">Section&nbsp;4.1</a>: the file’s benefit chain is a computation Mathematica runs for USDA on edited case records with the benefit formula of the FNS QC Minimodel, so parity certifies agreement with the benefit computation recorded in the file; it does not adjudicate case outcomes. Reaching it surfaced two defects in the encodings under test and left a residue in issued amounts, which <a href="#sec-decompose" class="quarto-xref">Section&nbsp;5</a> decomposes.</p>
-<p>Second, as raw material for decomposing measured error. QC’s own cause coding and a finding-nature classification give two estimates of how much of the error rate reflects computation failure versus wrong facts correctly processed, and a replay of reconstructed case inputs through the verified engine tests whether one wrong input reproduces each issued benefit (<a href="#sec-decompose" class="quarto-xref">Section&nbsp;5</a>).</p>
+<p>Second, as raw material for decomposing measured error. QC’s own cause coding and a finding-nature classification give two estimates of how much of the error rate reflects computation failure versus information failure, and a replay of reconstructed case inputs through the verified engine tests whether one wrong input reproduces each issued benefit (<a href="#sec-decompose" class="quarto-xref">Section&nbsp;5</a>).</p>
 <p>Third, as training and validation data for a case-level model of the error process (<a href="#sec-model" class="quarto-xref">Section&nbsp;6</a>) and the basis for a distributional simulator of measured rates under the cost-sharing formula (<a href="#sec-simulate" class="quarto-xref">Section&nbsp;7</a>), public at policyengine.org/us/snap-payment-error-simulator. The simulator’s central output is a measurement result: near tier boundaries, sampling noise alone gives several states nearly even odds of a different cost tier than their point rate implies.</p>
 <p>The analysis pipeline, simulator, and this manuscript live in one public repository; a fact catalog maps every quantitative claim to a committed artifact (<a href="#sec-availability" class="quarto-xref">Section&nbsp;11</a>). The pipeline is deterministic — independent clean-room reruns reproduce every analysis artifact byte-for-byte, and the headline simulation artifact was independently reproduced from the raw file to four decimals — and the manuscript has been through six rounds of adversarial review, archived unedited in the repository.</p>
 </section>
@@ -431,7 +431,7 @@ <h3 class="anchored" data-anchor-id="sec-oracle-scope">What the parity target is
 <p>Exactness claims invite overreading, so the scope deserves a precise statement — more precise than an earlier draft of this paper gave it.</p>
 <p>The compared benefit chain is the one recorded in the edited public-use file. Its benefit variable (<code>FSBEN</code>) is a constructed variable: Mathematica, which edits the file for USDA under contract, computes it from each edited case record with the benefit formula of the QC Minimodel, one of FNS’s SNAP microsimulation models; the file’s other chain values (gross income, the standard and excess shelter deductions, net income, maximum allotment) are constructed variables too <span class="citation" data-cites="fns2024techdoc">(<a href="#ref-fns2024techdoc" role="doc-biblioref">U.S. Department of Agriculture, Food and Nutrition Service 2025b</a>)</span>. Where the calculated benefit is more than $5 from the issued benefit adjusted for the reviewer’s error amount (Step 13a lists the exact match conditions), the editing tries a documented sequence of changes (setting the dependent care deduction to dependent care costs, setting the utility amount in the excess shelter deduction to a standard allowance or zero, and zeroing the medical deduction of standard medical deduction demonstration participants) and keeps a change only if it produces a match <span class="citation" data-cites="fns2024techdoc">(<a href="#ref-fns2024techdoc" role="doc-biblioref">U.S. Department of Agriculture, Food and Nutrition Service 2025b, III</a>, Steps 12–13)</span>. The sequence does not always close the gap: in Colorado, <code>FSBEN</code> ends within $5 of <code>BENFIX</code> in 797 of 856 cases. The comparison also supplies several intermediates directly from the file rather than deriving them from raw facts: the QC-calculated medical and child-support deduction amounts, the recorded utility allowance, and the recorded categorical-eligibility status; eligibility screens receive passing defaults, since every filed case was enrolled.</p>
 <p>Parity therefore certifies the following: that the encoded computation chain — income aggregation, deduction sequencing, net-income arithmetic, allotment lookup, rounding — agrees exactly with the benefit computation recorded in the edited file, over the full diversity of 6,081 real case configurations, parameter values, and boundary conditions. It is an agreement between two implementations of the same rules, closer in kind to the cross-model comparisons this method extends than to independent adjudication of case outcomes; what distinguishes it from ordinary cross-model comparison is the counterparty (a computation Mathematica runs for USDA on the edited case records), the case realism, and the zero-tolerance bar. It does not certify independent derivation of every state eligibility and deduction pathway from raw facts. Extending the harness to feed raw facts — recorded medical expenses rather than the computed deduction, energy-assistance facts rather than the utility tier — is the necessary next step before the engine can compute counterfactual intermediates under alternative policy (<a href="#sec-roadmap" class="quarto-xref">Section&nbsp;8</a>). The reviewer-adjudicated quantities in the file (<code>RAWBEN</code>, <code>BENFIX</code>, error amounts and findings) are not the parity target; they are the subject of <a href="#sec-decompose" class="quarto-xref">Section&nbsp;5</a>.</p>
-<p>Holding tolerance at zero is what made the exercise informative. Early runs did not match, and each mismatch class was a finding about somebody’s rules: two defects in the encodings under test (a regulatory dollar literal superseded by statute, and a missing whole-dollar rounding step — both fixed in the public repository, with the oracle as the detector) and, once the encodings reproduced the file’s chain exactly, a residue in <em>issued</em> amounts attributable to the issuing side, which <a href="#sec-decompose" class="quarto-xref">Section&nbsp;5</a> takes up. The corrections were global rule fixes, not case-specific accommodations; still, the exactness result is post-correction on the same case set, a reuse the certification protocol mitigates but does not eliminate. A confirmatory run on the fiscal 2025 file at its release is the committed next test.</p>
+<p>Holding tolerance at zero is what made the exercise informative. Early runs did not match, and each mismatch class traced to a defect in the encodings under test (a regulatory dollar literal superseded by statute, and a missing whole-dollar rounding step — both fixed in the public repository, with the oracle as the detector). Once the encodings reproduced the file’s chain exactly, what remained was the gap between <em>issued</em> amounts and that chain, which <a href="#sec-decompose" class="quarto-xref">Section&nbsp;5</a> takes up. The corrections were global rule fixes, not case-specific accommodations; still, the exactness result is post-correction on the same case set, a reuse the certification protocol mitigates but does not eliminate. A confirmatory run on the fiscal 2025 file at its release is the committed next test.</p>
 <p>The toolchain is certified for reproduction: a pinned engine build and encoding-repository commit reproduce the Colorado suite — 856 cases, 5,136 cells including the 856 benefit cells — under recorded hashes for every input, with an independent audit rebuilding the engine to a byte-identical binary and re-running the comparison to byte-identical evidence. Throughput is 230 cases per second end to end on a laptop, so verification at this scale is not compute-bound for this workload; details and timing calibration are in the repository’s certification report.</p>
 </section>
 </section>
@@ -483,8 +483,8 @@ <h2 class="anchored" data-anchor-id="sec-decompose">Decomposing measured error:
 </div>
 </figure>
 </div>
-<p>The third layer adapts the public reconstruction solver of <span class="citation" data-cites="giannella_snapqc">Giannella and Molin (<a href="#ref-giannella_snapqc" role="doc-biblioref">2026</a>)</span> to fiscal 2024 and replays its output through the verified engine. The solver starts from the file’s edited inputs and moves only the input named by the case’s first finding (<code>ELEMENT1</code>): earned or unearned income, rent, the utility allowance, or the medical, dependent care or child-support deduction in $3 steps, or household size by one person. It stops when its own recomputed benefit reaches or passes the issued benefit (<code>RAWBEN</code>), when the input hits zero or a cap, or after 1,000 steps; a moved utility amount is then reset to the nearest value above or below the file’s that more than five filtered cases in the same state and calendar year use. When the first finding names none of those inputs, nothing moves. Of 283 deviation cases surviving the solver’s consistency filters (22 of 305 are excluded), 246 reproduce the issued amount within the file’s $5 editing tolerance, which is consistent with correct arithmetic applied to a wrong input. In 230 of the 246 the solver moved an input; in the other 16 the issued benefit was already within $5 of <code>FSBEN</code>, so nothing moved and the match restates the parity result. The rate differs by threshold status: among the 97 above-threshold official error cases 76 reproduce (78.4%, binomial standard error about 4 points), and among the 186 sub-threshold deviations 170 do (91.4%). The solver’s own recomputation and the engine agree on the within-$5 classification for all 283 cases, two implementations with distinct codebases reading the same file.</p>
-<p>Neither outcome identifies a computation error. A reproduced case shows that one value of the first-listed input, run through correct arithmetic, yields the issued benefit; it does not show which input the agency had wrong, and a moved input can absorb a computation error. A non-reproduced case can reflect the solver’s limits: it tries one input in one direction, and in 17 of the 37 it moved nothing (14 cases whose first finding names an input it does not adjust, and 3 where it stopped before moving). Of the 26 cases that carry a layer-2 computational finding, 13 reproduce, 10 do not and 3 were not replayed. The complete per-case replay output (283 rows), the layer-1 and layer-2 classification outputs, the reconstruction scripts and <code>audit_claims.py</code>, which regenerates the replay counts into <code>claims_audit.json</code>, are committed with this paper.</p>
+<p>The third layer adapts the public reconstruction solver of <span class="citation" data-cites="giannella_snapqc">Giannella and Molin (<a href="#ref-giannella_snapqc" role="doc-biblioref">2026</a>)</span> to fiscal 2024 and replays its output through the verified engine. The solver starts from the file’s edited inputs and moves only the input named by the case’s first finding (<code>ELEMENT1</code>): earned or unearned income, rent, the utility allowance, or the medical, dependent care or child-support deduction in $3 steps, or household size by one person (element 150 with natures 7, 12, 14 or 16). It stops when its own recomputed benefit reaches or passes the issued benefit (<code>RAWBEN</code>), when the input hits zero or a cap, when the benefit reaches $0, or after 1,000 steps. Each case it labels as a utility error then has its utility amount set to the nearest value above (or below) the file’s utility amount that more than five filtered cases in the same state and calendar year use, or to zero if none lies below; in 2 cases this moves an amount the steps left unchanged. When the first finding falls outside those element and nature codes, nothing moves. Of 283 deviation cases surviving the solver’s consistency filters (22 of 305 are excluded), 246 reproduce the issued amount within the file’s $5 editing tolerance, which is consistent with correct arithmetic applied to a wrong input. In 230 of the 246 the solver moved an input; in the other 16 the issued benefit was already within $5 of <code>FSBEN</code>, so nothing moved and the match restates the parity result. The rate differs by threshold status: among the 97 above-threshold official error cases 76 reproduce (78.4%, binomial standard error about 4 points), and among the 186 sub-threshold deviations 170 do (91.4%). The solver’s own recomputation and the engine agree on the within-$5 classification for all 283 cases, two implementations with distinct codebases reading the same file.</p>
+<p>Neither outcome identifies a computation error. A reproduced case shows that one value of the first-listed input, run through correct arithmetic, yields the issued benefit; it does not show which input the agency had wrong, and a moved input can absorb a computation error. A non-reproduced case can reflect the solver’s limits: it tries one input in one direction, and in 17 of the 37 it moved nothing (14 cases whose first finding falls outside those codes, and 3 where it stopped before moving). Of the 26 cases that carry a layer-2 computational finding, 13 reproduce, 10 do not and 3 were not replayed. The complete per-case replay output (283 rows), the layer-1 and layer-2 classification outputs, the reconstruction scripts and <code>audit_claims.py</code>, which regenerates the replay counts into <code>claims_audit.json</code>, are committed with this paper.</p>
 <p>Layers 1 and 2 agree on the structure: most measured SNAP payment error among eligible cases is information failure, not calculation failure — consistent with the administrative-burden literature’s account of where compliance costs bind <span class="citation" data-cites="herd2018burden">(<a href="#ref-herd2018burden" role="doc-biblioref">Herd and Moynihan 2018</a>)</span>. Correct-computation guarantees address the smaller share directly — roughly 3 to 10% of error dollars by layers 1–2 — plus whatever share of information failures better tooling prevents indirectly through verification prompts and documentation-requirement computation at intake; <a href="#sec-adoption" class="quarto-xref">Section&nbsp;7.2</a> prices the direct shares through the formula’s tiers, elections, and delay clause. Ineligibility errors, excluded from the file, are predominantly facts-driven, which would push the information-failure share higher still; that inference, unlike the in-file shares, is not directly computed here. The indirect channel is where policy design enters: rules determine what a caseworker must collect and verify, which is the mechanism the modeling section takes up.</p>
 </section>
 <section id="sec-model" class="level2">
diff --git a/app/public/paper/web/index.pdf b/app/public/paper/web/index.pdf
index 74129df0491bdc5792051046b201826c6d98c556..781a6cbfe973e11bb7e4c6bfe84362cb0eb98370 100644
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diff --git a/paper/FACTS.md b/paper/FACTS.md
index 975bc8d..cc473a4 100644
--- a/paper/FACTS.md
+++ b/paper/FACTS.md
@@ -34,7 +34,7 @@ Numbers from the superseded pre-audit pipeline (before PR #7) are banned.
 | C4 | WITHDRAWN 2026-10-03: the claim that the process surfaced errata in FNS's FY2024 technical documentation. No source records such errata: neither version of paper/snapshot/labs/amterr/ANALYSIS.md (27e1b09, c4333b3), the axiom.org Colorado report (original and corrected text) nor the axiom-oracles SNAP QC playbook documents one, GitHub issue searches across TheAxiomFoundation and PolicyEngine found none, and round-1 and round-2 referees could not locate the "reported upstream" thread. The manuscript no longer makes the claim (abstract, introduction, @sec-oracle-scope) | searched 2026-10-03; paper/reviews/round-1/referee-redteam.md, round-2/round2-rounddiff.md |
 | C5 | A certification probe (2026-08-06) attested a reproducible toolchain: engine commit de0efdc7 (binary SHA-256 bb8ec236…) × rulespec-us b53ce208 reproduces CO 856/856 benefits and 5,136/5,136 stage cells at zero tolerance; 856 cases in 3.73 s end-to-end (230 cases/s; 279/s with compile amortized) | CERT_REPORT (paper/snapshot/cert/CERT_REPORT.md) |
 | C6 | Extrapolation at certified throughput: 6,081 cases ≈ 26 s; 44,800 × 5 scenarios ≈ 16 minutes single-threaded on a laptop | CERT_REPORT extrapolation table |
-| C7 | What the parity target is: `FSBEN` is a constructed variable ("Final calculated benefit") that Mathematica computes for USDA from each edited case record with the QC Minimodel's benefit formula; the QC Minimodel is "one of FNA's SNAP microsimulation models". The file's other chain values (FSGRINC, FSSTDDED, FSSLTDED, FSNETINC, BENMAX) are constructed too; RAWBEN and AMTERR are reported. Editing Steps 12–13 calculate the benefit and, where it is not within $5 of the issued benefit adjusted for the error amount (Step 13a conditions), try the dependent care deduction, the utility amount and the SMD-demonstration medical deduction, keeping a change only if it produces a match. Colorado: `FSBEN` within $5 of `BENFIX` in 797 of 856 cases (743 exactly equal; same under the May and August 2026 postings, which differ only in weights). Never "the agency's own benefit-calculation software" or "computational canon" | FY2024 QC technical documentation, August 2026 posting: title page and acknowledgments (Mathematica for FNA, contract 12-3198-24-Q-0029), PDF p.15 (Minimodel), pp.36–37 (Steps 12–13), p.65 (Minimodel's FSBEN calculation refers to the codebook entry), p.73 and p.91 (FSBEN constructed; formula); 797/856 recomputed from qc_pub_fy2024.csv (sha256 e871a8e9…), locked in tests/test_retired_claims.py |
+| C7 | What the parity target is: `FSBEN` is a constructed variable ("Final calculated benefit") that Mathematica computes for USDA from each edited case record with the QC Minimodel's benefit formula; the QC Minimodel is "one of FNA's SNAP microsimulation models". The file's other chain values (FSGRINC, FSSTDDED, FSSLTDED, FSNETINC, BENMAX) are constructed too; RAWBEN and AMTERR are reported. Editing Steps 12–13 calculate the benefit and, where it is not within $5 of the issued benefit adjusted for the error amount (Step 13a conditions), try the dependent care deduction, the utility amount and the SMD-demonstration medical deduction, keeping a change only if it produces a match. Colorado: `FSBEN` within $5 of `BENFIX` in 797 of 856 cases (743 exactly equal; same under the May and August 2026 postings, which differ only in weights). Never "the agency's own benefit-calculation software" or "computational canon" | FY2024 QC technical documentation, August 2026 posting: title page and acknowledgments (Mathematica for FNA, contract 12-3198-24-Q-0029), PDF p.15 (Minimodel), pp.36–37 (Steps 12–13), p.65 (Minimodel's FSBEN calculation refers to the codebook entry), p.71 (FSGRINC, FSNETINC constructed), p.73 (FSSLTDED, FSSTDDED, BENMAX, FSBEN constructed; RAWBEN, AMTERR reported) and p.91 (FSBEN formula); 797/856 recomputed from qc_pub_fy2024.csv (sha256 e871a8e9…), locked in tests/test_retired_claims.py |
 
 ## D. Decomposition of measured error (Colorado FY2024)
 
@@ -43,7 +43,7 @@ Numbers from the superseded pre-audit pipeline (before PR #7) are banned.
 | D1 | Colorado FY2024: 856 sampled cases, 305 with payment errors (STATUS 2/3); $112.6M/yr weighted error dollars on $1.268B issuance (8.88% file-derived; official regression-adjusted rate 9.97% = 7.91 over + 2.06 under) | paper/snapshot/labs/amterr/ANALYSIS.md; FNS FY2024 PER (https://fns-prod.azureedge.us/sites/default/files/resource-files/snap-fy24QC-PER.pdf) |
 | D2 | QC cause coding, SUPERSEDED by K1 (revision 5): the lab tabulation denominated over all weighted deviation dollars ($8.1B national incl. sub-threshold) and gave CO $8.2M/7.3% strict, $11.9M/10.5% broad; national $320M/3.9%, $1.5B/18.4%. The committed artifact denominates over official error dollars and differs (see K1) | paper/snapshot/labs/amterr/native_decomposition.json + ANALYSIS.md (archived) |
 | D3 | Finding-nature classification: pure-math 3.3%, input-with-system-cause 6.6%, mixed 4.2%, other input 86.0% of Colorado weighted deviation dollars. NOW COMMITTED as K3 (revision 5): the artifact reproduces these to the cent under the corrected field semantics (NATUREi carries nature codes; E_FINDGi is impact-only, a lab labeling error that did not affect the numbers) | analysis/cause_shares.json layer 2 (was: paper/snapshot/labs/phase_a_classification.json) |
-| D4 | Replay: the public solver (FY2024 adaptation of giannella/snap_qc) starts from the file's edited inputs and moves only the ELEMENT1 input ($3 steps; household size by one person for element 150) until its own recomputed benefit reaches or passes RAWBEN, the input hits zero or a cap, or 1,000 steps; a moved utility amount is then reset to the nearest value above or below the file's UTIL that more than five filtered cases in the state and calendar year use. It uses no disclosure-perturbation structure. The verified engine on the solver's inputs reproduces RAWBEN within $5 for 246 of 283 filtered error cases (86.9%), consistent with correct arithmetic on a wrong input: 230 after the solver moved an input, 16 with nothing moved (\|RAWBEN − FSBEN\| ≤ $5 already, so the match restates parity). 37 do not reproduce; in 17 nothing moved (14 `no_change` rows whose ELEMENT1 the solver does not adjust, 3 where it stopped before moving), and 10 carry a layer-2 computational finding (8 of them among the 17). Solver and engine agree on all 283. Neither class identifies computation errors: of 26 cases with a layer-2 computational finding, 13 reproduce, 10 do not, 3 were not replayed. SUPERSEDES the revision-9 wording "explains 246/283 … as correct arithmetic on wrong facts" and "37 cases form the computation-side upper bound" (withdrawn 2026-10-03, matching ANALYSIS.md's revision of the same date) | paper/snapshot/labs/amterr/claims_audit.json case_level.replay (regenerated by audit_claims.py); amterr_replay_results.json (283 rows); reconstruct_co_fy2024.R; amterr_replay.py; ANALYSIS.md layer 3 |
+| D4 | Replay: the public solver (FY2024 adaptation of giannella/snap_qc) starts from the file's edited inputs and moves only the ELEMENT1 input ($3 steps; household size by one person for element 150 with natures 7, 12, 14 or 16) until its own recomputed benefit reaches or passes RAWBEN, the input hits zero or a cap, the benefit reaches $0, or 1,000 steps. Each case labeled a utility error then has its UTIL set to the nearest value above (or below) the file's UTIL that more than five filtered cases in the state and calendar year use, or to zero if none lies below; in 2 cases (202403-40765, 202404-40803) this moves an amount the steps left unchanged. It uses no disclosure-perturbation structure. The verified engine on the solver's inputs reproduces RAWBEN within $5 for 246 of 283 filtered error cases (86.9%), consistent with correct arithmetic on a wrong input: 230 after the solver moved an input, 16 with nothing moved (\|RAWBEN − FSBEN\| ≤ $5 already, so the match restates parity). 37 do not reproduce; in 17 nothing moved (14 `no_change` rows whose first finding falls outside the solver's element and nature codes, 2 of them element 150 with nature 13; 3 where it stopped before moving), and 10 carry a layer-2 computational finding (8 of them among the 17). Solver and engine agree on all 283. Neither class identifies computation errors: of 26 cases with a layer-2 computational finding, 13 reproduce, 10 do not, 3 were not replayed. SUPERSEDES the revision-9 wording "explains 246/283 … as correct arithmetic on wrong facts" and "37 cases form the computation-side upper bound" (withdrawn 2026-10-03, matching ANALYSIS.md's revision of the same date) | paper/snapshot/labs/amterr/claims_audit.json case_level.replay (regenerated by audit_claims.py); amterr_replay_results.json (283 rows); reconstruct_co_fy2024.R; amterr_replay.py; ANALYSIS.md layer 3 |
 
 ## E. Error models (corrected pipeline: PRs #7–#8 as refit by #13)
 
@@ -105,7 +105,7 @@ Numbers from the superseded pre-audit pipeline (before PR #7) are banned.
 | H5 | OBBBA cut the federal share of SNAP administrative costs from 50% to 25% beginning FY2027 | OBBBA § 10106; Federal Register 2026-12696 |
 | H6 | OBBBA § 10103 restricts the LIHEAP-triggered heat-and-eat utility allowance to households with an elderly or disabled member | Pub. L. 119-21 § 10103; state implementation letters |
 | H7 | Colorado deviation universe: 305 = 110 above-threshold official errors + 195 sub-threshold deviations; sub-threshold dollars $18.5M of $112.6M (16%); official-gate file-derived rate 7.42% | recomputed from qc_pub_fy2024.csv (official gate); paper/snapshot/labs/amterr/amterr_replay_results.json |
-| H8 | Replay reproduction rates by threshold status: 76/97 = 78.4% above threshold; 170/186 = 91.4% below (reproduced within $5; consistent with a wrong input, never "explained"). SUPERSEDES "33 of 246 explained cases have deviations ≤ $5 (within comparison tolerance mechanically)": 33 of the 246 have AMTERR ≤ $5, but the solver moved an input in 17 of them; the matches that follow mechanically from FSBEN parity are the 16 with nothing moved (D4), all among the 33 | analysis/cause_shares.json colorado_replay_reconciliation.slices; claims_audit.json case_level.replay.reproduced_without_move_n; amterr_replay_results.json (amterr field); locked in tests/test_retired_claims.py |
+| H8 | Replay reproduction rates by threshold status: 76/97 = 78.4% above threshold; 170/186 = 91.4% below (reproduced within $5, which is consistent with a wrong input; G lists the retired wording). SUPERSEDES "33 of 246 explained cases have deviations ≤ $5 (within comparison tolerance mechanically)": 33 of the 246 have AMTERR ≤ $5, but the solver moved an input in 17 of them; the matches that follow mechanically from FSBEN parity are the 16 with nothing moved (D4), all among the 33 | analysis/cause_shares.json colorado_replay_reconciliation.slices; claims_audit.json case_level.replay.reproduced_without_move_n; amterr_replay_results.json (amterr field); locked in tests/test_retired_claims.py |
 
 ## I. Engine counterfactuals and the model-primary round (added in revision 3)
 
diff --git a/paper/index.qmd b/paper/index.qmd
index b0f2f8f..57708b7 100644
--- a/paper/index.qmd
+++ b/paper/index.qmd
@@ -115,8 +115,8 @@ the encodings under test and left a residue in issued amounts, which
 
 Second, as raw material for decomposing measured error. QC's own cause
 coding and a finding-nature classification give two estimates of
-how much of the error rate reflects computation failure versus wrong
-facts correctly processed, and a replay of reconstructed case inputs
+how much of the error rate reflects computation failure versus
+information failure, and a replay of reconstructed case inputs
 through the verified engine tests whether one wrong input reproduces
 each issued benefit (@sec-decompose).
 
@@ -393,13 +393,13 @@ engine can compute counterfactual intermediates under alternative policy
 target; they are the subject of @sec-decompose.
 
 Holding tolerance at zero is what made the exercise informative. Early
-runs did not match, and each mismatch class was a finding about
-somebody's rules: two defects in the encodings under test (a regulatory
-dollar literal superseded by statute, and a missing whole-dollar rounding
-step — both fixed in the public repository, with the oracle as the
-detector) and, once the encodings reproduced the file's chain exactly, a
-residue in *issued* amounts attributable to the issuing side, which
-@sec-decompose takes up. The corrections were global
+runs did not match, and each mismatch class traced to a defect in the
+encodings under test (a regulatory dollar literal superseded by statute,
+and a missing whole-dollar rounding step — both fixed in the public
+repository, with the oracle as the detector). Once the encodings
+reproduced the file's chain exactly, what remained was the gap between
+*issued* amounts and that chain, which @sec-decompose takes up. The
+corrections were global
 rule fixes, not case-specific accommodations; still, the exactness result
 is post-correction on the same case set, a reuse the certification
 protocol mitigates but does not eliminate. A confirmatory run on the
@@ -459,13 +459,17 @@ verified engine. The solver starts from the file's edited inputs and
 moves only the input named by the case's first finding (`ELEMENT1`):
 earned or unearned income, rent, the utility allowance, or the medical,
 dependent care or child-support deduction in $3 steps, or household size
-by one person. It stops when its own recomputed benefit reaches or passes
-the issued benefit (`RAWBEN`), when the input hits zero or a cap, or
-after 1,000 steps; a moved utility amount is then reset to the nearest
-value above or below the file's that more than five filtered cases in
-the same state and calendar year use. When the first finding names none
-of those inputs, nothing moves. Of 283 deviation cases surviving the solver's consistency
-filters (22 of 305 are excluded), 246 reproduce the issued amount within
+by one person (element 150 with natures 7, 12, 14 or 16). It stops when
+its own recomputed benefit reaches or passes the issued benefit
+(`RAWBEN`), when the input hits zero or a cap, when the benefit reaches
+$0, or after 1,000 steps. Each case it labels as a utility error then has
+its utility amount set to the nearest value above (or below) the file's
+utility amount that more than five filtered cases in the same state and
+calendar year use, or to zero if none lies below; in 2 cases this moves
+an amount the steps left unchanged. When the first finding falls outside
+those element and nature codes, nothing moves. Of 283 deviation cases
+surviving the solver's consistency filters (22 of 305 are excluded), 246
+reproduce the issued amount within
 the file's $5 editing tolerance, which is consistent with correct
 arithmetic applied to a wrong input. In 230 of the 246 the solver moved an
 input; in the other 16 the issued benefit was already within $5 of
@@ -483,8 +487,8 @@ arithmetic, yields the issued benefit; it does not show which input the
 agency had wrong, and a moved input can absorb a computation error. A
 non-reproduced case can reflect the solver's limits: it tries one input
 in one direction, and in 17 of the 37 it moved nothing (14 cases whose
-first finding names an input it does not adjust, and 3 where it stopped
-before moving). Of the 26 cases that carry a layer-2 computational
+first finding falls outside those codes, and 3 where it stopped before
+moving). Of the 26 cases that carry a layer-2 computational
 finding, 13 reproduce, 10 do not and 3 were not replayed. The complete
 per-case replay output (283 rows), the layer-1 and layer-2
 classification outputs, the reconstruction scripts and
diff --git a/paper/references.bib b/paper/references.bib
index 5c91db8..a245adb 100644
--- a/paper/references.bib
+++ b/paper/references.bib
@@ -17,7 +17,7 @@ @misc{giannella_snapqc
   title = {snap\_qc: reconstruction and analysis code for the SNAP Quality Control public-use files},
   year = {2026},
   url = {https://github.com/giannella/snap_qc},
-  note = {Software repository; includes the pre-edit value reconstruction solver and state policy-option registries. Commit 9b77be6, accessed 2026-08-07}
+  note = {Software repository; includes the reconstruction solver adapted here and state policy-option registries. Commit 9b77be6, accessed 2026-08-07}
 }
 
 @article{duan1983smearing,
diff --git a/tests/test_retired_claims.py b/tests/test_retired_claims.py
index ec1c2a1..10fb4eb 100644
--- a/tests/test_retired_claims.py
+++ b/tests/test_retired_claims.py
@@ -24,7 +24,7 @@
 
 import pandas as pd
 import pytest
-from hypothesis import given
+from hypothesis import given, settings
 from hypothesis import strategies as st
 
 ROOT = Path(__file__).resolve().parents[1]
@@ -35,6 +35,7 @@
 README = ROOT / "README.md"
 FACTS = ROOT / "paper/FACTS.md"
 SIMULATOR = ROOT / "app/public/index.html"
+SIMULATOR_JS = ROOT / "app/public/app.js"
 
 AUDIT = json.loads((LAB / "claims_audit.json").read_text(encoding="utf-8"))
 SLICES = json.loads((ROOT / "analysis/cause_shares.json").read_text(encoding="utf-8"))[
@@ -65,6 +66,9 @@
     "errata in the federal technical documentation",
     "errata in the technical documentation",
     "explained cases have deviations",
+    "minimodel's full-formula recomputation",
+    "minimodel-recomputed",
+    "fna qc minimodel",
 )
 
 
@@ -87,7 +91,9 @@ def _percent(numerator: int, denominator: int) -> str:
 # retired wording
 
 
-@pytest.mark.parametrize("path", [MANUSCRIPT, RENDERED, README, SIMULATOR])
+@pytest.mark.parametrize(
+    "path", [MANUSCRIPT, RENDERED, README, SIMULATOR, SIMULATOR_JS]
+)
 def test_retired_wording_is_gone(path):
     text = _read(path).lower()
     found = [phrase for phrase in RETIRED if phrase in text]
@@ -210,8 +216,8 @@ def replay_quotes(c: dict[str, int]) -> list[str]:
         (f"agree on the within-$5 classification for all {c['agree']} cases"),
         (
             f"in {unmoved_misses} of the {c['not_reproduced']} it moved nothing "
-            f"({c['miss_no_change']} cases whose first finding names an input it "
-            f"does not adjust, and {c['miss_stopped']} where it stopped before moving)"
+            f"({c['miss_no_change']} cases whose first finding falls outside those "
+            f"codes, and {c['miss_stopped']} where it stopped before moving)"
         ),
         (
             f"Of the {c['layer2']} cases that carry a layer-2 computational finding, "
@@ -415,6 +421,7 @@ def replay_counts(draw):
     }
 
 
+@settings(deadline=None)
 @given(replay_counts(), st.data())
 def test_lock_detects_any_changed_count(counts, data):
     """Every count is load-bearing: changing one changes the required prose."""
@@ -423,6 +430,7 @@ def test_lock_detects_any_changed_count(counts, data):
     assert replay_quotes(changed) != replay_quotes(counts)
 
 
+@settings(deadline=None)
 @given(replay_counts())
 def test_lock_is_deterministic_and_counts_render_verbatim(counts):
     quotes = replay_quotes(counts)
@@ -432,6 +440,7 @@ def test_lock_is_deterministic_and_counts_render_verbatim(counts):
         assert re.search(rf"(?<![\d.]){counts[key]}(?![\d.])", joined), key
 
 
+@settings(deadline=None)
 @given(st.integers(1, 10_000), st.data())
 def test_complementary_shares_sum_to_100(denominator, data):
     numerator = data.draw(st.integers(0, denominator))

From 9e34b2eb86cd28ca1109f11c3c22434e0acd2a0e Mon Sep 17 00:00:00 2001
From: Max Ghenis <mghenis@gmail.com>
Date: Sun, 4 Oct 2026 08:52:35 -0400
Subject: [PATCH 4/9] Address review round 3 on the retired-claim corrections

- Solver description: the body now says what moves and where (household
  size once, by one person, for element 150 with natures 7/12/14/16;
  income, rent, utility and deductions in $3 steps toward RAWBEN), and a
  footnote gives the stop rules as reconstruct_co_fy2024.R applies them:
  income steps stop only once the benefit passes RAWBEN, so they run to
  zero income or the step limit where RAWBEN is the maximum allotment;
  the cap is the shelter deduction's; the shelter-at-zero and
  uncapped-benefit-below-zero rules are included. The utility snap picks
  the common amount nearest the stepped amount, from those above or
  below the file's UTIL by the sign of RAWBEN - FSBEN. FACTS D4 mirrors
  it.
- The 16 unmoved matches: "it took no step, and the issued benefit was
  already within $5 of FSBEN", replacing a "so nothing moved" causal
  reading the code does not support.
- Abstract and table: the solver moves inputs "toward" the issued amount.
- Oracle scope: "reaching parity surfaced two defects in the encodings",
  matching FACTS C3. FACTS C1: "computed benefit chain".
- Tests: the JS scan rejoins concatenated string literals, so the retired
  app.js sentence (split across a template-literal join) is caught; a
  unit test covers the join.

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
---
 app/public/paper/web/index.html |  11 +++---
 app/public/paper/web/index.pdf  | Bin 212256 -> 213994 bytes
 paper/FACTS.md                  |   4 +--
 paper/index.qmd                 |  57 ++++++++++++++++++++------------
 tests/test_retired_claims.py    |  33 +++++++++++++++---
 5 files changed, 72 insertions(+), 33 deletions(-)

diff --git a/app/public/paper/web/index.html b/app/public/paper/web/index.html
index 4c0c4fc..071055f 100644
--- a/app/public/paper/web/index.html
+++ b/app/public/paper/web/index.html
@@ -108,7 +108,7 @@
 per case, zero dollar tolerance — a process that surfaced and fixed two
 defects in the encodings. For 283 of the 305 Colorado cases with a
 recorded payment deviation, a public solver moves the input named by each case's first finding, where
-it can, until its recomputed benefit reaches the issued amount; run on
+it can, toward the issued amount; run on
 those inputs, the verified engine reproduces the issued benefit within
 $5 for 78.4% of the above-threshold official error cases and 91.4% of
 the sub-threshold deviations, which is consistent with correct
@@ -218,7 +218,7 @@ <h1 class="title">Administrative quality control data as an oracle: verifying en
     <div>
       <div class="abstract">
         <div class="block-title">Abstract</div>
-        <p>Beginning in fiscal year 2028, each state’s Supplemental Nutrition Assistance Program (SNAP) payment error rate sets its share of benefit costs, in steps of five percentage points at rates of 6, 8, and 10 percent. A measurement system built for oversight now moves large sums, which raises two questions this paper treats together: whether the rules that produce benefit amounts can be executed correctly, and how the errors that remain should be measured and simulated. I use the USDA Quality Control (QC) public-use files as a verification oracle for independently encoded SNAP rules: across seven states, the encodings reproduce the file’s benefit-computation chain, from file-recorded intermediates, exactly — all 6,081 in-scope cases of the 6,194-case official universe (113 documented exclusions), six computation stages per case, zero dollar tolerance — a process that surfaced and fixed two defects in the encodings. For 283 of the 305 Colorado cases with a recorded payment deviation, a public solver moves the input named by each case’s first finding, where it can, until its recomputed benefit reaches the issued amount; run on those inputs, the verified engine reproduces the issued benefit within $5 for 78.4% of the above-threshold official error cases and 91.4% of the sub-threshold deviations, which is consistent with correct arithmetic on a wrong input. Pricing a cause-code decomposition through the formula’s tiers, elections, and delay clause — removing the error dollars the file’s cause codes assign to the computing apparatus, an accounting scenario rather than a causal estimate — moves summed expected fiscal 2028 bills from $7.7B to $7.6B a year under a strict class convention and $6.9B under a broad one, and not monotonically: a state just under the statute’s delay threshold sees its near-term expected bill rise as its measured rate falls. I also fit and validate a distributional model of case-level payment deviations, reporting its validation failures alongside its results, and serve one policy counterfactual from it in the deployed simulator: a standard medical deduction scenario computed on engine-recomputed case parameters, whose model-implied Colorado effect opposes the naive accounting bound in sign. The central finding, from an open Monte Carlo simulator resampling each state’s own QC cases, is about measurement: at regulatory sample sizes, several states face nearly even odds — 46 to 50 percent — of being assigned a different cost tier than their official point rate implies. The fiscal 2025 rates, published in June 2026, land where that mechanism says realized years should: 18 of 53 jurisdictions changed cost-sharing tiers in one year, and only seven of those changes exceed the 95% band that two years of sampling noise — estimated from fiscal 2024 microdata alone — implies.</p>
+        <p>Beginning in fiscal year 2028, each state’s Supplemental Nutrition Assistance Program (SNAP) payment error rate sets its share of benefit costs, in steps of five percentage points at rates of 6, 8, and 10 percent. A measurement system built for oversight now moves large sums, which raises two questions this paper treats together: whether the rules that produce benefit amounts can be executed correctly, and how the errors that remain should be measured and simulated. I use the USDA Quality Control (QC) public-use files as a verification oracle for independently encoded SNAP rules: across seven states, the encodings reproduce the file’s benefit-computation chain, from file-recorded intermediates, exactly — all 6,081 in-scope cases of the 6,194-case official universe (113 documented exclusions), six computation stages per case, zero dollar tolerance — a process that surfaced and fixed two defects in the encodings. For 283 of the 305 Colorado cases with a recorded payment deviation, a public solver moves the input named by each case’s first finding, where it can, toward the issued amount; run on those inputs, the verified engine reproduces the issued benefit within $5 for 78.4% of the above-threshold official error cases and 91.4% of the sub-threshold deviations, which is consistent with correct arithmetic on a wrong input. Pricing a cause-code decomposition through the formula’s tiers, elections, and delay clause — removing the error dollars the file’s cause codes assign to the computing apparatus, an accounting scenario rather than a causal estimate — moves summed expected fiscal 2028 bills from $7.7B to $7.6B a year under a strict class convention and $6.9B under a broad one, and not monotonically: a state just under the statute’s delay threshold sees its near-term expected bill rise as its measured rate falls. I also fit and validate a distributional model of case-level payment deviations, reporting its validation failures alongside its results, and serve one policy counterfactual from it in the deployed simulator: a standard medical deduction scenario computed on engine-recomputed case parameters, whose model-implied Colorado effect opposes the naive accounting bound in sign. The central finding, from an open Monte Carlo simulator resampling each state’s own QC cases, is about measurement: at regulatory sample sizes, several states face nearly even odds — 46 to 50 percent — of being assigned a different cost tier than their official point rate implies. The fiscal 2025 rates, published in June 2026, land where that mechanism says realized years should: 18 of 53 jurisdictions changed cost-sharing tiers in one year, and only seven of those changes exceed the 95% band that two years of sampling noise — estimated from fiscal 2024 microdata alone — implies.</p>
       </div>
     </div>
 
@@ -431,7 +431,7 @@ <h3 class="anchored" data-anchor-id="sec-oracle-scope">What the parity target is
 <p>Exactness claims invite overreading, so the scope deserves a precise statement — more precise than an earlier draft of this paper gave it.</p>
 <p>The compared benefit chain is the one recorded in the edited public-use file. Its benefit variable (<code>FSBEN</code>) is a constructed variable: Mathematica, which edits the file for USDA under contract, computes it from each edited case record with the benefit formula of the QC Minimodel, one of FNS’s SNAP microsimulation models; the file’s other chain values (gross income, the standard and excess shelter deductions, net income, maximum allotment) are constructed variables too <span class="citation" data-cites="fns2024techdoc">(<a href="#ref-fns2024techdoc" role="doc-biblioref">U.S. Department of Agriculture, Food and Nutrition Service 2025b</a>)</span>. Where the calculated benefit is more than $5 from the issued benefit adjusted for the reviewer’s error amount (Step 13a lists the exact match conditions), the editing tries a documented sequence of changes (setting the dependent care deduction to dependent care costs, setting the utility amount in the excess shelter deduction to a standard allowance or zero, and zeroing the medical deduction of standard medical deduction demonstration participants) and keeps a change only if it produces a match <span class="citation" data-cites="fns2024techdoc">(<a href="#ref-fns2024techdoc" role="doc-biblioref">U.S. Department of Agriculture, Food and Nutrition Service 2025b, III</a>, Steps 12–13)</span>. The sequence does not always close the gap: in Colorado, <code>FSBEN</code> ends within $5 of <code>BENFIX</code> in 797 of 856 cases. The comparison also supplies several intermediates directly from the file rather than deriving them from raw facts: the QC-calculated medical and child-support deduction amounts, the recorded utility allowance, and the recorded categorical-eligibility status; eligibility screens receive passing defaults, since every filed case was enrolled.</p>
 <p>Parity therefore certifies the following: that the encoded computation chain — income aggregation, deduction sequencing, net-income arithmetic, allotment lookup, rounding — agrees exactly with the benefit computation recorded in the edited file, over the full diversity of 6,081 real case configurations, parameter values, and boundary conditions. It is an agreement between two implementations of the same rules, closer in kind to the cross-model comparisons this method extends than to independent adjudication of case outcomes; what distinguishes it from ordinary cross-model comparison is the counterparty (a computation Mathematica runs for USDA on the edited case records), the case realism, and the zero-tolerance bar. It does not certify independent derivation of every state eligibility and deduction pathway from raw facts. Extending the harness to feed raw facts — recorded medical expenses rather than the computed deduction, energy-assistance facts rather than the utility tier — is the necessary next step before the engine can compute counterfactual intermediates under alternative policy (<a href="#sec-roadmap" class="quarto-xref">Section&nbsp;8</a>). The reviewer-adjudicated quantities in the file (<code>RAWBEN</code>, <code>BENFIX</code>, error amounts and findings) are not the parity target; they are the subject of <a href="#sec-decompose" class="quarto-xref">Section&nbsp;5</a>.</p>
-<p>Holding tolerance at zero is what made the exercise informative. Early runs did not match, and each mismatch class traced to a defect in the encodings under test (a regulatory dollar literal superseded by statute, and a missing whole-dollar rounding step — both fixed in the public repository, with the oracle as the detector). Once the encodings reproduced the file’s chain exactly, what remained was the gap between <em>issued</em> amounts and that chain, which <a href="#sec-decompose" class="quarto-xref">Section&nbsp;5</a> takes up. The corrections were global rule fixes, not case-specific accommodations; still, the exactness result is post-correction on the same case set, a reuse the certification protocol mitigates but does not eliminate. A confirmatory run on the fiscal 2025 file at its release is the committed next test.</p>
+<p>Holding tolerance at zero is what made the exercise informative. Early runs did not match; reaching parity surfaced two defects in the encodings under test (a regulatory dollar literal superseded by statute, and a missing whole-dollar rounding step — both fixed in the public repository, with the oracle as the detector). Once the encodings reproduced the file’s chain exactly, what remained was the gap between <em>issued</em> amounts and that chain, which <a href="#sec-decompose" class="quarto-xref">Section&nbsp;5</a> takes up. The corrections were global rule fixes, not case-specific accommodations; still, the exactness result is post-correction on the same case set, a reuse the certification protocol mitigates but does not eliminate. A confirmatory run on the fiscal 2025 file at its release is the committed next test.</p>
 <p>The toolchain is certified for reproduction: a pinned engine build and encoding-repository commit reproduce the Colorado suite — 856 cases, 5,136 cells including the 856 benefit cells — under recorded hashes for every input, with an independent audit rebuilding the engine to a byte-identical binary and re-running the comparison to byte-identical evidence. Throughput is 230 cases per second end to end on a laptop, so verification at this scale is not compute-bound for this workload; details and timing calibration are in the repository’s certification report.</p>
 </section>
 </section>
@@ -474,7 +474,7 @@ <h2 class="anchored" data-anchor-id="sec-decompose">Decomposing measured error:
 </tr>
 <tr class="odd">
 <td>3. Engine replay</td>
-<td>A public solver moves the input named by each case’s first finding until its recomputed benefit reaches the issued amount; the verified engine replays the solver’s inputs</td>
+<td>A public solver moves the input named by each case’s first finding toward the issued amount; the verified engine replays the solver’s inputs</td>
 <td>Not identified. 78.4% of above-threshold official errors and 91.4% of sub-threshold deviations (86.9% blended) reproduce the issued benefit within $5, consistent with a wrong input; of the 37 that do not, the solver moved nothing in 17 and 10 carry a computational finding (8 are both)</td>
 <td>283 filtered Colorado deviation cases (97 above threshold, 186 below); case shares, not dollar shares</td>
 </tr>
@@ -483,7 +483,7 @@ <h2 class="anchored" data-anchor-id="sec-decompose">Decomposing measured error:
 </div>
 </figure>
 </div>
-<p>The third layer adapts the public reconstruction solver of <span class="citation" data-cites="giannella_snapqc">Giannella and Molin (<a href="#ref-giannella_snapqc" role="doc-biblioref">2026</a>)</span> to fiscal 2024 and replays its output through the verified engine. The solver starts from the file’s edited inputs and moves only the input named by the case’s first finding (<code>ELEMENT1</code>): earned or unearned income, rent, the utility allowance, or the medical, dependent care or child-support deduction in $3 steps, or household size by one person (element 150 with natures 7, 12, 14 or 16). It stops when its own recomputed benefit reaches or passes the issued benefit (<code>RAWBEN</code>), when the input hits zero or a cap, when the benefit reaches $0, or after 1,000 steps. Each case it labels as a utility error then has its utility amount set to the nearest value above (or below) the file’s utility amount that more than five filtered cases in the same state and calendar year use, or to zero if none lies below; in 2 cases this moves an amount the steps left unchanged. When the first finding falls outside those element and nature codes, nothing moves. Of 283 deviation cases surviving the solver’s consistency filters (22 of 305 are excluded), 246 reproduce the issued amount within the file’s $5 editing tolerance, which is consistent with correct arithmetic applied to a wrong input. In 230 of the 246 the solver moved an input; in the other 16 the issued benefit was already within $5 of <code>FSBEN</code>, so nothing moved and the match restates the parity result. The rate differs by threshold status: among the 97 above-threshold official error cases 76 reproduce (78.4%, binomial standard error about 4 points), and among the 186 sub-threshold deviations 170 do (91.4%). The solver’s own recomputation and the engine agree on the within-$5 classification for all 283 cases, two implementations with distinct codebases reading the same file.</p>
+<p>The third layer adapts the public reconstruction solver of <span class="citation" data-cites="giannella_snapqc">Giannella and Molin (<a href="#ref-giannella_snapqc" role="doc-biblioref">2026</a>)</span> to fiscal 2024 and replays its output through the verified engine. The solver starts from the file’s edited inputs and moves only the input named by the case’s first finding (<code>ELEMENT1</code>). Household size moves once, by one person (element 150 with natures 7, 12, 14 or 16). Earned or unearned income, rent, the utility allowance, and the medical, dependent care or child-support deduction move in $3 steps toward the issued benefit (<code>RAWBEN</code>) until a stop rule binds, and a stepped utility amount is then reset to a commonly reported value.<a href="#fn3" class="footnote-ref" id="fnref3" role="doc-noteref"><sup>3</sup></a> When the first finding falls outside those element and nature codes, nothing moves. Of 283 deviation cases surviving the solver’s consistency filters (22 of 305 are excluded), 246 reproduce the issued amount within the file’s $5 editing tolerance, which is consistent with correct arithmetic applied to a wrong input. In 230 of the 246 the solver moved an input; in the other 16 it took no step, and the issued benefit was already within $5 of <code>FSBEN</code>, so the match restates the parity result. The rate differs by threshold status: among the 97 above-threshold official error cases 76 reproduce (78.4%, binomial standard error about 4 points), and among the 186 sub-threshold deviations 170 do (91.4%). The solver’s own recomputation and the engine agree on the within-$5 classification for all 283 cases, two implementations with distinct codebases reading the same file.</p>
 <p>Neither outcome identifies a computation error. A reproduced case shows that one value of the first-listed input, run through correct arithmetic, yields the issued benefit; it does not show which input the agency had wrong, and a moved input can absorb a computation error. A non-reproduced case can reflect the solver’s limits: it tries one input in one direction, and in 17 of the 37 it moved nothing (14 cases whose first finding falls outside those codes, and 3 where it stopped before moving). Of the 26 cases that carry a layer-2 computational finding, 13 reproduce, 10 do not and 3 were not replayed. The complete per-case replay output (283 rows), the layer-1 and layer-2 classification outputs, the reconstruction scripts and <code>audit_claims.py</code>, which regenerates the replay counts into <code>claims_audit.json</code>, are committed with this paper.</p>
 <p>Layers 1 and 2 agree on the structure: most measured SNAP payment error among eligible cases is information failure, not calculation failure — consistent with the administrative-burden literature’s account of where compliance costs bind <span class="citation" data-cites="herd2018burden">(<a href="#ref-herd2018burden" role="doc-biblioref">Herd and Moynihan 2018</a>)</span>. Correct-computation guarantees address the smaller share directly — roughly 3 to 10% of error dollars by layers 1–2 — plus whatever share of information failures better tooling prevents indirectly through verification prompts and documentation-requirement computation at intake; <a href="#sec-adoption" class="quarto-xref">Section&nbsp;7.2</a> prices the direct shares through the formula’s tiers, elections, and delay clause. Ineligibility errors, excluded from the file, are predominantly facts-driven, which would push the information-failure share higher still; that inference, unlike the in-file shares, is not directly computed here. The indirect channel is where policy design enters: rules determine what a caseworker must collect and verify, which is the mechanism the modeling section takes up.</p>
 </section>
@@ -757,6 +757,7 @@ <h2 class="anchored" data-anchor-id="acknowledgments">Acknowledgments</h2>
 <ol>
 <li id="fn1"><p>7 U.S.C. § 2013(a)(2)(B)(iii). The test applies mechanically to each year’s rate and is independent of the state’s election. The fiscal 2025 rates published in June 2026 settle the first leg: seven jurisdictions — Alaska (23.15%), the District of Columbia, Delaware, Georgia, Illinois, New Mexico, and Oregon — crossed on fiscal 2025 and owe nothing in FY2028, with no bill before FY2029 — keyed to the fiscal 2026 rate unless that rate also crosses, which pushes the start to FY2030 and is nearly certain for Alaska. Fiscal 2024 rates would have qualified ten, including New York (14.09%); New York’s fiscal 2025 rate of 13.18% misses the 13.33% threshold, so its delay now rides on the fiscal 2026 measurement. Any projection of FY2028 outlays must model the delay, and the simulator does: delayed years enter expected bills as zero.<a href="#fnref1" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 <li id="fn2"><p>The published thresholds — $48, $54, $56, $57, $58 for fiscal 2022–26 — are uniquely reproduced by flooring the indexed value, which puts the fiscal 2027 threshold at an estimated $59: $59.58 unrounded on the May 2026 Thrifty Food Plan cost of $1,018.20, with $60 requiring a June figure of at least $1,025.40. The estimate hardens when the June figure publishes.<a href="#fnref2" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
+<li id="fn3"><p>Income steps stop once the solver’s recomputed benefit passes <code>RAWBEN</code>; rent, utility and deduction steps also stop within $3 of it. Steps that lower an input stop at zero; rent and utility steps also stop when the shelter deduction reaches its cap (when raising) or zero (when lowering); income-raising steps stop when the uncapped benefit falls below zero; and every step stops when the benefit reaches $0 or after 1,000 steps. Where <code>RAWBEN</code> is the maximum allotment, which the benefit cannot pass, income steps continue until the income reaches zero or the step limit. Each case whose first finding is the utility allowance and whose <code>RAWBEN</code> differs from <code>FSBEN</code> then has its utility amount reset among the amounts that more than five filtered cases of any review status in the same state and calendar year report: those above the file’s amount when <code>RAWBEN</code> exceeds <code>FSBEN</code>, those below it when <code>RAWBEN</code> is lower. The solver takes the candidate nearest its stepped amount, keeps the stepped amount if none lies above, and sets zero if none lies below; in 2 cases this moves an amount the steps left unchanged (<code>reconstruct_co_fy2024.R</code>).<a href="#fnref3" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main>
 <!-- /main column -->
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zM6m{zVBWoL#7vcRh!A||gyreX+dOt0-ioKwyNl~W4#!_?2FTl6uB(pR{4m;CzxpgU
zM0U;$P6EE3;6_Q=h&aFk|7Zb)PNmK+>L4av$E%&Xj$dW}30mfaikoN0F*bIB{xKW3
zkDl)9mh-Q#bImppPXd1$=vipZ<gaj7Y9ADR`<E~6yjJd53gpp@5-F!n`4*%ey)3q=
zMHJ)l3e#{KC_)_RkzUi%pxY@#sB$tFr=-nW@!-flAs&i@chY(#!BiORTpXMb1_o(W
H8OVPD=WM`;

diff --git a/paper/FACTS.md b/paper/FACTS.md
index cc473a4..3df315a 100644
--- a/paper/FACTS.md
+++ b/paper/FACTS.md
@@ -28,7 +28,7 @@ Numbers from the superseded pre-audit pipeline (before PR #7) are banned.
 
 | # | Fact | Source |
 |---|---|---|
-| C1 | Seven states' encoded benefit computations reproduce the QC file's Minimodel chain exactly at zero tolerance for every in-scope case: 6,081 of 6,194 official-universe cases (CO 856/856, NY 847/885, CA 883/883, AZ 922/925, GA 945/945, MD 722/745, TX 906/955; 113 exclusions are enumerated program-structure classes); six asserted values per case | axiom-oracles PRs #244, #268, #269; committed suite reports; per-state suite configs ("in-scope") |
+| C1 | Seven states' encoded benefit computations reproduce the QC file's computed benefit chain exactly at zero tolerance for every in-scope case: 6,081 of 6,194 official-universe cases (CO 856/856, NY 847/885, CA 883/883, AZ 922/925, GA 945/945, MD 722/745, TX 906/955; 113 exclusions are enumerated program-structure classes); six asserted values per case | axiom-oracles PRs #244, #268, #269; committed suite reports; per-state suite configs ("in-scope") |
 | C2 | Scope caveat: the comparison bridge supplies several QC-derived intermediates as inputs (QC-calculated medical and child-support deductions, QC utility amount, QC categorical-eligibility status), so parity certifies the downstream benefit arithmetic given those intermediates, not independent derivation of every state-policy path | axiom-oracles bridge (snap_qc_compare.py); engine-leg recon 2026-08-06 |
 | C3 | Reaching parity surfaced and fixed two defects in the encodings themselves: a stale regulatory dollar literal superseded by statute, and a missing whole-dollar rounding step in the benefit computation | rulespec-us PRs (COLA backfill; #826 rounding); axiom-oracles #268 history |
 | C4 | WITHDRAWN 2026-10-03: the claim that the process surfaced errata in FNS's FY2024 technical documentation. No source records such errata: neither version of paper/snapshot/labs/amterr/ANALYSIS.md (27e1b09, c4333b3), the axiom.org Colorado report (original and corrected text) nor the axiom-oracles SNAP QC playbook documents one, GitHub issue searches across TheAxiomFoundation and PolicyEngine found none, and round-1 and round-2 referees could not locate the "reported upstream" thread. The manuscript no longer makes the claim (abstract, introduction, @sec-oracle-scope) | searched 2026-10-03; paper/reviews/round-1/referee-redteam.md, round-2/round2-rounddiff.md |
@@ -43,7 +43,7 @@ Numbers from the superseded pre-audit pipeline (before PR #7) are banned.
 | D1 | Colorado FY2024: 856 sampled cases, 305 with payment errors (STATUS 2/3); $112.6M/yr weighted error dollars on $1.268B issuance (8.88% file-derived; official regression-adjusted rate 9.97% = 7.91 over + 2.06 under) | paper/snapshot/labs/amterr/ANALYSIS.md; FNS FY2024 PER (https://fns-prod.azureedge.us/sites/default/files/resource-files/snap-fy24QC-PER.pdf) |
 | D2 | QC cause coding, SUPERSEDED by K1 (revision 5): the lab tabulation denominated over all weighted deviation dollars ($8.1B national incl. sub-threshold) and gave CO $8.2M/7.3% strict, $11.9M/10.5% broad; national $320M/3.9%, $1.5B/18.4%. The committed artifact denominates over official error dollars and differs (see K1) | paper/snapshot/labs/amterr/native_decomposition.json + ANALYSIS.md (archived) |
 | D3 | Finding-nature classification: pure-math 3.3%, input-with-system-cause 6.6%, mixed 4.2%, other input 86.0% of Colorado weighted deviation dollars. NOW COMMITTED as K3 (revision 5): the artifact reproduces these to the cent under the corrected field semantics (NATUREi carries nature codes; E_FINDGi is impact-only, a lab labeling error that did not affect the numbers) | analysis/cause_shares.json layer 2 (was: paper/snapshot/labs/phase_a_classification.json) |
-| D4 | Replay: the public solver (FY2024 adaptation of giannella/snap_qc) starts from the file's edited inputs and moves only the ELEMENT1 input ($3 steps; household size by one person for element 150 with natures 7, 12, 14 or 16) until its own recomputed benefit reaches or passes RAWBEN, the input hits zero or a cap, the benefit reaches $0, or 1,000 steps. Each case labeled a utility error then has its UTIL set to the nearest value above (or below) the file's UTIL that more than five filtered cases in the state and calendar year use, or to zero if none lies below; in 2 cases (202403-40765, 202404-40803) this moves an amount the steps left unchanged. It uses no disclosure-perturbation structure. The verified engine on the solver's inputs reproduces RAWBEN within $5 for 246 of 283 filtered error cases (86.9%), consistent with correct arithmetic on a wrong input: 230 after the solver moved an input, 16 with nothing moved (\|RAWBEN − FSBEN\| ≤ $5 already, so the match restates parity). 37 do not reproduce; in 17 nothing moved (14 `no_change` rows whose first finding falls outside the solver's element and nature codes, 2 of them element 150 with nature 13; 3 where it stopped before moving), and 10 carry a layer-2 computational finding (8 of them among the 17). Solver and engine agree on all 283. Neither class identifies computation errors: of 26 cases with a layer-2 computational finding, 13 reproduce, 10 do not, 3 were not replayed. SUPERSEDES the revision-9 wording "explains 246/283 … as correct arithmetic on wrong facts" and "37 cases form the computation-side upper bound" (withdrawn 2026-10-03, matching ANALYSIS.md's revision of the same date) | paper/snapshot/labs/amterr/claims_audit.json case_level.replay (regenerated by audit_claims.py); amterr_replay_results.json (283 rows); reconstruct_co_fy2024.R; amterr_replay.py; ANALYSIS.md layer 3 |
+| D4 | Replay: the public solver (FY2024 adaptation of giannella/snap_qc) starts from the file's edited inputs and moves only the ELEMENT1 input. Household size moves once, by one person, for element 150 with natures 7, 12, 14 or 16. Income, rent, utility and deduction inputs move in $3 steps toward RAWBEN: income steps stop once the recomputed benefit passes RAWBEN, the others also within $3 of it; steps also stop when a lowered input reaches zero, when the shelter deduction reaches its cap (raising) or zero (lowering), when the uncapped benefit falls below zero as income rises, when the benefit reaches $0, or after 1,000 steps. Where RAWBEN is the maximum allotment, income steps run until the income reaches zero or the step limit. Each case whose first finding is the utility allowance and whose RAWBEN differs from FSBEN then has its utility input reset to the candidate nearest its stepped amount, among amounts that more than five filtered cases of any review status in the state and calendar year report (above the file's UTIL when RAWBEN > FSBEN, below it when RAWBEN < FSBEN; the stepped amount is kept if none lies above, zero if none lies below); in 2 cases (202403-40765, 202404-40803) this moves an amount the steps left unchanged. It uses no disclosure-perturbation structure. The verified engine on the solver's inputs reproduces RAWBEN within $5 for 246 of 283 filtered error cases (86.9%), consistent with correct arithmetic on a wrong input: 230 after the solver moved an input, 16 with nothing moved (\|RAWBEN − FSBEN\| ≤ $5 already, so the match restates parity). 37 do not reproduce; in 17 nothing moved (14 `no_change` rows whose first finding falls outside the solver's element and nature codes, 2 of them element 150 with nature 13; 3 where it stopped before moving), and 10 carry a layer-2 computational finding (8 of them among the 17). Solver and engine agree on all 283. Neither class identifies computation errors: of 26 cases with a layer-2 computational finding, 13 reproduce, 10 do not, 3 were not replayed. SUPERSEDES the revision-9 wording "explains 246/283 … as correct arithmetic on wrong facts" and "37 cases form the computation-side upper bound" (withdrawn 2026-10-03, matching ANALYSIS.md's revision of the same date) | paper/snapshot/labs/amterr/claims_audit.json case_level.replay (regenerated by audit_claims.py); amterr_replay_results.json (283 rows); reconstruct_co_fy2024.R; amterr_replay.py; ANALYSIS.md layer 3 |
 
 ## E. Error models (corrected pipeline: PRs #7–#8 as refit by #13)
 
diff --git a/paper/index.qmd b/paper/index.qmd
index 57708b7..fb7b726 100644
--- a/paper/index.qmd
+++ b/paper/index.qmd
@@ -23,7 +23,7 @@ abstract: |
   per case, zero dollar tolerance — a process that surfaced and fixed two
   defects in the encodings. For 283 of the 305 Colorado cases with a
   recorded payment deviation, a public solver moves the input named by each case's first finding, where
-  it can, until its recomputed benefit reaches the issued amount; run on
+  it can, toward the issued amount; run on
   those inputs, the verified engine reproduces the issued benefit within
   $5 for 78.4% of the above-threshold official error cases and 91.4% of
   the sub-threshold deviations, which is consistent with correct
@@ -393,7 +393,7 @@ engine can compute counterfactual intermediates under alternative policy
 target; they are the subject of @sec-decompose.
 
 Holding tolerance at zero is what made the exercise informative. Early
-runs did not match, and each mismatch class traced to a defect in the
+runs did not match; reaching parity surfaced two defects in the
 encodings under test (a regulatory dollar literal superseded by statute,
 and a missing whole-dollar rounding step — both fixed in the public
 repository, with the oracle as the detector). Once the encodings
@@ -435,7 +435,7 @@ construction of the file, that ineligible-case error.
 |---|---|---|---|
 | 1. Cause codes | QC agency-cause codes naming the computing apparatus (programming, arithmetic, mass change), any-presence across a case's nine cause slots, share of official error dollars (HWGT × `AMTERR`) | 7.4% strict ($6.9M/yr); 10.6% ($10.0M/yr) adding policy-misapplied, budgeted-wrong, computer-user | Colorado; nationally 3.5% strict ($232M/yr), 19.3% broad ($1.27B of $6.59B) |
 | 2. Finding natures | `NATURE` codes inherently describing computation (rounding, conversion, wrong standard, benefit miscomputed) plus the arithmetic `ELEMENT`; deduction-amount natures count only when the paired cause is in layer 1's broad computing-apparatus set | 3.3% pure-computation; 6.6% system-caused input; 4.2% mixed; 86.0% other input | Colorado weighted deviation dollars; nationally 4.6% / 11.1% / 6.5% / 77.7% |
-| 3. Engine replay | A public solver moves the input named by each case's first finding until its recomputed benefit reaches the issued amount; the verified engine replays the solver's inputs | Not identified. 78.4% of above-threshold official errors and 91.4% of sub-threshold deviations (86.9% blended) reproduce the issued benefit within $5, consistent with a wrong input; of the 37 that do not, the solver moved nothing in 17 and 10 carry a computational finding (8 are both) | 283 filtered Colorado deviation cases (97 above threshold, 186 below); case shares, not dollar shares |
+| 3. Engine replay | A public solver moves the input named by each case's first finding toward the issued amount; the verified engine replays the solver's inputs | Not identified. 78.4% of above-threshold official errors and 91.4% of sub-threshold deviations (86.9% blended) reproduce the issued benefit within $5, consistent with a wrong input; of the 37 that do not, the solver moved nothing in 17 and 10 carry a computational finding (8 are both) | 283 filtered Colorado deviation cases (97 above threshold, 186 below); case shares, not dollar shares |
 
 : Three decompositions of Colorado fiscal 2024 measured error.
 Layers 1–2 quote the committed cause-share artifact
@@ -456,24 +456,20 @@ to the finding-code sets. {#tbl-decompose}
 The third layer adapts the public reconstruction solver of
 @giannella_snapqc to fiscal 2024 and replays its output through the
 verified engine. The solver starts from the file's edited inputs and
-moves only the input named by the case's first finding (`ELEMENT1`):
-earned or unearned income, rent, the utility allowance, or the medical,
-dependent care or child-support deduction in $3 steps, or household size
-by one person (element 150 with natures 7, 12, 14 or 16). It stops when
-its own recomputed benefit reaches or passes the issued benefit
-(`RAWBEN`), when the input hits zero or a cap, when the benefit reaches
-$0, or after 1,000 steps. Each case it labels as a utility error then has
-its utility amount set to the nearest value above (or below) the file's
-utility amount that more than five filtered cases in the same state and
-calendar year use, or to zero if none lies below; in 2 cases this moves
-an amount the steps left unchanged. When the first finding falls outside
-those element and nature codes, nothing moves. Of 283 deviation cases
-surviving the solver's consistency filters (22 of 305 are excluded), 246
-reproduce the issued amount within
-the file's $5 editing tolerance, which is consistent with correct
-arithmetic applied to a wrong input. In 230 of the 246 the solver moved an
-input; in the other 16 the issued benefit was already within $5 of
-`FSBEN`, so nothing moved and the match restates the parity result. The
+moves only the input named by the case's first finding (`ELEMENT1`).
+Household size moves once, by one person (element 150 with natures 7,
+12, 14 or 16). Earned or unearned income, rent, the utility allowance,
+and the medical, dependent care or child-support deduction move in $3
+steps toward the issued benefit (`RAWBEN`) until a stop rule binds, and
+a stepped utility amount is then reset to a commonly reported
+value.[^solver-rules] When the first finding falls outside those element
+and nature codes, nothing moves. Of 283 deviation cases surviving the
+solver's consistency filters (22 of 305 are excluded), 246 reproduce the
+issued amount within the file's $5 editing tolerance, which is
+consistent with correct arithmetic applied to a wrong input. In 230 of
+the 246 the solver moved an input; in the other 16 it took no step, and
+the issued benefit was already within $5 of `FSBEN`, so the match
+restates the parity result. The
 rate differs by threshold status: among the 97 above-threshold official
 error cases 76 reproduce (78.4%, binomial standard error about 4
 points), and among the 186 sub-threshold deviations 170 do (91.4%). The
@@ -495,6 +491,25 @@ classification outputs, the reconstruction scripts and
 `audit_claims.py`, which regenerates the replay counts into
 `claims_audit.json`, are committed with this paper.
 
+[^solver-rules]: Income steps stop once the solver's recomputed benefit
+    passes `RAWBEN`; rent, utility and deduction steps also stop within
+    $3 of it. Steps that lower an input stop at zero; rent and utility
+    steps also stop when the shelter deduction reaches its cap (when
+    raising) or zero (when lowering); income-raising steps stop when the
+    uncapped benefit falls below zero; and every step stops when the
+    benefit reaches $0 or after 1,000 steps. Where `RAWBEN` is the
+    maximum allotment, which the benefit cannot pass, income steps
+    continue until the income reaches zero or the step limit. Each case
+    whose first finding is the utility allowance and whose `RAWBEN`
+    differs from `FSBEN` then has its utility amount reset among the
+    amounts that more than five filtered cases of any review status in
+    the same state and calendar year report: those above the file's
+    amount when `RAWBEN` exceeds `FSBEN`, those below it when `RAWBEN` is
+    lower. The solver takes the candidate nearest its stepped amount,
+    keeps the stepped amount if none lies above, and sets zero if none
+    lies below; in 2 cases this moves an amount the steps left unchanged
+    (`reconstruct_co_fy2024.R`).
+
 Layers 1 and 2 agree on the structure: most measured SNAP payment error among eligible cases is
 information failure, not calculation failure — consistent with the
 administrative-burden literature's account of where compliance costs
diff --git a/tests/test_retired_claims.py b/tests/test_retired_claims.py
index 10fb4eb..bcafb32 100644
--- a/tests/test_retired_claims.py
+++ b/tests/test_retired_claims.py
@@ -72,15 +72,26 @@
 )
 
 
-def _normalize(text: str, *, is_html: bool = False) -> str:
-    """Collapse whitespace and typographic apostrophes; for HTML, drop tags."""
+def _normalize(text: str, *, is_html: bool = False, is_js: bool = False) -> str:
+    """Collapse whitespace and typographic apostrophes.
+
+    For HTML, drop tags. For JavaScript, rejoin string and template-literal
+    concatenations broken across source lines, so a phrase split over two
+    lines still reads as one.
+    """
     if is_html:
         text = html.unescape(re.sub(r"<[^>]+>", " ", text))
+    if is_js:
+        text = re.sub(r"[`\"']\s*\+\s*[`\"']", "", text)
     return " ".join(text.replace("’", "'").split())
 
 
 def _read(path: Path) -> str:
-    return _normalize(path.read_text(encoding="utf-8"), is_html=path.suffix == ".html")
+    return _normalize(
+        path.read_text(encoding="utf-8"),
+        is_html=path.suffix == ".html",
+        is_js=path.suffix == ".js",
+    )
 
 
 def _percent(numerator: int, denominator: int) -> str:
@@ -114,6 +125,18 @@ def test_retired_wording_is_gone_from_the_pdf():
     assert not found, f"index.pdf still carries {found}"
 
 
+def test_js_scan_sees_phrases_split_across_concatenated_lines():
+    """The retired app.js sentence was split over a template-literal join."""
+    retired_js = (
+        "`the allotment corrected for the reviewer's findings (BENFIX), and the "
+        "Minimodel's full-formula ` +\n    `recomputation (FSBEN). |RAWBEN - "
+        "BENFIX| equals`"
+    )
+    text = _normalize(retired_js, is_js=True).lower()
+    assert "minimodel's full-formula recomputation" in text
+    assert "minimodel's full-formula recomputation" in RETIRED
+
+
 def test_facts_records_each_withdrawal():
     """The catalog keeps the retired wording only as a superseded record."""
     facts = _read(FACTS)
@@ -202,8 +225,8 @@ def replay_quotes(c: dict[str, int]) -> list[str]:
         ),
         (
             f"In {c['moved']} of the {c['reproduced']} the solver moved an input; in "
-            f"the other {c['unmoved']} the issued benefit was already within $5 of "
-            "`FSBEN`, so nothing moved and the match restates the parity result."
+            f"the other {c['unmoved']} it took no step, and the issued benefit was "
+            "already within $5 of `FSBEN`, so the match restates the parity result."
         ),
         (
             f"among the {c['above_n']} above-threshold official error cases "

From 326a381c1af96326f3a7c9f08c167fc45d766b84 Mon Sep 17 00:00:00 2001
From: Max Ghenis <mghenis@gmail.com>
Date: Sun, 4 Oct 2026 09:43:30 -0400
Subject: [PATCH 5/9] Address review round 4 on the retired-claim corrections

- Abstract: the solver "changes" the named input; "toward the issued
  amount" was false for household-size moves, whose direction comes from
  the nature code (2 of 7 Colorado moves go away from RAWBEN). The table
  cell keeps "toward" for the $3-stepped inputs only, and the body says
  which natures remove or add a person.
- FACTS D4: the shelter-deduction stop rule applies to rent and utility
  steps only, matching the footnote.
- Footnote and D4: the maximum-allotment sentence is about
  income-lowering steps; "all steps stop" replaces "every step stops".

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
---
 app/public/paper/web/index.html |  16 ++++++++--------
 app/public/paper/web/index.pdf  | Bin 213994 -> 214134 bytes
 paper/FACTS.md                  |   2 +-
 paper/index.qmd                 |  20 ++++++++++----------
 4 files changed, 19 insertions(+), 19 deletions(-)

diff --git a/app/public/paper/web/index.html b/app/public/paper/web/index.html
index 071055f..92e6d0a 100644
--- a/app/public/paper/web/index.html
+++ b/app/public/paper/web/index.html
@@ -107,10 +107,10 @@
 official universe (113 documented exclusions), six computation stages
 per case, zero dollar tolerance — a process that surfaced and fixed two
 defects in the encodings. For 283 of the 305 Colorado cases with a
-recorded payment deviation, a public solver moves the input named by each case's first finding, where
-it can, toward the issued amount; run on
-those inputs, the verified engine reproduces the issued benefit within
-$5 for 78.4% of the above-threshold official error cases and 91.4% of
+recorded payment deviation, a public solver changes the input named by
+each case's first finding, where it can; run on those inputs, the
+verified engine reproduces the issued benefit within $5 for 78.4% of
+the above-threshold official error cases and 91.4% of
 the sub-threshold deviations, which is consistent with correct
 arithmetic on a wrong input. Pricing a cause-code decomposition through
 the formula's tiers, elections, and delay clause — removing the error
@@ -218,7 +218,7 @@ <h1 class="title">Administrative quality control data as an oracle: verifying en
     <div>
       <div class="abstract">
         <div class="block-title">Abstract</div>
-        <p>Beginning in fiscal year 2028, each state’s Supplemental Nutrition Assistance Program (SNAP) payment error rate sets its share of benefit costs, in steps of five percentage points at rates of 6, 8, and 10 percent. A measurement system built for oversight now moves large sums, which raises two questions this paper treats together: whether the rules that produce benefit amounts can be executed correctly, and how the errors that remain should be measured and simulated. I use the USDA Quality Control (QC) public-use files as a verification oracle for independently encoded SNAP rules: across seven states, the encodings reproduce the file’s benefit-computation chain, from file-recorded intermediates, exactly — all 6,081 in-scope cases of the 6,194-case official universe (113 documented exclusions), six computation stages per case, zero dollar tolerance — a process that surfaced and fixed two defects in the encodings. For 283 of the 305 Colorado cases with a recorded payment deviation, a public solver moves the input named by each case’s first finding, where it can, toward the issued amount; run on those inputs, the verified engine reproduces the issued benefit within $5 for 78.4% of the above-threshold official error cases and 91.4% of the sub-threshold deviations, which is consistent with correct arithmetic on a wrong input. Pricing a cause-code decomposition through the formula’s tiers, elections, and delay clause — removing the error dollars the file’s cause codes assign to the computing apparatus, an accounting scenario rather than a causal estimate — moves summed expected fiscal 2028 bills from $7.7B to $7.6B a year under a strict class convention and $6.9B under a broad one, and not monotonically: a state just under the statute’s delay threshold sees its near-term expected bill rise as its measured rate falls. I also fit and validate a distributional model of case-level payment deviations, reporting its validation failures alongside its results, and serve one policy counterfactual from it in the deployed simulator: a standard medical deduction scenario computed on engine-recomputed case parameters, whose model-implied Colorado effect opposes the naive accounting bound in sign. The central finding, from an open Monte Carlo simulator resampling each state’s own QC cases, is about measurement: at regulatory sample sizes, several states face nearly even odds — 46 to 50 percent — of being assigned a different cost tier than their official point rate implies. The fiscal 2025 rates, published in June 2026, land where that mechanism says realized years should: 18 of 53 jurisdictions changed cost-sharing tiers in one year, and only seven of those changes exceed the 95% band that two years of sampling noise — estimated from fiscal 2024 microdata alone — implies.</p>
+        <p>Beginning in fiscal year 2028, each state’s Supplemental Nutrition Assistance Program (SNAP) payment error rate sets its share of benefit costs, in steps of five percentage points at rates of 6, 8, and 10 percent. A measurement system built for oversight now moves large sums, which raises two questions this paper treats together: whether the rules that produce benefit amounts can be executed correctly, and how the errors that remain should be measured and simulated. I use the USDA Quality Control (QC) public-use files as a verification oracle for independently encoded SNAP rules: across seven states, the encodings reproduce the file’s benefit-computation chain, from file-recorded intermediates, exactly — all 6,081 in-scope cases of the 6,194-case official universe (113 documented exclusions), six computation stages per case, zero dollar tolerance — a process that surfaced and fixed two defects in the encodings. For 283 of the 305 Colorado cases with a recorded payment deviation, a public solver changes the input named by each case’s first finding, where it can; run on those inputs, the verified engine reproduces the issued benefit within $5 for 78.4% of the above-threshold official error cases and 91.4% of the sub-threshold deviations, which is consistent with correct arithmetic on a wrong input. Pricing a cause-code decomposition through the formula’s tiers, elections, and delay clause — removing the error dollars the file’s cause codes assign to the computing apparatus, an accounting scenario rather than a causal estimate — moves summed expected fiscal 2028 bills from $7.7B to $7.6B a year under a strict class convention and $6.9B under a broad one, and not monotonically: a state just under the statute’s delay threshold sees its near-term expected bill rise as its measured rate falls. I also fit and validate a distributional model of case-level payment deviations, reporting its validation failures alongside its results, and serve one policy counterfactual from it in the deployed simulator: a standard medical deduction scenario computed on engine-recomputed case parameters, whose model-implied Colorado effect opposes the naive accounting bound in sign. The central finding, from an open Monte Carlo simulator resampling each state’s own QC cases, is about measurement: at regulatory sample sizes, several states face nearly even odds — 46 to 50 percent — of being assigned a different cost tier than their official point rate implies. The fiscal 2025 rates, published in June 2026, land where that mechanism says realized years should: 18 of 53 jurisdictions changed cost-sharing tiers in one year, and only seven of those changes exceed the 95% band that two years of sampling noise — estimated from fiscal 2024 microdata alone — implies.</p>
       </div>
     </div>
 
@@ -474,7 +474,7 @@ <h2 class="anchored" data-anchor-id="sec-decompose">Decomposing measured error:
 </tr>
 <tr class="odd">
 <td>3. Engine replay</td>
-<td>A public solver moves the input named by each case’s first finding toward the issued amount; the verified engine replays the solver’s inputs</td>
+<td>A public solver changes the input named by each case’s first finding (household size by one person; other inputs in $3 steps toward the issued amount); the verified engine replays the solver’s inputs</td>
 <td>Not identified. 78.4% of above-threshold official errors and 91.4% of sub-threshold deviations (86.9% blended) reproduce the issued benefit within $5, consistent with a wrong input; of the 37 that do not, the solver moved nothing in 17 and 10 carry a computational finding (8 are both)</td>
 <td>283 filtered Colorado deviation cases (97 above threshold, 186 below); case shares, not dollar shares</td>
 </tr>
@@ -483,7 +483,7 @@ <h2 class="anchored" data-anchor-id="sec-decompose">Decomposing measured error:
 </div>
 </figure>
 </div>
-<p>The third layer adapts the public reconstruction solver of <span class="citation" data-cites="giannella_snapqc">Giannella and Molin (<a href="#ref-giannella_snapqc" role="doc-biblioref">2026</a>)</span> to fiscal 2024 and replays its output through the verified engine. The solver starts from the file’s edited inputs and moves only the input named by the case’s first finding (<code>ELEMENT1</code>). Household size moves once, by one person (element 150 with natures 7, 12, 14 or 16). Earned or unearned income, rent, the utility allowance, and the medical, dependent care or child-support deduction move in $3 steps toward the issued benefit (<code>RAWBEN</code>) until a stop rule binds, and a stepped utility amount is then reset to a commonly reported value.<a href="#fn3" class="footnote-ref" id="fnref3" role="doc-noteref"><sup>3</sup></a> When the first finding falls outside those element and nature codes, nothing moves. Of 283 deviation cases surviving the solver’s consistency filters (22 of 305 are excluded), 246 reproduce the issued amount within the file’s $5 editing tolerance, which is consistent with correct arithmetic applied to a wrong input. In 230 of the 246 the solver moved an input; in the other 16 it took no step, and the issued benefit was already within $5 of <code>FSBEN</code>, so the match restates the parity result. The rate differs by threshold status: among the 97 above-threshold official error cases 76 reproduce (78.4%, binomial standard error about 4 points), and among the 186 sub-threshold deviations 170 do (91.4%). The solver’s own recomputation and the engine agree on the within-$5 classification for all 283 cases, two implementations with distinct codebases reading the same file.</p>
+<p>The third layer adapts the public reconstruction solver of <span class="citation" data-cites="giannella_snapqc">Giannella and Molin (<a href="#ref-giannella_snapqc" role="doc-biblioref">2026</a>)</span> to fiscal 2024 and replays its output through the verified engine. The solver starts from the file’s edited inputs and moves only the input named by the case’s first finding (<code>ELEMENT1</code>). Household size moves once, by one person: one fewer for natures 12, 14 or 16 and one more for nature 7 (element 150). Earned or unearned income, rent, the utility allowance, and the medical, dependent care or child-support deduction move in $3 steps toward the issued benefit (<code>RAWBEN</code>) until a stop rule binds, and a stepped utility amount is then reset to a commonly reported value.<a href="#fn3" class="footnote-ref" id="fnref3" role="doc-noteref"><sup>3</sup></a> When the first finding falls outside those element and nature codes, nothing moves. Of 283 deviation cases surviving the solver’s consistency filters (22 of 305 are excluded), 246 reproduce the issued amount within the file’s $5 editing tolerance, which is consistent with correct arithmetic applied to a wrong input. In 230 of the 246 the solver moved an input; in the other 16 it took no step, and the issued benefit was already within $5 of <code>FSBEN</code>, so the match restates the parity result. The rate differs by threshold status: among the 97 above-threshold official error cases 76 reproduce (78.4%, binomial standard error about 4 points), and among the 186 sub-threshold deviations 170 do (91.4%). The solver’s own recomputation and the engine agree on the within-$5 classification for all 283 cases, two implementations with distinct codebases reading the same file.</p>
 <p>Neither outcome identifies a computation error. A reproduced case shows that one value of the first-listed input, run through correct arithmetic, yields the issued benefit; it does not show which input the agency had wrong, and a moved input can absorb a computation error. A non-reproduced case can reflect the solver’s limits: it tries one input in one direction, and in 17 of the 37 it moved nothing (14 cases whose first finding falls outside those codes, and 3 where it stopped before moving). Of the 26 cases that carry a layer-2 computational finding, 13 reproduce, 10 do not and 3 were not replayed. The complete per-case replay output (283 rows), the layer-1 and layer-2 classification outputs, the reconstruction scripts and <code>audit_claims.py</code>, which regenerates the replay counts into <code>claims_audit.json</code>, are committed with this paper.</p>
 <p>Layers 1 and 2 agree on the structure: most measured SNAP payment error among eligible cases is information failure, not calculation failure — consistent with the administrative-burden literature’s account of where compliance costs bind <span class="citation" data-cites="herd2018burden">(<a href="#ref-herd2018burden" role="doc-biblioref">Herd and Moynihan 2018</a>)</span>. Correct-computation guarantees address the smaller share directly — roughly 3 to 10% of error dollars by layers 1–2 — plus whatever share of information failures better tooling prevents indirectly through verification prompts and documentation-requirement computation at intake; <a href="#sec-adoption" class="quarto-xref">Section&nbsp;7.2</a> prices the direct shares through the formula’s tiers, elections, and delay clause. Ineligibility errors, excluded from the file, are predominantly facts-driven, which would push the information-failure share higher still; that inference, unlike the in-file shares, is not directly computed here. The indirect channel is where policy design enters: rules determine what a caseworker must collect and verify, which is the mechanism the modeling section takes up.</p>
 </section>
@@ -757,7 +757,7 @@ <h2 class="anchored" data-anchor-id="acknowledgments">Acknowledgments</h2>
 <ol>
 <li id="fn1"><p>7 U.S.C. § 2013(a)(2)(B)(iii). The test applies mechanically to each year’s rate and is independent of the state’s election. The fiscal 2025 rates published in June 2026 settle the first leg: seven jurisdictions — Alaska (23.15%), the District of Columbia, Delaware, Georgia, Illinois, New Mexico, and Oregon — crossed on fiscal 2025 and owe nothing in FY2028, with no bill before FY2029 — keyed to the fiscal 2026 rate unless that rate also crosses, which pushes the start to FY2030 and is nearly certain for Alaska. Fiscal 2024 rates would have qualified ten, including New York (14.09%); New York’s fiscal 2025 rate of 13.18% misses the 13.33% threshold, so its delay now rides on the fiscal 2026 measurement. Any projection of FY2028 outlays must model the delay, and the simulator does: delayed years enter expected bills as zero.<a href="#fnref1" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 <li id="fn2"><p>The published thresholds — $48, $54, $56, $57, $58 for fiscal 2022–26 — are uniquely reproduced by flooring the indexed value, which puts the fiscal 2027 threshold at an estimated $59: $59.58 unrounded on the May 2026 Thrifty Food Plan cost of $1,018.20, with $60 requiring a June figure of at least $1,025.40. The estimate hardens when the June figure publishes.<a href="#fnref2" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
-<li id="fn3"><p>Income steps stop once the solver’s recomputed benefit passes <code>RAWBEN</code>; rent, utility and deduction steps also stop within $3 of it. Steps that lower an input stop at zero; rent and utility steps also stop when the shelter deduction reaches its cap (when raising) or zero (when lowering); income-raising steps stop when the uncapped benefit falls below zero; and every step stops when the benefit reaches $0 or after 1,000 steps. Where <code>RAWBEN</code> is the maximum allotment, which the benefit cannot pass, income steps continue until the income reaches zero or the step limit. Each case whose first finding is the utility allowance and whose <code>RAWBEN</code> differs from <code>FSBEN</code> then has its utility amount reset among the amounts that more than five filtered cases of any review status in the same state and calendar year report: those above the file’s amount when <code>RAWBEN</code> exceeds <code>FSBEN</code>, those below it when <code>RAWBEN</code> is lower. The solver takes the candidate nearest its stepped amount, keeps the stepped amount if none lies above, and sets zero if none lies below; in 2 cases this moves an amount the steps left unchanged (<code>reconstruct_co_fy2024.R</code>).<a href="#fnref3" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
+<li id="fn3"><p>Income steps stop once the solver’s recomputed benefit passes <code>RAWBEN</code>; rent, utility and deduction steps also stop within $3 of it. Steps that lower an input stop at zero; rent and utility steps also stop when the shelter deduction reaches its cap (when raising) or zero (when lowering); income-raising steps stop when the uncapped benefit falls below zero; and all steps stop when the benefit reaches $0 or after 1,000 steps. Where <code>RAWBEN</code> is the maximum allotment, which the benefit cannot pass, income-lowering steps continue until the income reaches zero or the step limit. Each case whose first finding is the utility allowance and whose <code>RAWBEN</code> differs from <code>FSBEN</code> then has its utility amount reset among the amounts that more than five filtered cases of any review status in the same state and calendar year report: those above the file’s amount when <code>RAWBEN</code> exceeds <code>FSBEN</code>, those below it when <code>RAWBEN</code> is lower. The solver takes the candidate nearest its stepped amount, keeps the stepped amount if none lies above, and sets zero if none lies below; in 2 cases this moves an amount the steps left unchanged (<code>reconstruct_co_fy2024.R</code>).<a href="#fnref3" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main>
 <!-- /main column -->
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diff --git a/paper/FACTS.md b/paper/FACTS.md
index 3df315a..d1015d7 100644
--- a/paper/FACTS.md
+++ b/paper/FACTS.md
@@ -43,7 +43,7 @@ Numbers from the superseded pre-audit pipeline (before PR #7) are banned.
 | D1 | Colorado FY2024: 856 sampled cases, 305 with payment errors (STATUS 2/3); $112.6M/yr weighted error dollars on $1.268B issuance (8.88% file-derived; official regression-adjusted rate 9.97% = 7.91 over + 2.06 under) | paper/snapshot/labs/amterr/ANALYSIS.md; FNS FY2024 PER (https://fns-prod.azureedge.us/sites/default/files/resource-files/snap-fy24QC-PER.pdf) |
 | D2 | QC cause coding, SUPERSEDED by K1 (revision 5): the lab tabulation denominated over all weighted deviation dollars ($8.1B national incl. sub-threshold) and gave CO $8.2M/7.3% strict, $11.9M/10.5% broad; national $320M/3.9%, $1.5B/18.4%. The committed artifact denominates over official error dollars and differs (see K1) | paper/snapshot/labs/amterr/native_decomposition.json + ANALYSIS.md (archived) |
 | D3 | Finding-nature classification: pure-math 3.3%, input-with-system-cause 6.6%, mixed 4.2%, other input 86.0% of Colorado weighted deviation dollars. NOW COMMITTED as K3 (revision 5): the artifact reproduces these to the cent under the corrected field semantics (NATUREi carries nature codes; E_FINDGi is impact-only, a lab labeling error that did not affect the numbers) | analysis/cause_shares.json layer 2 (was: paper/snapshot/labs/phase_a_classification.json) |
-| D4 | Replay: the public solver (FY2024 adaptation of giannella/snap_qc) starts from the file's edited inputs and moves only the ELEMENT1 input. Household size moves once, by one person, for element 150 with natures 7, 12, 14 or 16. Income, rent, utility and deduction inputs move in $3 steps toward RAWBEN: income steps stop once the recomputed benefit passes RAWBEN, the others also within $3 of it; steps also stop when a lowered input reaches zero, when the shelter deduction reaches its cap (raising) or zero (lowering), when the uncapped benefit falls below zero as income rises, when the benefit reaches $0, or after 1,000 steps. Where RAWBEN is the maximum allotment, income steps run until the income reaches zero or the step limit. Each case whose first finding is the utility allowance and whose RAWBEN differs from FSBEN then has its utility input reset to the candidate nearest its stepped amount, among amounts that more than five filtered cases of any review status in the state and calendar year report (above the file's UTIL when RAWBEN > FSBEN, below it when RAWBEN < FSBEN; the stepped amount is kept if none lies above, zero if none lies below); in 2 cases (202403-40765, 202404-40803) this moves an amount the steps left unchanged. It uses no disclosure-perturbation structure. The verified engine on the solver's inputs reproduces RAWBEN within $5 for 246 of 283 filtered error cases (86.9%), consistent with correct arithmetic on a wrong input: 230 after the solver moved an input, 16 with nothing moved (\|RAWBEN − FSBEN\| ≤ $5 already, so the match restates parity). 37 do not reproduce; in 17 nothing moved (14 `no_change` rows whose first finding falls outside the solver's element and nature codes, 2 of them element 150 with nature 13; 3 where it stopped before moving), and 10 carry a layer-2 computational finding (8 of them among the 17). Solver and engine agree on all 283. Neither class identifies computation errors: of 26 cases with a layer-2 computational finding, 13 reproduce, 10 do not, 3 were not replayed. SUPERSEDES the revision-9 wording "explains 246/283 … as correct arithmetic on wrong facts" and "37 cases form the computation-side upper bound" (withdrawn 2026-10-03, matching ANALYSIS.md's revision of the same date) | paper/snapshot/labs/amterr/claims_audit.json case_level.replay (regenerated by audit_claims.py); amterr_replay_results.json (283 rows); reconstruct_co_fy2024.R; amterr_replay.py; ANALYSIS.md layer 3 |
+| D4 | Replay: the public solver (FY2024 adaptation of giannella/snap_qc) starts from the file's edited inputs and moves only the ELEMENT1 input. Household size moves once, by one person, for element 150: one fewer for natures 12, 14 or 16, one more for nature 7 (the nature code sets the direction, so the benefit can move away from RAWBEN). Income, rent, utility and deduction inputs move in $3 steps toward RAWBEN: income steps stop once the recomputed benefit passes RAWBEN, the others also within $3 of it; steps that lower an input stop at zero; rent and utility steps also stop when the shelter deduction reaches its cap (raising) or zero (lowering); income-raising steps stop when the uncapped benefit falls below zero; and all steps stop when the benefit reaches $0 or after 1,000 steps. Where RAWBEN is the maximum allotment, income-lowering steps run until the income reaches zero or the step limit. Each case whose first finding is the utility allowance and whose RAWBEN differs from FSBEN then has its utility input reset to the candidate nearest its stepped amount, among amounts that more than five filtered cases of any review status in the state and calendar year report (above the file's UTIL when RAWBEN > FSBEN, below it when RAWBEN < FSBEN; the stepped amount is kept if none lies above, zero if none lies below); in 2 cases (202403-40765, 202404-40803) this moves an amount the steps left unchanged. It uses no disclosure-perturbation structure. The verified engine on the solver's inputs reproduces RAWBEN within $5 for 246 of 283 filtered error cases (86.9%), consistent with correct arithmetic on a wrong input: 230 after the solver moved an input, 16 with nothing moved (\|RAWBEN − FSBEN\| ≤ $5 already, so the match restates parity). 37 do not reproduce; in 17 nothing moved (14 `no_change` rows whose first finding falls outside the solver's element and nature codes, 2 of them element 150 with nature 13; 3 where it stopped before moving), and 10 carry a layer-2 computational finding (8 of them among the 17). Solver and engine agree on all 283. Neither class identifies computation errors: of 26 cases with a layer-2 computational finding, 13 reproduce, 10 do not, 3 were not replayed. SUPERSEDES the revision-9 wording "explains 246/283 … as correct arithmetic on wrong facts" and "37 cases form the computation-side upper bound" (withdrawn 2026-10-03, matching ANALYSIS.md's revision of the same date) | paper/snapshot/labs/amterr/claims_audit.json case_level.replay (regenerated by audit_claims.py); amterr_replay_results.json (283 rows); reconstruct_co_fy2024.R; amterr_replay.py; ANALYSIS.md layer 3 |
 
 ## E. Error models (corrected pipeline: PRs #7–#8 as refit by #13)
 
diff --git a/paper/index.qmd b/paper/index.qmd
index fb7b726..804751a 100644
--- a/paper/index.qmd
+++ b/paper/index.qmd
@@ -22,10 +22,10 @@ abstract: |
   official universe (113 documented exclusions), six computation stages
   per case, zero dollar tolerance — a process that surfaced and fixed two
   defects in the encodings. For 283 of the 305 Colorado cases with a
-  recorded payment deviation, a public solver moves the input named by each case's first finding, where
-  it can, toward the issued amount; run on
-  those inputs, the verified engine reproduces the issued benefit within
-  $5 for 78.4% of the above-threshold official error cases and 91.4% of
+  recorded payment deviation, a public solver changes the input named by
+  each case's first finding, where it can; run on those inputs, the
+  verified engine reproduces the issued benefit within $5 for 78.4% of
+  the above-threshold official error cases and 91.4% of
   the sub-threshold deviations, which is consistent with correct
   arithmetic on a wrong input. Pricing a cause-code decomposition through
   the formula's tiers, elections, and delay clause — removing the error
@@ -435,7 +435,7 @@ construction of the file, that ineligible-case error.
 |---|---|---|---|
 | 1. Cause codes | QC agency-cause codes naming the computing apparatus (programming, arithmetic, mass change), any-presence across a case's nine cause slots, share of official error dollars (HWGT × `AMTERR`) | 7.4% strict ($6.9M/yr); 10.6% ($10.0M/yr) adding policy-misapplied, budgeted-wrong, computer-user | Colorado; nationally 3.5% strict ($232M/yr), 19.3% broad ($1.27B of $6.59B) |
 | 2. Finding natures | `NATURE` codes inherently describing computation (rounding, conversion, wrong standard, benefit miscomputed) plus the arithmetic `ELEMENT`; deduction-amount natures count only when the paired cause is in layer 1's broad computing-apparatus set | 3.3% pure-computation; 6.6% system-caused input; 4.2% mixed; 86.0% other input | Colorado weighted deviation dollars; nationally 4.6% / 11.1% / 6.5% / 77.7% |
-| 3. Engine replay | A public solver moves the input named by each case's first finding toward the issued amount; the verified engine replays the solver's inputs | Not identified. 78.4% of above-threshold official errors and 91.4% of sub-threshold deviations (86.9% blended) reproduce the issued benefit within $5, consistent with a wrong input; of the 37 that do not, the solver moved nothing in 17 and 10 carry a computational finding (8 are both) | 283 filtered Colorado deviation cases (97 above threshold, 186 below); case shares, not dollar shares |
+| 3. Engine replay | A public solver changes the input named by each case's first finding (household size by one person; other inputs in $3 steps toward the issued amount); the verified engine replays the solver's inputs | Not identified. 78.4% of above-threshold official errors and 91.4% of sub-threshold deviations (86.9% blended) reproduce the issued benefit within $5, consistent with a wrong input; of the 37 that do not, the solver moved nothing in 17 and 10 carry a computational finding (8 are both) | 283 filtered Colorado deviation cases (97 above threshold, 186 below); case shares, not dollar shares |
 
 : Three decompositions of Colorado fiscal 2024 measured error.
 Layers 1–2 quote the committed cause-share artifact
@@ -457,8 +457,8 @@ The third layer adapts the public reconstruction solver of
 @giannella_snapqc to fiscal 2024 and replays its output through the
 verified engine. The solver starts from the file's edited inputs and
 moves only the input named by the case's first finding (`ELEMENT1`).
-Household size moves once, by one person (element 150 with natures 7,
-12, 14 or 16). Earned or unearned income, rent, the utility allowance,
+Household size moves once, by one person: one fewer for natures 12, 14
+or 16 and one more for nature 7 (element 150). Earned or unearned income, rent, the utility allowance,
 and the medical, dependent care or child-support deduction move in $3
 steps toward the issued benefit (`RAWBEN`) until a stop rule binds, and
 a stepped utility amount is then reset to a commonly reported
@@ -496,10 +496,10 @@ classification outputs, the reconstruction scripts and
     $3 of it. Steps that lower an input stop at zero; rent and utility
     steps also stop when the shelter deduction reaches its cap (when
     raising) or zero (when lowering); income-raising steps stop when the
-    uncapped benefit falls below zero; and every step stops when the
+    uncapped benefit falls below zero; and all steps stop when the
     benefit reaches $0 or after 1,000 steps. Where `RAWBEN` is the
-    maximum allotment, which the benefit cannot pass, income steps
-    continue until the income reaches zero or the step limit. Each case
+    maximum allotment, which the benefit cannot pass, income-lowering
+    steps continue until the income reaches zero or the step limit. Each case
     whose first finding is the utility allowance and whose `RAWBEN`
     differs from `FSBEN` then has its utility amount reset among the
     amounts that more than five filtered cases of any review status in

From 696e4f9647a993d540eaba5ed94fe72dfbbbc864 Mon Sep 17 00:00:00 2001
From: Max Ghenis <mghenis@gmail.com>
Date: Sun, 4 Oct 2026 10:25:20 -0400
Subject: [PATCH 6/9] Address review round 5: drop the mechanism clause from
 the layer-3 cell

The cell's parenthetical ("other inputs in $3 steps toward the issued
amount") left out the utility reset, which sets the final input in all 15
Colorado utility cases. The cell now says only that the solver changes
the named input where it can, as the abstract does; the body and its
footnote carry the mechanism.

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
---
 app/public/paper/web/index.html |   2 +-
 app/public/paper/web/index.pdf  | Bin 214134 -> 214079 bytes
 paper/index.qmd                 |   2 +-
 3 files changed, 2 insertions(+), 2 deletions(-)

diff --git a/app/public/paper/web/index.html b/app/public/paper/web/index.html
index 92e6d0a..f64962a 100644
--- a/app/public/paper/web/index.html
+++ b/app/public/paper/web/index.html
@@ -474,7 +474,7 @@ <h2 class="anchored" data-anchor-id="sec-decompose">Decomposing measured error:
 </tr>
 <tr class="odd">
 <td>3. Engine replay</td>
-<td>A public solver changes the input named by each case’s first finding (household size by one person; other inputs in $3 steps toward the issued amount); the verified engine replays the solver’s inputs</td>
+<td>A public solver changes the input named by each case’s first finding, where it can; the verified engine replays the solver’s inputs</td>
 <td>Not identified. 78.4% of above-threshold official errors and 91.4% of sub-threshold deviations (86.9% blended) reproduce the issued benefit within $5, consistent with a wrong input; of the 37 that do not, the solver moved nothing in 17 and 10 carry a computational finding (8 are both)</td>
 <td>283 filtered Colorado deviation cases (97 above threshold, 186 below); case shares, not dollar shares</td>
 </tr>
diff --git a/app/public/paper/web/index.pdf b/app/public/paper/web/index.pdf
index 74525710cdeeceba2d69c033fd5c4d66e93c575a..bc44d486bb6c983746297943d2b3190b5e38f374 100644
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diff --git a/paper/index.qmd b/paper/index.qmd
index 804751a..fd11ff4 100644
--- a/paper/index.qmd
+++ b/paper/index.qmd
@@ -435,7 +435,7 @@ construction of the file, that ineligible-case error.
 |---|---|---|---|
 | 1. Cause codes | QC agency-cause codes naming the computing apparatus (programming, arithmetic, mass change), any-presence across a case's nine cause slots, share of official error dollars (HWGT × `AMTERR`) | 7.4% strict ($6.9M/yr); 10.6% ($10.0M/yr) adding policy-misapplied, budgeted-wrong, computer-user | Colorado; nationally 3.5% strict ($232M/yr), 19.3% broad ($1.27B of $6.59B) |
 | 2. Finding natures | `NATURE` codes inherently describing computation (rounding, conversion, wrong standard, benefit miscomputed) plus the arithmetic `ELEMENT`; deduction-amount natures count only when the paired cause is in layer 1's broad computing-apparatus set | 3.3% pure-computation; 6.6% system-caused input; 4.2% mixed; 86.0% other input | Colorado weighted deviation dollars; nationally 4.6% / 11.1% / 6.5% / 77.7% |
-| 3. Engine replay | A public solver changes the input named by each case's first finding (household size by one person; other inputs in $3 steps toward the issued amount); the verified engine replays the solver's inputs | Not identified. 78.4% of above-threshold official errors and 91.4% of sub-threshold deviations (86.9% blended) reproduce the issued benefit within $5, consistent with a wrong input; of the 37 that do not, the solver moved nothing in 17 and 10 carry a computational finding (8 are both) | 283 filtered Colorado deviation cases (97 above threshold, 186 below); case shares, not dollar shares |
+| 3. Engine replay | A public solver changes the input named by each case's first finding, where it can; the verified engine replays the solver's inputs | Not identified. 78.4% of above-threshold official errors and 91.4% of sub-threshold deviations (86.9% blended) reproduce the issued benefit within $5, consistent with a wrong input; of the 37 that do not, the solver moved nothing in 17 and 10 carry a computational finding (8 are both) | 283 filtered Colorado deviation cases (97 above threshold, 186 below); case shares, not dollar shares |
 
 : Three decompositions of Colorado fiscal 2024 measured error.
 Layers 1–2 quote the committed cause-share artifact

From 34d15d83d53729524f8c964c34c9b08a589d3952 Mon Sep 17 00:00:00 2001
From: Max Ghenis <mghenis@gmail.com>
Date: Sun, 4 Oct 2026 11:43:25 -0400
Subject: [PATCH 7/9] Address review round 6 on the retired-claim corrections

- The old "33 within comparison tolerance mechanically" was right in
  substance: all 33, and 2 more with AMTERR of $6, started with RAWBEN
  within $5 of FSBEN, so none of those 35 matches needed a move. The
  solver took no step in 16 and moved an input anyway in 19. The paper
  now says so for the 19, and FACTS H8 retires only the word "explained".
- "yields a benefit within $5 of the issued benefit" (only 51 of 246
  matches are exact).
- 797 of 856: 21 of the other 59 are prorated allotments (ALLADJ 2),
  which the full-month FSBEN does not reflect; FACTS C7 mirrors it.
- Intro: the replay tests "changing the input named by each case's first
  finding"; limitations: the solver moves "at most one" input; the table
  caption matches the section's "three layers of evidence".
- references.bib: the tech doc is the August 2026 Mathematica report by
  Leftin et al., not an FNS 2025 publication.
- Tests lock the 35/16/19 split, the new sentence and the 21 prorated
  cases.

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
---
 app/public/paper/web/index.html |  32 ++++++++---------
 app/public/paper/web/index.pdf  | Bin 214079 -> 214102 bytes
 paper/FACTS.md                  |   4 +--
 paper/index.qmd                 |  17 +++++----
 paper/references.bib            |  10 ++++--
 tests/test_retired_claims.py    |  60 +++++++++++++++++++++++++++-----
 6 files changed, 87 insertions(+), 36 deletions(-)

diff --git a/app/public/paper/web/index.html b/app/public/paper/web/index.html
index f64962a..314e464 100644
--- a/app/public/paper/web/index.html
+++ b/app/public/paper/web/index.html
@@ -143,7 +143,7 @@
 <meta name="citation_year" content="2026">
 <meta name="citation_online_date" content="2026-10-03">
 <meta name="citation_language" content="en">
-<meta name="citation_reference" content="citation_title=Technical documentation for the fiscal year 2024 supplemental nutrition assistance program quality control database and the QC minimodel;,citation_author=U.S. Department of Agriculture, Food and Nutrition Service;,citation_publication_date=2025;,citation_cover_date=2025;,citation_year=2025;,citation_fulltext_html_url=https://snapqcdata.net/datafiles;">
+<meta name="citation_reference" content="citation_title=Technical documentation for the fiscal year 2024 supplemental nutrition assistance program quality control database and the QC minimodel;,citation_author=Joshua Leftin;,citation_author=Mia Monkovic;,citation_author=Francisco Yang;,citation_author=Sam Bonelli;,citation_author=John Carlo Maula;,citation_publication_date=2026-08;,citation_cover_date=2026-08;,citation_year=2026;,citation_fulltext_html_url=https://snapqcdata.net/datafiles;,citation_technical_report_institution=Mathematica, for the U.S. Department of Agriculture;">
 <meta name="citation_reference" content="citation_title=Supplemental nutrition assistance program payment error rates, fiscal year 2024;,citation_author=U.S. Department of Agriculture, Food and Nutrition Service;,citation_publication_date=2025;,citation_cover_date=2025;,citation_year=2025;,citation_fulltext_html_url=https://www.fns.usda.gov/snap/qc/per;">
 <meta name="citation_reference" content="citation_title=Snap_qc: Reconstruction and analysis code for the SNAP quality control public-use files;,citation_author=Eric Giannella;,citation_author=Ben Molin;,citation_publication_date=2026;,citation_cover_date=2026;,citation_year=2026;,citation_fulltext_html_url=https://github.com/giannella/snap_qc;">
 <meta name="citation_reference" content="citation_title=Smearing estimate: A nonparametric retransformation method;,citation_author=Naihua Duan;,citation_publication_date=1983;,citation_cover_date=1983;,citation_year=1983;,citation_fulltext_html_url=https://doi.org/10.1080/01621459.1983.10478017;,citation_issue=383;,citation_volume=78;,citation_journal_title=Journal of the American Statistical Association;">
@@ -268,10 +268,10 @@ <h2 id="toc-title">Table of contents</h2>
 <section id="introduction" class="level2">
 <h2 class="anchored" data-anchor-id="introduction">Introduction</h2>
 <p>Under 7 U.S.C. § 2013(a)(2), as amended by the One Big Beautiful Bill Act (Pub. L. No.&nbsp;119-21, 2025; OBBBA), a state whose SNAP payment error rate reaches 6% will pay 5% of benefit costs beginning in fiscal year 2028; at 8%, the share is 10%; at 10%, it is 15%. The rate that determines the FY2028 share is the state’s fiscal 2025 or 2026 rate, at its election; later years use the third preceding year. Implementation is delayed for the highest-error states: a state whose fiscal 2025 rate times 1.5 reaches 20% — that is, a rate above 13.33% (13.34% at the published two-decimal precision) — starts in FY2029 instead, and a state whose fiscal 2026 rate crosses the same test starts in FY2030.<a href="#fn1" class="footnote-ref" id="fnref1" role="doc-noteref"><sup>1</sup></a> For a mid-sized state, one tier is worth tens of millions of dollars a year; for the largest states, more than a hundred million.</p>
-<p>The error rate is estimated by the Quality Control system: each state re-reviews a probability sample of its own active cases — by formula in 7 C.F.R. § 275.11, between 300 and 2,400 annually, or 300 to 1,020 under an optional reduced schedule that every state elected in fiscal 2024 <span class="citation" data-cites="fns2024techdoc">(<a href="#ref-fns2024techdoc" role="doc-biblioref">U.S. Department of Agriculture, Food and Nutrition Service 2025b</a>)</span> — and federal reviewers re-review a subsample, from which the Food and Nutrition Service (FNS; renamed the Food and Nutrition Administration in 2026 — the fiscal 2025 rate publication carries the new letterhead <span class="citation" data-cites="fna2025per">(<a href="#ref-fna2025per" role="doc-biblioref">U.S. Department of Agriculture, Food and Nutrition Administration 2026</a>)</span>, and this paper keeps FNS, the name on every document the fiscal 2024 data cite) estimates official state rates with adjustments including federal re-review integration. The public-use files that result record, for roughly 45,000 cases a year, what the agency issued and what the review concluded the case should have received, with coded findings for every discrepancy.</p>
+<p>The error rate is estimated by the Quality Control system: each state re-reviews a probability sample of its own active cases — by formula in 7 C.F.R. § 275.11, between 300 and 2,400 annually, or 300 to 1,020 under an optional reduced schedule that every state elected in fiscal 2024 <span class="citation" data-cites="fns2024techdoc">(<a href="#ref-fns2024techdoc" role="doc-biblioref">Leftin et al. 2026</a>)</span> — and federal reviewers re-review a subsample, from which the Food and Nutrition Service (FNS; renamed the Food and Nutrition Administration in 2026 — the fiscal 2025 rate publication carries the new letterhead <span class="citation" data-cites="fna2025per">(<a href="#ref-fna2025per" role="doc-biblioref">U.S. Department of Agriculture, Food and Nutrition Administration 2026</a>)</span>, and this paper keeps FNS, the name on every document the fiscal 2024 data cite) estimates official state rates with adjustments including federal re-review integration. The public-use files that result record, for roughly 45,000 cases a year, what the agency issued and what the review concluded the case should have received, with coded findings for every discrepancy.</p>
 <p>This paper treats those files as a research instrument with three uses, the first in the software-testing sense of an <em>oracle</em>: an external source of ground-truth outputs for given inputs.</p>
 <p>First, as a verification target for rules as code. I test independently encoded SNAP rules — statute, regulations, and state policy manuals encoded as executable rule modules — against the QC file’s recorded benefit computations in seven states, holding the bar at exact reproduction: zero dollar tolerance across six computation stages per case (<a href="#sec-oracle" class="quarto-xref">Section&nbsp;4</a>). The target’s precise character matters and is stated in <a href="#sec-oracle-scope" class="quarto-xref">Section&nbsp;4.1</a>: the file’s benefit chain is a computation Mathematica runs for USDA on edited case records with the benefit formula of the FNS QC Minimodel, so parity certifies agreement with the benefit computation recorded in the file; it does not adjudicate case outcomes. Reaching it surfaced two defects in the encodings under test and left a residue in issued amounts, which <a href="#sec-decompose" class="quarto-xref">Section&nbsp;5</a> decomposes.</p>
-<p>Second, as raw material for decomposing measured error. QC’s own cause coding and a finding-nature classification give two estimates of how much of the error rate reflects computation failure versus information failure, and a replay of reconstructed case inputs through the verified engine tests whether one wrong input reproduces each issued benefit (<a href="#sec-decompose" class="quarto-xref">Section&nbsp;5</a>).</p>
+<p>Second, as raw material for decomposing measured error. QC’s own cause coding and a finding-nature classification give two estimates of how much of the error rate reflects computation failure versus information failure, and a replay of reconstructed case inputs through the verified engine tests whether changing the input named by each case’s first finding reproduces the issued benefit (<a href="#sec-decompose" class="quarto-xref">Section&nbsp;5</a>).</p>
 <p>Third, as training and validation data for a case-level model of the error process (<a href="#sec-model" class="quarto-xref">Section&nbsp;6</a>) and the basis for a distributional simulator of measured rates under the cost-sharing formula (<a href="#sec-simulate" class="quarto-xref">Section&nbsp;7</a>), public at policyengine.org/us/snap-payment-error-simulator. The simulator’s central output is a measurement result: near tier boundaries, sampling noise alone gives several states nearly even odds of a different cost tier than their point rate implies.</p>
 <p>The analysis pipeline, simulator, and this manuscript live in one public repository; a fact catalog maps every quantitative claim to a committed artifact (<a href="#sec-availability" class="quarto-xref">Section&nbsp;11</a>). The pipeline is deterministic — independent clean-room reruns reproduce every analysis artifact byte-for-byte, and the headline simulation artifact was independently reproduced from the raw file to four decimals — and the manuscript has been through six rounds of adversarial review, archived unedited in the repository.</p>
 </section>
@@ -284,7 +284,7 @@ <h2 class="anchored" data-anchor-id="sec-related">Related work</h2>
 </section>
 <section id="sec-qc" class="level2">
 <h2 class="anchored" data-anchor-id="sec-qc">The QC system and the money attached to it</h2>
-<p>QC reviews produce, for each sampled active case, the issued allotment and a corrected allotment constructed from the reviewer’s findings; when they differ by more than a tolerance threshold ($56 in fiscal 2024, indexed from a $37 statutory base under 7 U.S.C. § 2025(c)),<a href="#fn2" class="footnote-ref" id="fnref2" role="doc-noteref"><sup>2</sup></a> the case is a payment error, and findings are coded for sub-threshold deviations as well. <a href="#tbl-codebook" class="quarto-xref">Table&nbsp;1</a> lists the public-use file variables this paper works with; variable names in the text refer to these columns. The official state rate combines over- and underpayment rates <span class="citation" data-cites="fns2024per">(<a href="#ref-fns2024per" role="doc-biblioref">U.S. Department of Agriculture, Food and Nutrition Service 2025a</a>)</span>: Colorado’s 9.97% is 7.91 over plus 2.06 under, and Maryland’s 13.64 includes the nation’s highest underpayment rate, 4.79 — so cost sharing partly bills states for underpaying their own residents.</p>
+<p>QC reviews produce, for each sampled active case, the issued allotment and a corrected allotment constructed from the reviewer’s findings; when they differ by more than a tolerance threshold ($56 in fiscal 2024, indexed from a $37 statutory base under 7 U.S.C. § 2025(c)),<a href="#fn2" class="footnote-ref" id="fnref2" role="doc-noteref"><sup>2</sup></a> the case is a payment error, and findings are coded for sub-threshold deviations as well. <a href="#tbl-codebook" class="quarto-xref">Table&nbsp;1</a> lists the public-use file variables this paper works with; variable names in the text refer to these columns. The official state rate combines over- and underpayment rates <span class="citation" data-cites="fns2024per">(<a href="#ref-fns2024per" role="doc-biblioref">U.S. Department of Agriculture, Food and Nutrition Service 2025</a>)</span>: Colorado’s 9.97% is 7.91 over plus 2.06 under, and Maryland’s 13.64 includes the nation’s highest underpayment rate, 4.79 — so cost sharing partly bills states for underpaying their own residents.</p>
 <div id="tbl-codebook" class="quarto-float quarto-figure quarto-figure-center anchored">
 <figure class="quarto-float quarto-float-tbl figure">
 <figcaption class="quarto-float-caption-top quarto-float-caption quarto-float-tbl" id="tbl-codebook-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
@@ -341,10 +341,10 @@ <h2 class="anchored" data-anchor-id="sec-qc">The QC system and the money attache
 </figure>
 </div>
 <p>Four features of the system frame everything that follows.</p>
-<p><strong>The public file is not the full review universe.</strong> The edited public-use file excludes cases with ineligible findings — for which the official methodology scores the <em>entire</em> issued benefit as error dollars — cases whose overissuance equals or exceeds the issued benefit, incomplete reviews, and internal-consistency drops <span class="citation" data-cites="fns2024techdoc">(<a href="#ref-fns2024techdoc" role="doc-biblioref">U.S. Department of Agriculture, Food and Nutrition Service 2025b</a>)</span>. In fiscal 2024 this removed 1,037 ineligible-finding and 406 full-overissuance cases nationally. The file is therefore truncated at the top of the error-magnitude distribution. Every file-based estimate in this paper inherits that truncation: it is part of why Colorado’s file-derived rate (8.88%) sits below its official 9.97%. Restoring the excluded cases would add a large between-component variance term — their per-case error contributions are the full issued benefit — so the truncation biases simulated sampling variability downward; the i.i.d. approximation to states’ month-stratified sampling designs plausibly biases it upward, so the tier-noise results of <a href="#sec-simulate" class="quarto-xref">Section&nbsp;7</a> are likely, though not certainly, understated.</p>
+<p><strong>The public file is not the full review universe.</strong> The edited public-use file excludes cases with ineligible findings — for which the official methodology scores the <em>entire</em> issued benefit as error dollars — cases whose overissuance equals or exceeds the issued benefit, incomplete reviews, and internal-consistency drops <span class="citation" data-cites="fns2024techdoc">(<a href="#ref-fns2024techdoc" role="doc-biblioref">Leftin et al. 2026</a>)</span>. In fiscal 2024 this removed 1,037 ineligible-finding and 406 full-overissuance cases nationally. The file is therefore truncated at the top of the error-magnitude distribution. Every file-based estimate in this paper inherits that truncation: it is part of why Colorado’s file-derived rate (8.88%) sits below its official 9.97%. Restoring the excluded cases would add a large between-component variance term — their per-case error contributions are the full issued benefit — so the truncation biases simulated sampling variability downward; the i.i.d. approximation to states’ month-stratified sampling designs plausibly biases it upward, so the tier-noise results of <a href="#sec-simulate" class="quarto-xref">Section&nbsp;7</a> are likely, though not certainly, understated.</p>
 <p><strong>The measured rate is a property of the measurement system.</strong> Between roughly 2009 and 2016, several states used consultant-driven error-review practices that biased QC findings downward; the Department of Justice recovered more than $67 million in False Claims Act settlements from eight states over 2017–2021 <span class="citation" data-cites="doj2021snapqc">(<a href="#ref-doj2021snapqc" role="doc-biblioref">U.S. Department of Justice 2021</a>)</span>, FNS issued formal anti-bias guidance, and national rates for fiscal 2015–16 were not published <span class="citation" data-cites="crs2018errors">(<a href="#ref-crs2018errors" role="doc-biblioref">Congressional Research Service 2018</a>)</span>. This history is why the present analysis trains no earlier than fiscal 2017 — and why the fiscal 2025–27 rates now being measured, which carry orders of magnitude more money than the settlements-era rates did, cannot be treated as exogenous to the incentive they price.</p>
 <p><strong>Only active cases are priced.</strong> The QC system also reviews negative actions — denials, terminations, suspensions — feeding a separate case-and-procedure error rate that OBBBA does not price. A state can therefore lower its priced error rate by tightening the front door, shifting error into the unpriced metric. This paper analyzes the priced, active-case system; the asymmetry belongs on any list of the formula’s incentive properties.</p>
-<p><strong>Precision was contractually waived.</strong> A state electing the reduced sampling schedule agrees not to dispute later error-rate findings on the basis of the precision of the estimates <span class="citation" data-cites="fns2024techdoc">(<a href="#ref-fns2024techdoc" role="doc-biblioref">U.S. Department of Agriculture, Food and Nutrition Service 2025b</a>)</span>. All 53 jurisdictions had so elected as of fiscal 2024 — a defensible economy when QC reviews were an administrative cost with no tier attached, and a striking one now that precision determines tier placement worth tens of millions of dollars. The cost calculus has since moved <em>against</em> precision even as its value rose: OBBBA cut the federal share of SNAP administrative costs — QC review staffing included — from 50% to 25% beginning fiscal 2027 (Pub. L. No.&nbsp;119-21 § 10106), so each additional review now costs a state three dollars where it used to cost two. Whether states revert to the standard schedule — regaining precision and dispute rights at that higher marginal price — is a live sampling-plan question, and exactly the one <a href="#sec-simulate" class="quarto-xref">Section&nbsp;7</a> prices with its additional-reviews lever.</p>
+<p><strong>Precision was contractually waived.</strong> A state electing the reduced sampling schedule agrees not to dispute later error-rate findings on the basis of the precision of the estimates <span class="citation" data-cites="fns2024techdoc">(<a href="#ref-fns2024techdoc" role="doc-biblioref">Leftin et al. 2026</a>)</span>. All 53 jurisdictions had so elected as of fiscal 2024 — a defensible economy when QC reviews were an administrative cost with no tier attached, and a striking one now that precision determines tier placement worth tens of millions of dollars. The cost calculus has since moved <em>against</em> precision even as its value rose: OBBBA cut the federal share of SNAP administrative costs — QC review staffing included — from 50% to 25% beginning fiscal 2027 (Pub. L. No.&nbsp;119-21 § 10106), so each additional review now costs a state three dollars where it used to cost two. Whether states revert to the standard schedule — regaining precision and dispute rights at that higher marginal price — is a live sampling-plan question, and exactly the one <a href="#sec-simulate" class="quarto-xref">Section&nbsp;7</a> prices with its additional-reviews lever.</p>
 </section>
 <section id="sec-oracle" class="level2">
 <h2 class="anchored" data-anchor-id="sec-oracle">The QC file as a verification oracle</h2>
@@ -429,7 +429,7 @@ <h2 class="anchored" data-anchor-id="sec-oracle">The QC file as a verification o
 <section id="sec-oracle-scope" class="level3">
 <h3 class="anchored" data-anchor-id="sec-oracle-scope">What the parity target is</h3>
 <p>Exactness claims invite overreading, so the scope deserves a precise statement — more precise than an earlier draft of this paper gave it.</p>
-<p>The compared benefit chain is the one recorded in the edited public-use file. Its benefit variable (<code>FSBEN</code>) is a constructed variable: Mathematica, which edits the file for USDA under contract, computes it from each edited case record with the benefit formula of the QC Minimodel, one of FNS’s SNAP microsimulation models; the file’s other chain values (gross income, the standard and excess shelter deductions, net income, maximum allotment) are constructed variables too <span class="citation" data-cites="fns2024techdoc">(<a href="#ref-fns2024techdoc" role="doc-biblioref">U.S. Department of Agriculture, Food and Nutrition Service 2025b</a>)</span>. Where the calculated benefit is more than $5 from the issued benefit adjusted for the reviewer’s error amount (Step 13a lists the exact match conditions), the editing tries a documented sequence of changes (setting the dependent care deduction to dependent care costs, setting the utility amount in the excess shelter deduction to a standard allowance or zero, and zeroing the medical deduction of standard medical deduction demonstration participants) and keeps a change only if it produces a match <span class="citation" data-cites="fns2024techdoc">(<a href="#ref-fns2024techdoc" role="doc-biblioref">U.S. Department of Agriculture, Food and Nutrition Service 2025b, III</a>, Steps 12–13)</span>. The sequence does not always close the gap: in Colorado, <code>FSBEN</code> ends within $5 of <code>BENFIX</code> in 797 of 856 cases. The comparison also supplies several intermediates directly from the file rather than deriving them from raw facts: the QC-calculated medical and child-support deduction amounts, the recorded utility allowance, and the recorded categorical-eligibility status; eligibility screens receive passing defaults, since every filed case was enrolled.</p>
+<p>The compared benefit chain is the one recorded in the edited public-use file. Its benefit variable (<code>FSBEN</code>) is a constructed variable: Mathematica, which edits the file for USDA under contract, computes it from each edited case record with the benefit formula of the QC Minimodel, one of FNS’s SNAP microsimulation models; the file’s other chain values (gross income, the standard and excess shelter deductions, net income, maximum allotment) are constructed variables too <span class="citation" data-cites="fns2024techdoc">(<a href="#ref-fns2024techdoc" role="doc-biblioref">Leftin et al. 2026</a>)</span>. Where the calculated benefit is more than $5 from the issued benefit adjusted for the reviewer’s error amount (Step 13a lists the exact match conditions), the editing tries a documented sequence of changes (setting the dependent care deduction to dependent care costs, setting the utility amount in the excess shelter deduction to a standard allowance or zero, and zeroing the medical deduction of standard medical deduction demonstration participants) and keeps a change only if it produces a match <span class="citation" data-cites="fns2024techdoc">(<a href="#ref-fns2024techdoc" role="doc-biblioref">Leftin et al. 2026, III</a>, Steps 12–13)</span>. The sequence does not always close the gap: in Colorado, <code>FSBEN</code> ends within $5 of <code>BENFIX</code> in 797 of 856 cases; 21 of the other 59 are prorated allotments, which the full-month <code>FSBEN</code> does not reflect. The comparison also supplies several intermediates directly from the file rather than deriving them from raw facts: the QC-calculated medical and child-support deduction amounts, the recorded utility allowance, and the recorded categorical-eligibility status; eligibility screens receive passing defaults, since every filed case was enrolled.</p>
 <p>Parity therefore certifies the following: that the encoded computation chain — income aggregation, deduction sequencing, net-income arithmetic, allotment lookup, rounding — agrees exactly with the benefit computation recorded in the edited file, over the full diversity of 6,081 real case configurations, parameter values, and boundary conditions. It is an agreement between two implementations of the same rules, closer in kind to the cross-model comparisons this method extends than to independent adjudication of case outcomes; what distinguishes it from ordinary cross-model comparison is the counterparty (a computation Mathematica runs for USDA on the edited case records), the case realism, and the zero-tolerance bar. It does not certify independent derivation of every state eligibility and deduction pathway from raw facts. Extending the harness to feed raw facts — recorded medical expenses rather than the computed deduction, energy-assistance facts rather than the utility tier — is the necessary next step before the engine can compute counterfactual intermediates under alternative policy (<a href="#sec-roadmap" class="quarto-xref">Section&nbsp;8</a>). The reviewer-adjudicated quantities in the file (<code>RAWBEN</code>, <code>BENFIX</code>, error amounts and findings) are not the parity target; they are the subject of <a href="#sec-decompose" class="quarto-xref">Section&nbsp;5</a>.</p>
 <p>Holding tolerance at zero is what made the exercise informative. Early runs did not match; reaching parity surfaced two defects in the encodings under test (a regulatory dollar literal superseded by statute, and a missing whole-dollar rounding step — both fixed in the public repository, with the oracle as the detector). Once the encodings reproduced the file’s chain exactly, what remained was the gap between <em>issued</em> amounts and that chain, which <a href="#sec-decompose" class="quarto-xref">Section&nbsp;5</a> takes up. The corrections were global rule fixes, not case-specific accommodations; still, the exactness result is post-correction on the same case set, a reuse the certification protocol mitigates but does not eliminate. A confirmatory run on the fiscal 2025 file at its release is the committed next test.</p>
 <p>The toolchain is certified for reproduction: a pinned engine build and encoding-repository commit reproduce the Colorado suite — 856 cases, 5,136 cells including the 856 benefit cells — under recorded hashes for every input, with an independent audit rebuilding the engine to a byte-identical binary and re-running the comparison to byte-identical evidence. Throughput is 230 cases per second end to end on a laptop, so verification at this scale is not compute-bound for this workload; details and timing calibration are in the repository’s certification report.</p>
@@ -441,7 +441,7 @@ <h2 class="anchored" data-anchor-id="sec-decompose">Decomposing measured error:
 <div id="tbl-decompose" class="quarto-float quarto-figure quarto-figure-center anchored">
 <figure class="quarto-float quarto-float-tbl figure">
 <figcaption class="quarto-float-caption-top quarto-float-caption quarto-float-tbl" id="tbl-decompose-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
-Table&nbsp;3: Three decompositions of Colorado fiscal 2024 measured error. Layers 1–2 quote the committed cause-share artifact (<code>analysis/cause_shares.json</code>, all 53 jurisdictions, conventions and code-to-class maps serialized in the file); layer 3 is a share of filtered cases. Any-presence attribution counts a case’s dollars toward every cause class present in its slots, so layer-1 classes overlap. Layer 2 partitions cases; its system-side test reuses layer 1’s broad computing-apparatus cause set — the artifact also carries a wider agency-responsibility convention under which the system-side shares are several times larger (Colorado system-caused input 31% rather than 6.6%), answering a responsibility question rather than this table’s computation question. Cause and nature coding reflect state reviewer judgment under handbook guidance <span class="citation" data-cites="fns310handbook">(<a href="#ref-fns310handbook" role="doc-biblioref">U.S. Department of Agriculture, Food and Nutrition Service 2023</a>)</span> and carry cross-state coding variation; fiscal 2024 also brought minor revisions to the finding-code sets.
+Table&nbsp;3: Three layers of evidence on Colorado fiscal 2024 measured error. Layers 1–2 quote the committed cause-share artifact (<code>analysis/cause_shares.json</code>, all 53 jurisdictions, conventions and code-to-class maps serialized in the file); layer 3 is a share of filtered cases. Any-presence attribution counts a case’s dollars toward every cause class present in its slots, so layer-1 classes overlap. Layer 2 partitions cases; its system-side test reuses layer 1’s broad computing-apparatus cause set — the artifact also carries a wider agency-responsibility convention under which the system-side shares are several times larger (Colorado system-caused input 31% rather than 6.6%), answering a responsibility question rather than this table’s computation question. Cause and nature coding reflect state reviewer judgment under handbook guidance <span class="citation" data-cites="fns310handbook">(<a href="#ref-fns310handbook" role="doc-biblioref">U.S. Department of Agriculture, Food and Nutrition Service 2023</a>)</span> and carry cross-state coding variation; fiscal 2024 also brought minor revisions to the finding-code sets.
 </figcaption>
 <div aria-describedby="tbl-decompose-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
 <table class="caption-top table">
@@ -483,8 +483,8 @@ <h2 class="anchored" data-anchor-id="sec-decompose">Decomposing measured error:
 </div>
 </figure>
 </div>
-<p>The third layer adapts the public reconstruction solver of <span class="citation" data-cites="giannella_snapqc">Giannella and Molin (<a href="#ref-giannella_snapqc" role="doc-biblioref">2026</a>)</span> to fiscal 2024 and replays its output through the verified engine. The solver starts from the file’s edited inputs and moves only the input named by the case’s first finding (<code>ELEMENT1</code>). Household size moves once, by one person: one fewer for natures 12, 14 or 16 and one more for nature 7 (element 150). Earned or unearned income, rent, the utility allowance, and the medical, dependent care or child-support deduction move in $3 steps toward the issued benefit (<code>RAWBEN</code>) until a stop rule binds, and a stepped utility amount is then reset to a commonly reported value.<a href="#fn3" class="footnote-ref" id="fnref3" role="doc-noteref"><sup>3</sup></a> When the first finding falls outside those element and nature codes, nothing moves. Of 283 deviation cases surviving the solver’s consistency filters (22 of 305 are excluded), 246 reproduce the issued amount within the file’s $5 editing tolerance, which is consistent with correct arithmetic applied to a wrong input. In 230 of the 246 the solver moved an input; in the other 16 it took no step, and the issued benefit was already within $5 of <code>FSBEN</code>, so the match restates the parity result. The rate differs by threshold status: among the 97 above-threshold official error cases 76 reproduce (78.4%, binomial standard error about 4 points), and among the 186 sub-threshold deviations 170 do (91.4%). The solver’s own recomputation and the engine agree on the within-$5 classification for all 283 cases, two implementations with distinct codebases reading the same file.</p>
-<p>Neither outcome identifies a computation error. A reproduced case shows that one value of the first-listed input, run through correct arithmetic, yields the issued benefit; it does not show which input the agency had wrong, and a moved input can absorb a computation error. A non-reproduced case can reflect the solver’s limits: it tries one input in one direction, and in 17 of the 37 it moved nothing (14 cases whose first finding falls outside those codes, and 3 where it stopped before moving). Of the 26 cases that carry a layer-2 computational finding, 13 reproduce, 10 do not and 3 were not replayed. The complete per-case replay output (283 rows), the layer-1 and layer-2 classification outputs, the reconstruction scripts and <code>audit_claims.py</code>, which regenerates the replay counts into <code>claims_audit.json</code>, are committed with this paper.</p>
+<p>The third layer adapts the public reconstruction solver of <span class="citation" data-cites="giannella_snapqc">Giannella and Molin (<a href="#ref-giannella_snapqc" role="doc-biblioref">2026</a>)</span> to fiscal 2024 and replays its output through the verified engine. The solver starts from the file’s edited inputs and moves only the input named by the case’s first finding (<code>ELEMENT1</code>). Household size moves once, by one person: one fewer for natures 12, 14 or 16 and one more for nature 7 (element 150). Earned or unearned income, rent, the utility allowance, and the medical, dependent care or child-support deduction move in $3 steps toward the issued benefit (<code>RAWBEN</code>) until a stop rule binds, and a stepped utility amount is then reset to a commonly reported value.<a href="#fn3" class="footnote-ref" id="fnref3" role="doc-noteref"><sup>3</sup></a> When the first finding falls outside those element and nature codes, nothing moves. Of 283 deviation cases surviving the solver’s consistency filters (22 of 305 are excluded), 246 reproduce the issued amount within the file’s $5 editing tolerance, which is consistent with correct arithmetic applied to a wrong input. In 230 of the 246 the solver moved an input; in the other 16 it took no step, and the issued benefit was already within $5 of <code>FSBEN</code>, so the match restates the parity result. In 19 of the 230 the issued benefit was also within $5 of <code>FSBEN</code> before the solver moved, so those matches did not need the move. The rate differs by threshold status: among the 97 above-threshold official error cases 76 reproduce (78.4%, binomial standard error about 4 points), and among the 186 sub-threshold deviations 170 do (91.4%). The solver’s own recomputation and the engine agree on the within-$5 classification for all 283 cases, two implementations with distinct codebases reading the same file.</p>
+<p>Neither outcome identifies a computation error. A reproduced case shows that one value of the first-listed input, run through correct arithmetic, yields a benefit within $5 of the issued benefit; it does not show which input the agency had wrong, and a moved input can absorb a computation error. A non-reproduced case can reflect the solver’s limits: it tries one input in one direction, and in 17 of the 37 it moved nothing (14 cases whose first finding falls outside those codes, and 3 where it stopped before moving). Of the 26 cases that carry a layer-2 computational finding, 13 reproduce, 10 do not and 3 were not replayed. The complete per-case replay output (283 rows), the layer-1 and layer-2 classification outputs, the reconstruction scripts and <code>audit_claims.py</code>, which regenerates the replay counts into <code>claims_audit.json</code>, are committed with this paper.</p>
 <p>Layers 1 and 2 agree on the structure: most measured SNAP payment error among eligible cases is information failure, not calculation failure — consistent with the administrative-burden literature’s account of where compliance costs bind <span class="citation" data-cites="herd2018burden">(<a href="#ref-herd2018burden" role="doc-biblioref">Herd and Moynihan 2018</a>)</span>. Correct-computation guarantees address the smaller share directly — roughly 3 to 10% of error dollars by layers 1–2 — plus whatever share of information failures better tooling prevents indirectly through verification prompts and documentation-requirement computation at intake; <a href="#sec-adoption" class="quarto-xref">Section&nbsp;7.2</a> prices the direct shares through the formula’s tiers, elections, and delay clause. Ineligibility errors, excluded from the file, are predominantly facts-driven, which would push the information-failure share higher still; that inference, unlike the in-file shares, is not directly computed here. The indirect channel is where policy design enters: rules determine what a caseworker must collect and verify, which is the mechanism the modeling section takes up.</p>
 </section>
 <section id="sec-model" class="level2">
@@ -649,7 +649,7 @@ <h2 class="anchored" data-anchor-id="sec-roadmap">Toward engine-computed counter
 </section>
 <section id="sec-limitations" class="level2">
 <h2 class="anchored" data-anchor-id="sec-limitations">Limitations</h2>
-<p>The parity result certifies agreement with the edited file’s computed benefit chain from supplied intermediates — not independent derivation from raw facts, and not the reviewer-adjudicated outcome fields (<a href="#sec-oracle-scope" class="quarto-xref">Section&nbsp;4.1</a>); it is also post-correction on the same case set, pending a fiscal 2025 confirmatory run. The public file excludes ineligible-case and full-overissuance error, truncating the error distribution every file-based estimate inherits. The replay depends on a reconstruction solver whose filters exclude 22 of 305 deviation cases and which moves one input per case; its shares are case shares, and neither its reproduced nor its non-reproduced class identifies computation errors. The error models predict an adjudicated, thresholded label — a property of the measurement system, with a documented manipulation history (<a href="#sec-qc" class="quarto-xref">Section&nbsp;3</a>). Cross-state model transport is imperfect: the model’s raw level underpredicts high-error states, all nine coverage gaps remain one-sided negative after the tail and level-factor fixes — and after the committed location repair, which improves every level but closes only part of the gap — and the deployed simulator accordingly serves the model only for its level-gated standard-medical-deduction scenario, with seven jurisdictions disabled outright; every simulation result in this paper uses the observed-resample engine. That mode resamples i.i.d. from a probability sample whose actual design varies by state, treats the official rate’s federal re-review adjustment as a fixed level with no variance contribution, and holds the error process fixed — no behavioral response of agencies or households to cost sharing, no corrective-feedback channel from audit volume, both live questions the fiscal 2025–27 files will begin to answer under the new incentives. Adoption contrasts are descriptive contrasts of policy bundles; nothing here identifies causal effects of burden policies on error.</p>
+<p>The parity result certifies agreement with the edited file’s computed benefit chain from supplied intermediates — not independent derivation from raw facts, and not the reviewer-adjudicated outcome fields (<a href="#sec-oracle-scope" class="quarto-xref">Section&nbsp;4.1</a>); it is also post-correction on the same case set, pending a fiscal 2025 confirmatory run. The public file excludes ineligible-case and full-overissuance error, truncating the error distribution every file-based estimate inherits. The replay depends on a reconstruction solver whose filters exclude 22 of 305 deviation cases and which moves at most one input per case; its shares are case shares, and neither its reproduced nor its non-reproduced class identifies computation errors. The error models predict an adjudicated, thresholded label — a property of the measurement system, with a documented manipulation history (<a href="#sec-qc" class="quarto-xref">Section&nbsp;3</a>). Cross-state model transport is imperfect: the model’s raw level underpredicts high-error states, all nine coverage gaps remain one-sided negative after the tail and level-factor fixes — and after the committed location repair, which improves every level but closes only part of the gap — and the deployed simulator accordingly serves the model only for its level-gated standard-medical-deduction scenario, with seven jurisdictions disabled outright; every simulation result in this paper uses the observed-resample engine. That mode resamples i.i.d. from a probability sample whose actual design varies by state, treats the official rate’s federal re-review adjustment as a fixed level with no variance contribution, and holds the error process fixed — no behavioral response of agencies or households to cost sharing, no corrective-feedback channel from audit volume, both live questions the fiscal 2025–27 files will begin to answer under the new incentives. Adoption contrasts are descriptive contrasts of policy bundles; nothing here identifies causal effects of burden policies on error.</p>
 </section>
 <section id="conclusion" class="level2">
 <h2 class="anchored" data-anchor-id="conclusion">Conclusion</h2>
@@ -657,7 +657,7 @@ <h2 class="anchored" data-anchor-id="conclusion">Conclusion</h2>
 </section>
 <section id="sec-availability" class="level2">
 <h2 class="anchored" data-anchor-id="sec-availability">Data and code availability</h2>
-<p>The repository <code>PolicyEngine/snap-qc-sim</code> contains the analysis pipeline (deterministic; independent clean-clone reruns reproduce all five pipeline artifacts byte-for-byte on the pinned interpreter version, recorded in <code>.python-version</code> and each artifact’s provenance block), the simulator, frozen simulation, decomposition, and certification artifacts under <code>paper/snapshot/</code> (including the complete per-case replay output, the layer-1 and layer-2 classification outputs, and the reconstruction scripts), the archived adversarial review reports — six rounds — and the round-1 editorial synthesis under <code>paper/reviews/</code>, the fact catalog (<code>paper/FACTS.md</code>), the fiscal 2026 pre-registration (<code>analysis/PREREGISTRATION_OBBBA_BOUNDARY.md</code>, with its hash-pinned predictions artifact and deterministic generator), and this manuscript. The verification harness and per-case suite reports are public in <code>TheAxiomFoundation/axiom-oracles</code>; the encodings in <code>TheAxiomFoundation/rulespec-us</code>; the engine in <code>TheAxiomFoundation/axiom-rules-engine</code>. QC public-use files are available from USDA FNS <span class="citation" data-cites="fns2024techdoc">(<a href="#ref-fns2024techdoc" role="doc-biblioref">U.S. Department of Agriculture, Food and Nutrition Service 2025b</a>)</span>, and the training files are tracked in the public <span class="citation" data-cites="giannella_snapqc">Giannella and Molin (<a href="#ref-giannella_snapqc" role="doc-biblioref">2026</a>)</span> repository. Reproduction commands are in the repository README.</p>
+<p>The repository <code>PolicyEngine/snap-qc-sim</code> contains the analysis pipeline (deterministic; independent clean-clone reruns reproduce all five pipeline artifacts byte-for-byte on the pinned interpreter version, recorded in <code>.python-version</code> and each artifact’s provenance block), the simulator, frozen simulation, decomposition, and certification artifacts under <code>paper/snapshot/</code> (including the complete per-case replay output, the layer-1 and layer-2 classification outputs, and the reconstruction scripts), the archived adversarial review reports — six rounds — and the round-1 editorial synthesis under <code>paper/reviews/</code>, the fact catalog (<code>paper/FACTS.md</code>), the fiscal 2026 pre-registration (<code>analysis/PREREGISTRATION_OBBBA_BOUNDARY.md</code>, with its hash-pinned predictions artifact and deterministic generator), and this manuscript. The verification harness and per-case suite reports are public in <code>TheAxiomFoundation/axiom-oracles</code>; the encodings in <code>TheAxiomFoundation/rulespec-us</code>; the engine in <code>TheAxiomFoundation/axiom-rules-engine</code>. QC public-use files are available from USDA FNS <span class="citation" data-cites="fns2024techdoc">(<a href="#ref-fns2024techdoc" role="doc-biblioref">Leftin et al. 2026</a>)</span>, and the training files are tracked in the public <span class="citation" data-cites="giannella_snapqc">Giannella and Molin (<a href="#ref-giannella_snapqc" role="doc-biblioref">2026</a>)</span> repository. Reproduction commands are in the repository README.</p>
 </section>
 <section id="disclosure" class="level2">
 <h2 class="anchored" data-anchor-id="disclosure">Disclosure</h2>
@@ -713,6 +713,9 @@ <h2 class="anchored" data-anchor-id="acknowledgments">Acknowledgments</h2>
 <div id="ref-koenker1978quantile" class="csl-entry" role="listitem">
 Koenker, Roger, and Gilbert Bassett. 1978. <span>“Regression Quantiles.”</span> <em>Econometrica</em> 46 (1): 33–50. <a href="https://doi.org/10.2307/1913643">https://doi.org/10.2307/1913643</a>.
 </div>
+<div id="ref-fns2024techdoc" class="csl-entry" role="listitem">
+Leftin, Joshua, Mia Monkovic, Francisco Yang, Sam Bonelli, and John Carlo Maula. 2026. <em>Technical Documentation for the Fiscal Year 2024 Supplemental Nutrition Assistance Program Quality Control Database and the QC Minimodel</em>. Final report. Mathematica, for the U.S. Department of Agriculture. <a href="https://snapqcdata.net/datafiles">https://snapqcdata.net/datafiles</a>.
+</div>
 <div id="ref-merigoux2021catala" class="csl-entry" role="listitem">
 Mérigoux, Denis, Nicolas Chataing, and Jonathan Protzenko. 2021. <span>“Catala: A Programming Language for the Law.”</span> <em>Proceedings of the ACM on Programming Languages</em> 5 (ICFP): 1–29. <a href="https://doi.org/10.1145/3473582">https://doi.org/10.1145/3473582</a>.
 </div>
@@ -735,10 +738,7 @@ <h2 class="anchored" data-anchor-id="acknowledgments">Acknowledgments</h2>
 U.S. Department of Agriculture, Food and Nutrition Service. 2023. <em>FNS Handbook 310: SNAP Quality Control Review Handbook</em>. <a href="https://www.fns.usda.gov/snap/fns-handbook-310">https://www.fns.usda.gov/snap/fns-handbook-310</a>.
 </div>
 <div id="ref-fns2024per" class="csl-entry" role="listitem">
-U.S. Department of Agriculture, Food and Nutrition Service. 2025a. <em>Supplemental Nutrition Assistance Program Payment Error Rates, Fiscal Year 2024</em>. <a href="https://www.fns.usda.gov/snap/qc/per">https://www.fns.usda.gov/snap/qc/per</a>.
-</div>
-<div id="ref-fns2024techdoc" class="csl-entry" role="listitem">
-U.S. Department of Agriculture, Food and Nutrition Service. 2025b. <em>Technical Documentation for the Fiscal Year 2024 Supplemental Nutrition Assistance Program Quality Control Database and the QC Minimodel</em>. <a href="https://snapqcdata.net/datafiles">https://snapqcdata.net/datafiles</a>.
+U.S. Department of Agriculture, Food and Nutrition Service. 2025. <em>Supplemental Nutrition Assistance Program Payment Error Rates, Fiscal Year 2024</em>. <a href="https://www.fns.usda.gov/snap/qc/per">https://www.fns.usda.gov/snap/qc/per</a>.
 </div>
 <div id="ref-doj2019texas" class="csl-entry" role="listitem">
 U.S. Department of Justice. 2019. <em>Texas <span class="nocase">Health and Human Services Commission</span> Agrees to Pay over $15 Million to Resolve <span>False Claims Act</span> Liability for Deficient <span>SNAP</span> Quality Control</em>. <a href="https://www.justice.gov/opa/pr/texas-health-and-human-services-commission-agrees-pay-over-15-million-resolve-false-claims">https://www.justice.gov/opa/pr/texas-health-and-human-services-commission-agrees-pay-over-15-million-resolve-false-claims</a>.
diff --git a/app/public/paper/web/index.pdf b/app/public/paper/web/index.pdf
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diff --git a/paper/FACTS.md b/paper/FACTS.md
index d1015d7..1bef288 100644
--- a/paper/FACTS.md
+++ b/paper/FACTS.md
@@ -34,7 +34,7 @@ Numbers from the superseded pre-audit pipeline (before PR #7) are banned.
 | C4 | WITHDRAWN 2026-10-03: the claim that the process surfaced errata in FNS's FY2024 technical documentation. No source records such errata: neither version of paper/snapshot/labs/amterr/ANALYSIS.md (27e1b09, c4333b3), the axiom.org Colorado report (original and corrected text) nor the axiom-oracles SNAP QC playbook documents one, GitHub issue searches across TheAxiomFoundation and PolicyEngine found none, and round-1 and round-2 referees could not locate the "reported upstream" thread. The manuscript no longer makes the claim (abstract, introduction, @sec-oracle-scope) | searched 2026-10-03; paper/reviews/round-1/referee-redteam.md, round-2/round2-rounddiff.md |
 | C5 | A certification probe (2026-08-06) attested a reproducible toolchain: engine commit de0efdc7 (binary SHA-256 bb8ec236…) × rulespec-us b53ce208 reproduces CO 856/856 benefits and 5,136/5,136 stage cells at zero tolerance; 856 cases in 3.73 s end-to-end (230 cases/s; 279/s with compile amortized) | CERT_REPORT (paper/snapshot/cert/CERT_REPORT.md) |
 | C6 | Extrapolation at certified throughput: 6,081 cases ≈ 26 s; 44,800 × 5 scenarios ≈ 16 minutes single-threaded on a laptop | CERT_REPORT extrapolation table |
-| C7 | What the parity target is: `FSBEN` is a constructed variable ("Final calculated benefit") that Mathematica computes for USDA from each edited case record with the QC Minimodel's benefit formula; the QC Minimodel is "one of FNA's SNAP microsimulation models". The file's other chain values (FSGRINC, FSSTDDED, FSSLTDED, FSNETINC, BENMAX) are constructed too; RAWBEN and AMTERR are reported. Editing Steps 12–13 calculate the benefit and, where it is not within $5 of the issued benefit adjusted for the error amount (Step 13a conditions), try the dependent care deduction, the utility amount and the SMD-demonstration medical deduction, keeping a change only if it produces a match. Colorado: `FSBEN` within $5 of `BENFIX` in 797 of 856 cases (743 exactly equal; same under the May and August 2026 postings, which differ only in weights). Never "the agency's own benefit-calculation software" or "computational canon" | FY2024 QC technical documentation, August 2026 posting: title page and acknowledgments (Mathematica for FNA, contract 12-3198-24-Q-0029), PDF p.15 (Minimodel), pp.36–37 (Steps 12–13), p.65 (Minimodel's FSBEN calculation refers to the codebook entry), p.71 (FSGRINC, FSNETINC constructed), p.73 (FSSLTDED, FSSTDDED, BENMAX, FSBEN constructed; RAWBEN, AMTERR reported) and p.91 (FSBEN formula); 797/856 recomputed from qc_pub_fy2024.csv (sha256 e871a8e9…), locked in tests/test_retired_claims.py |
+| C7 | What the parity target is: `FSBEN` is a constructed variable ("Final calculated benefit") that Mathematica computes for USDA from each edited case record with the QC Minimodel's benefit formula; the QC Minimodel is "one of FNA's SNAP microsimulation models". The file's other chain values (FSGRINC, FSSTDDED, FSSLTDED, FSNETINC, BENMAX) are constructed too; RAWBEN and AMTERR are reported. Editing Steps 12–13 calculate the benefit and, where it is not within $5 of the issued benefit adjusted for the error amount (Step 13a conditions), try the dependent care deduction, the utility amount and the SMD-demonstration medical deduction, keeping a change only if it produces a match. Colorado: `FSBEN` within $5 of `BENFIX` in 797 of 856 cases (743 exactly equal; 21 of the other 59 are prorated allotments, ALLADJ = 2, PDF p.76, which the full-month FSBEN formula does not reflect; same under the May and August 2026 postings, which differ only in weights). Never "the agency's own benefit-calculation software" or "computational canon" | FY2024 QC technical documentation, August 2026 posting: title page and acknowledgments (Mathematica for FNA, contract 12-3198-24-Q-0029), PDF p.15 (Minimodel), pp.36–37 (Steps 12–13), p.65 (Minimodel's FSBEN calculation refers to the codebook entry), p.71 (FSGRINC, FSNETINC constructed), p.73 (FSSLTDED, FSSTDDED, BENMAX, FSBEN constructed; RAWBEN, AMTERR reported) and p.91 (FSBEN formula); 797/856 recomputed from qc_pub_fy2024.csv (sha256 e871a8e9…), locked in tests/test_retired_claims.py |
 
 ## D. Decomposition of measured error (Colorado FY2024)
 
@@ -105,7 +105,7 @@ Numbers from the superseded pre-audit pipeline (before PR #7) are banned.
 | H5 | OBBBA cut the federal share of SNAP administrative costs from 50% to 25% beginning FY2027 | OBBBA § 10106; Federal Register 2026-12696 |
 | H6 | OBBBA § 10103 restricts the LIHEAP-triggered heat-and-eat utility allowance to households with an elderly or disabled member | Pub. L. 119-21 § 10103; state implementation letters |
 | H7 | Colorado deviation universe: 305 = 110 above-threshold official errors + 195 sub-threshold deviations; sub-threshold dollars $18.5M of $112.6M (16%); official-gate file-derived rate 7.42% | recomputed from qc_pub_fy2024.csv (official gate); paper/snapshot/labs/amterr/amterr_replay_results.json |
-| H8 | Replay reproduction rates by threshold status: 76/97 = 78.4% above threshold; 170/186 = 91.4% below (reproduced within $5, which is consistent with a wrong input; G lists the retired wording). SUPERSEDES "33 of 246 explained cases have deviations ≤ $5 (within comparison tolerance mechanically)": 33 of the 246 have AMTERR ≤ $5, but the solver moved an input in 17 of them; the matches that follow mechanically from FSBEN parity are the 16 with nothing moved (D4), all among the 33 | analysis/cause_shares.json colorado_replay_reconciliation.slices; claims_audit.json case_level.replay.reproduced_without_move_n; amterr_replay_results.json (amterr field); locked in tests/test_retired_claims.py |
+| H8 | Replay reproduction rates by threshold status: 76/97 = 78.4% above threshold; 170/186 = 91.4% below (reproduced within $5, which is consistent with a wrong input; G lists the retired wording). SUPERSEDES "33 of 246 explained cases have deviations ≤ $5 (within comparison tolerance mechanically)": the word "explained" is retired, and the count stands. 33 of the 246 have AMTERR ≤ $5; in all 33, and in 2 more with AMTERR of $6, RAWBEN was within $5 of FSBEN before any step, so none of these 35 matches needed a move. The solver took no step in 16 (D4) and moved an input anyway in 19 (7 rent, whose steps stop only within $3; 10 income, which have no within-$3 stop; 2 utility resets). All 35 are sub-threshold | analysis/cause_shares.json colorado_replay_reconciliation.slices; claims_audit.json case_level.replay.reproduced_without_move_n and replay_inputs_unchanged_keys; amterr_replay_results.json (rawben, fsben, amterr, within5); locked in tests/test_retired_claims.py |
 
 ## I. Engine counterfactuals and the model-primary round (added in revision 3)
 
diff --git a/paper/index.qmd b/paper/index.qmd
index fd11ff4..bc79dc7 100644
--- a/paper/index.qmd
+++ b/paper/index.qmd
@@ -117,8 +117,8 @@ Second, as raw material for decomposing measured error. QC's own cause
 coding and a finding-nature classification give two estimates of
 how much of the error rate reflects computation failure versus
 information failure, and a replay of reconstructed case inputs
-through the verified engine tests whether one wrong input reproduces
-each issued benefit (@sec-decompose).
+through the verified engine tests whether changing the input named by
+each case's first finding reproduces the issued benefit (@sec-decompose).
 
 Third, as training and validation data for a case-level model of the
 error process (@sec-model) and the basis for a distributional simulator
@@ -365,7 +365,8 @@ allowance or zero, and zeroing the medical deduction of standard medical
 deduction demonstration participants) and keeps a change only if it
 produces a match [@fns2024techdoc, chap. III, Steps 12–13]. The
 sequence does not always close the gap: in Colorado, `FSBEN` ends within
-$5 of `BENFIX` in 797 of 856 cases. The comparison also supplies several
+$5 of `BENFIX` in 797 of 856 cases; 21 of the other 59 are prorated
+allotments, which the full-month `FSBEN` does not reflect. The comparison also supplies several
 intermediates directly from the file rather than deriving them from raw
 facts: the QC-calculated medical and
 child-support deduction amounts, the recorded utility allowance, and the
@@ -437,7 +438,7 @@ construction of the file, that ineligible-case error.
 | 2. Finding natures | `NATURE` codes inherently describing computation (rounding, conversion, wrong standard, benefit miscomputed) plus the arithmetic `ELEMENT`; deduction-amount natures count only when the paired cause is in layer 1's broad computing-apparatus set | 3.3% pure-computation; 6.6% system-caused input; 4.2% mixed; 86.0% other input | Colorado weighted deviation dollars; nationally 4.6% / 11.1% / 6.5% / 77.7% |
 | 3. Engine replay | A public solver changes the input named by each case's first finding, where it can; the verified engine replays the solver's inputs | Not identified. 78.4% of above-threshold official errors and 91.4% of sub-threshold deviations (86.9% blended) reproduce the issued benefit within $5, consistent with a wrong input; of the 37 that do not, the solver moved nothing in 17 and 10 carry a computational finding (8 are both) | 283 filtered Colorado deviation cases (97 above threshold, 186 below); case shares, not dollar shares |
 
-: Three decompositions of Colorado fiscal 2024 measured error.
+: Three layers of evidence on Colorado fiscal 2024 measured error.
 Layers 1–2 quote the committed cause-share artifact
 (`analysis/cause_shares.json`, all 53 jurisdictions, conventions and
 code-to-class maps serialized in the file); layer 3 is a share of
@@ -469,7 +470,9 @@ issued amount within the file's $5 editing tolerance, which is
 consistent with correct arithmetic applied to a wrong input. In 230 of
 the 246 the solver moved an input; in the other 16 it took no step, and
 the issued benefit was already within $5 of `FSBEN`, so the match
-restates the parity result. The
+restates the parity result. In 19 of the 230 the issued benefit was also
+within $5 of `FSBEN` before the solver moved, so those matches did not
+need the move. The
 rate differs by threshold status: among the 97 above-threshold official
 error cases 76 reproduce (78.4%, binomial standard error about 4
 points), and among the 186 sub-threshold deviations 170 do (91.4%). The
@@ -479,7 +482,7 @@ codebases reading the same file.
 
 Neither outcome identifies a computation error. A reproduced case shows
 that one value of the first-listed input, run through correct
-arithmetic, yields the issued benefit; it does not show which input the
+arithmetic, yields a benefit within $5 of the issued benefit; it does not show which input the
 agency had wrong, and a moved input can absorb a computation error. A
 non-reproduced case can reflect the solver's limits: it tries one input
 in one direction, and in 17 of the 37 it moved nothing (14 cases whose
@@ -1122,7 +1125,7 @@ set, pending a fiscal 2025 confirmatory run. The public file excludes
 ineligible-case and full-overissuance error, truncating the error
 distribution every file-based estimate inherits. The replay depends on
 a reconstruction solver whose filters exclude 22 of 305 deviation cases and
-which moves one input per case; its shares are case shares, and neither
+which moves at most one input per case; its shares are case shares, and neither
 its reproduced nor its non-reproduced class identifies computation
 errors. The error models predict an adjudicated,
 thresholded label — a property of the measurement system, with a
diff --git a/paper/references.bib b/paper/references.bib
index a245adb..23a0ca0 100644
--- a/paper/references.bib
+++ b/paper/references.bib
@@ -1,7 +1,11 @@
-@misc{fns2024techdoc,
-  author = {{U.S. Department of Agriculture, Food and Nutrition Service}},
+@techreport{fns2024techdoc,
+  author = {Leftin, Joshua and Monkovic, Mia and Yang, Francisco and Bonelli, Sam and Maula, John Carlo},
   title = {Technical documentation for the fiscal year 2024 Supplemental Nutrition Assistance Program Quality Control database and the QC minimodel},
-  year = {2025},
+  institution = {Mathematica, for the U.S. Department of Agriculture},
+  type = {Final report},
+  year = {2026},
+  month = aug,
+  note = {August 2026 revision of the May 2026 report; contract 12-3198-24-Q-0029},
   url = {https://snapqcdata.net/datafiles}
 }
 
diff --git a/tests/test_retired_claims.py b/tests/test_retired_claims.py
index bcafb32..757368e 100644
--- a/tests/test_retired_claims.py
+++ b/tests/test_retired_claims.py
@@ -149,6 +149,23 @@ def test_facts_records_each_withdrawal():
 # replay figures: manuscript prose against claims_audit.json and the slices
 
 
+def _key(row: dict) -> str:
+    return f"{row['yrmonth']}-{row['hhldno']}"
+
+
+def _near_fsben_rows() -> list[dict]:
+    """Replay rows whose issued benefit was within $5 of FSBEN before any step."""
+    return [r for r in REPLAY_ROWS if abs(r["rawben"] - r["fsben"]) <= 5]
+
+
+def _moved_near_matches() -> int:
+    """Reproduced cases the solver moved although RAWBEN was already near FSBEN."""
+    unchanged = set(AUDIT["case_level"]["replay_inputs_unchanged_keys"])
+    return sum(
+        1 for r in _near_fsben_rows() if r["within5"] and _key(r) not in unchanged
+    )
+
+
 def _unmoved_computational_misses() -> int:
     """Non-reproduced cases that both moved nothing and carry a computational finding."""
     cases = AUDIT["case_level"]
@@ -181,6 +198,7 @@ def _replay_counts() -> dict[str, int]:
             "not_reproduced_with_computational_finding"
         ]["n"],
         "miss_overlap": _unmoved_computational_misses(),
+        "moved_near": _moved_near_matches(),
         "above_n": above["total"]["n"],
         "above_reproduced": above["outcomes"]["reproduced"]["n"],
         "sub_n": sub["total"]["n"],
@@ -228,6 +246,11 @@ def replay_quotes(c: dict[str, int]) -> list[str]:
             f"the other {c['unmoved']} it took no step, and the issued benefit was "
             "already within $5 of `FSBEN`, so the match restates the parity result."
         ),
+        (
+            f"In {c['moved_near']} of the {c['moved']} the issued benefit was also "
+            "within $5 of `FSBEN` before the solver moved, so those matches did not "
+            "need the move."
+        ),
         (
             f"among the {c['above_n']} above-threshold official error cases "
             f"{c['above_reproduced']} reproduce ({above}, binomial standard error"
@@ -300,16 +323,33 @@ def test_unmoved_matches_restate_parity():
         assert abs(r["rawben"] - r["fsben"]) <= 5
 
 
-def test_superseded_33_reconciles_with_the_16():
-    """FACTS H8: 33 reproduced cases have AMTERR <= $5; 16 of them moved nothing."""
+def test_superseded_33_reconciles_with_the_near_fsben_matches():
+    """FACTS H8: every match that started within $5 of FSBEN needed no move.
+
+    The old "33 within comparison tolerance mechanically" counted AMTERR <= $5.
+    All 33 started with RAWBEN within $5 of FSBEN, as did 2 more, and on the
+    file's own inputs the engine returns FSBEN, so all 35 match without a move.
+    The solver took no step in 16 and moved an input anyway in the other 19.
+    """
     unchanged = set(AUDIT["case_level"]["replay_inputs_unchanged_keys"])
     small = [r for r in REPLAY_ROWS if r["within5"] and r["amterr"] <= 5]
-    unmoved = [r for r in small if f"{r['yrmonth']}-{r['hhldno']}" in unchanged]
-    assert (len(small), len(unmoved)) == (33, 16)
+    near = _near_fsben_rows()
+    assert all(abs(r["rawben"] - r["fsben"]) <= 5 for r in small)
+    assert all(r["within5"] for r in near)
+    unmoved = [r for r in near if _key(r) in unchanged]
+    moved = [r for r in near if _key(r) not in unchanged]
+    assert (len(small), len(near), len(unmoved), len(moved)) == (33, 35, 16, 19)
     assert len(unmoved) == _replay_counts()["unmoved"]
+    assert len(moved) == _replay_counts()["moved_near"]
+    extra = sorted({r["amterr"] for r in near if r["amterr"] > 5})
     facts = _read(FACTS)
     assert f"{len(small)} of the 246 have AMTERR ≤ $5" in facts
-    assert f"the solver moved an input in {len(small) - len(unmoved)} of them" in facts
+    assert (
+        f"in {len(near) - len(small)} more with AMTERR of ${extra[0]:.0f}" in facts
+        and len(extra) == 1
+    )
+    assert f"none of these {len(near)} matches needed a move" in facts
+    assert f"moved an input anyway in {len(moved)}" in facts
 
 
 # ---------------------------------------------------------------------------
@@ -384,15 +424,18 @@ def _available_postings() -> list[str]:
 def test_fsben_within_five_of_benfix_in_colorado(posting):
     frame = pd.read_csv(
         audit.posting_path(posting),
-        usecols=["STATE", "FSBEN", "BENFIX"],
+        usecols=["STATE", "FSBEN", "BENFIX", "ALLADJ"],
         low_memory=False,
     )
     co = frame[frame["STATE"] == 8]
     within = int(((co["FSBEN"] - co["BENFIX"]).abs() <= 5).sum())
     exact = int((co["FSBEN"] == co["BENFIX"]).sum())
-    assert (within, len(co), exact) == (797, 856, 743)
+    off = co[(co["FSBEN"] - co["BENFIX"]).abs() > 5]
+    prorated = int((off["ALLADJ"] == 2).sum())  # ALLADJ 2 = prorated benefit
+    assert (within, len(co), exact, prorated) == (797, 856, 743, 21)
     assert (
-        f"`FSBEN` ends within $5 of `BENFIX` in {within} of {len(co)} cases"
+        f"`FSBEN` ends within $5 of `BENFIX` in {within} of {len(co)} cases; "
+        f"{prorated} of the other {len(off)} are prorated allotments"
     ) in _read(MANUSCRIPT)
     facts = _read(FACTS)
     assert (
@@ -437,6 +480,7 @@ def replay_counts(draw):
         "layer2_reproduced": layer2_reproduced,
         "layer2_not_reproduced": layer2_not_reproduced,
         "miss_overlap": miss_overlap,
+        "moved_near": draw(st.integers(0, reproduced - unmoved)),
         "above_n": above_n,
         "above_reproduced": above_reproduced,
         "sub_n": sub_n,

From dfc1ddc90df8a170e13912f2a352e91439833cf6 Mon Sep 17 00:00:00 2001
From: Max Ghenis <mghenis@gmail.com>
Date: Sun, 4 Oct 2026 12:46:17 -0400
Subject: [PATCH 8/9] Address review round 7 on the retired-claim corrections

- The paper keeps FNS as "the agency's name during fiscal 2024"; "the name
  on every document the fiscal 2024 data cite" became false once the
  citation moved to the FNA-lettered August 2026 tech doc.
- amterr_replay.py's docstring no longer calls the solver's inputs
  "pre-edit original values"; the engine_on_original field name stays,
  since the pinned results JSON uses it.
- FACTS H8: rent's "closeness stop is $3". The C7 proration count is now
  locked in the data-gated test.

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
---
 app/public/paper/web/index.html             |   2 +-
 app/public/paper/web/index.pdf              | Bin 214102 -> 214069 bytes
 paper/FACTS.md                              |   2 +-
 paper/index.qmd                             |   2 +-
 paper/snapshot/labs/amterr/amterr_replay.py |  16 +++++++++-------
 tests/test_retired_claims.py                |   6 +++---
 6 files changed, 15 insertions(+), 13 deletions(-)

diff --git a/app/public/paper/web/index.html b/app/public/paper/web/index.html
index 314e464..f7c3df6 100644
--- a/app/public/paper/web/index.html
+++ b/app/public/paper/web/index.html
@@ -268,7 +268,7 @@ <h2 id="toc-title">Table of contents</h2>
 <section id="introduction" class="level2">
 <h2 class="anchored" data-anchor-id="introduction">Introduction</h2>
 <p>Under 7 U.S.C. § 2013(a)(2), as amended by the One Big Beautiful Bill Act (Pub. L. No.&nbsp;119-21, 2025; OBBBA), a state whose SNAP payment error rate reaches 6% will pay 5% of benefit costs beginning in fiscal year 2028; at 8%, the share is 10%; at 10%, it is 15%. The rate that determines the FY2028 share is the state’s fiscal 2025 or 2026 rate, at its election; later years use the third preceding year. Implementation is delayed for the highest-error states: a state whose fiscal 2025 rate times 1.5 reaches 20% — that is, a rate above 13.33% (13.34% at the published two-decimal precision) — starts in FY2029 instead, and a state whose fiscal 2026 rate crosses the same test starts in FY2030.<a href="#fn1" class="footnote-ref" id="fnref1" role="doc-noteref"><sup>1</sup></a> For a mid-sized state, one tier is worth tens of millions of dollars a year; for the largest states, more than a hundred million.</p>
-<p>The error rate is estimated by the Quality Control system: each state re-reviews a probability sample of its own active cases — by formula in 7 C.F.R. § 275.11, between 300 and 2,400 annually, or 300 to 1,020 under an optional reduced schedule that every state elected in fiscal 2024 <span class="citation" data-cites="fns2024techdoc">(<a href="#ref-fns2024techdoc" role="doc-biblioref">Leftin et al. 2026</a>)</span> — and federal reviewers re-review a subsample, from which the Food and Nutrition Service (FNS; renamed the Food and Nutrition Administration in 2026 — the fiscal 2025 rate publication carries the new letterhead <span class="citation" data-cites="fna2025per">(<a href="#ref-fna2025per" role="doc-biblioref">U.S. Department of Agriculture, Food and Nutrition Administration 2026</a>)</span>, and this paper keeps FNS, the name on every document the fiscal 2024 data cite) estimates official state rates with adjustments including federal re-review integration. The public-use files that result record, for roughly 45,000 cases a year, what the agency issued and what the review concluded the case should have received, with coded findings for every discrepancy.</p>
+<p>The error rate is estimated by the Quality Control system: each state re-reviews a probability sample of its own active cases — by formula in 7 C.F.R. § 275.11, between 300 and 2,400 annually, or 300 to 1,020 under an optional reduced schedule that every state elected in fiscal 2024 <span class="citation" data-cites="fns2024techdoc">(<a href="#ref-fns2024techdoc" role="doc-biblioref">Leftin et al. 2026</a>)</span> — and federal reviewers re-review a subsample, from which the Food and Nutrition Service (FNS; renamed the Food and Nutrition Administration in 2026 — the fiscal 2025 rate publication carries the new letterhead <span class="citation" data-cites="fna2025per">(<a href="#ref-fna2025per" role="doc-biblioref">U.S. Department of Agriculture, Food and Nutrition Administration 2026</a>)</span>, and this paper keeps FNS, the agency’s name during fiscal 2024) estimates official state rates with adjustments including federal re-review integration. The public-use files that result record, for roughly 45,000 cases a year, what the agency issued and what the review concluded the case should have received, with coded findings for every discrepancy.</p>
 <p>This paper treats those files as a research instrument with three uses, the first in the software-testing sense of an <em>oracle</em>: an external source of ground-truth outputs for given inputs.</p>
 <p>First, as a verification target for rules as code. I test independently encoded SNAP rules — statute, regulations, and state policy manuals encoded as executable rule modules — against the QC file’s recorded benefit computations in seven states, holding the bar at exact reproduction: zero dollar tolerance across six computation stages per case (<a href="#sec-oracle" class="quarto-xref">Section&nbsp;4</a>). The target’s precise character matters and is stated in <a href="#sec-oracle-scope" class="quarto-xref">Section&nbsp;4.1</a>: the file’s benefit chain is a computation Mathematica runs for USDA on edited case records with the benefit formula of the FNS QC Minimodel, so parity certifies agreement with the benefit computation recorded in the file; it does not adjudicate case outcomes. Reaching it surfaced two defects in the encodings under test and left a residue in issued amounts, which <a href="#sec-decompose" class="quarto-xref">Section&nbsp;5</a> decomposes.</p>
 <p>Second, as raw material for decomposing measured error. QC’s own cause coding and a finding-nature classification give two estimates of how much of the error rate reflects computation failure versus information failure, and a replay of reconstructed case inputs through the verified engine tests whether changing the input named by each case’s first finding reproduces the issued benefit (<a href="#sec-decompose" class="quarto-xref">Section&nbsp;5</a>).</p>
diff --git a/app/public/paper/web/index.pdf b/app/public/paper/web/index.pdf
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diff --git a/paper/FACTS.md b/paper/FACTS.md
index 1bef288..fbe4a0a 100644
--- a/paper/FACTS.md
+++ b/paper/FACTS.md
@@ -105,7 +105,7 @@ Numbers from the superseded pre-audit pipeline (before PR #7) are banned.
 | H5 | OBBBA cut the federal share of SNAP administrative costs from 50% to 25% beginning FY2027 | OBBBA § 10106; Federal Register 2026-12696 |
 | H6 | OBBBA § 10103 restricts the LIHEAP-triggered heat-and-eat utility allowance to households with an elderly or disabled member | Pub. L. 119-21 § 10103; state implementation letters |
 | H7 | Colorado deviation universe: 305 = 110 above-threshold official errors + 195 sub-threshold deviations; sub-threshold dollars $18.5M of $112.6M (16%); official-gate file-derived rate 7.42% | recomputed from qc_pub_fy2024.csv (official gate); paper/snapshot/labs/amterr/amterr_replay_results.json |
-| H8 | Replay reproduction rates by threshold status: 76/97 = 78.4% above threshold; 170/186 = 91.4% below (reproduced within $5, which is consistent with a wrong input; G lists the retired wording). SUPERSEDES "33 of 246 explained cases have deviations ≤ $5 (within comparison tolerance mechanically)": the word "explained" is retired, and the count stands. 33 of the 246 have AMTERR ≤ $5; in all 33, and in 2 more with AMTERR of $6, RAWBEN was within $5 of FSBEN before any step, so none of these 35 matches needed a move. The solver took no step in 16 (D4) and moved an input anyway in 19 (7 rent, whose steps stop only within $3; 10 income, which have no within-$3 stop; 2 utility resets). All 35 are sub-threshold | analysis/cause_shares.json colorado_replay_reconciliation.slices; claims_audit.json case_level.replay.reproduced_without_move_n and replay_inputs_unchanged_keys; amterr_replay_results.json (rawben, fsben, amterr, within5); locked in tests/test_retired_claims.py |
+| H8 | Replay reproduction rates by threshold status: 76/97 = 78.4% above threshold; 170/186 = 91.4% below (reproduced within $5, which is consistent with a wrong input; G lists the retired wording). SUPERSEDES "33 of 246 explained cases have deviations ≤ $5 (within comparison tolerance mechanically)": the word "explained" is retired, and the count stands. 33 of the 246 have AMTERR ≤ $5; in all 33, and in 2 more with AMTERR of $6, RAWBEN was within $5 of FSBEN before any step, so none of these 35 matches needed a move. The solver took no step in 16 (D4) and moved an input anyway in 19 (7 rent, whose closeness stop is $3; 10 income, which have no within-$3 stop; 2 utility resets). All 35 are sub-threshold | analysis/cause_shares.json colorado_replay_reconciliation.slices; claims_audit.json case_level.replay.reproduced_without_move_n and replay_inputs_unchanged_keys; amterr_replay_results.json (rawben, fsben, amterr, within5); locked in tests/test_retired_claims.py |
 
 ## I. Engine counterfactuals and the model-primary round (added in revision 3)
 
diff --git a/paper/index.qmd b/paper/index.qmd
index bc79dc7..7e7d996 100644
--- a/paper/index.qmd
+++ b/paper/index.qmd
@@ -90,7 +90,7 @@ an optional reduced schedule that every state elected in fiscal 2024
 which the Food and Nutrition Service (FNS; renamed the Food and
 Nutrition Administration in 2026 — the fiscal 2025 rate publication
 carries the new letterhead [@fna2025per], and this paper keeps FNS, the
-name on every document the fiscal 2024 data cite) estimates official
+agency's name during fiscal 2024) estimates official
 state rates with adjustments including federal re-review integration. The public-use
 files that result record, for roughly 45,000 cases a year, what the
 agency issued and what the review concluded the case should have
diff --git a/paper/snapshot/labs/amterr/amterr_replay.py b/paper/snapshot/labs/amterr/amterr_replay.py
index a117549..deb32d6 100644
--- a/paper/snapshot/labs/amterr/amterr_replay.py
+++ b/paper/snapshot/labs/amterr/amterr_replay.py
@@ -1,10 +1,12 @@
-"""AMTERR replay: engine over reconstructed ORIGINAL inputs vs RAWBEN.
+"""AMTERR replay: engine over the solver's reconstructed inputs vs RAWBEN.
 
-For each Colorado FY2024 error case (STATUS 2/3) whose pre-edit original
-values the Giannella/Molin solver reconstructed (co_fy2024_reconstruction.csv),
-build the case through the proven map_qc_unit — but from a proxy unit carrying
-the ORIGINAL values — run the engine, and compare the allotment to RAWBEN
-(the benefit the agency actually issued).
+For each Colorado FY2024 error case (STATUS 2/3) the Giannella/Molin solver
+processed (co_fy2024_reconstruction.csv), build the case through the proven
+map_qc_unit, but from a proxy unit carrying the solver's raw* inputs; run the
+engine, and compare the allotment to RAWBEN (the benefit the agency actually
+issued). The solver changes at most the input named by the case's first finding
+(reconstruct_co_fy2024.R); the output field keeps its historical name,
+``engine_on_original``.
 
 Reading the result:
   match  -> the engine on the solver's inputs is within $5 of RAWBEN.
@@ -207,7 +209,7 @@ def main() -> None:
     n = len(summary)
     for tol, key in [("exact", "exact"), ("<=$1", "within1"), ("<=$5", "within5")]:
         k = sum(1 for s in summary if s[key])
-        print(f"engine(original) vs RAWBEN {tol}: {k}/{n} ({100*k/n:.1f}%)")
+        print(f"engine(solver inputs) vs RAWBEN {tol}: {k}/{n} ({100*k/n:.1f}%)")
     solver_ok = sum(1 for s in summary if s["solver_within5"])
     both = sum(1 for s in summary if s["solver_within5"] and s["within5"])
     print(f"solver within $5: {solver_ok}/{n}; engine agrees (<=$5) on {both} of those")
diff --git a/tests/test_retired_claims.py b/tests/test_retired_claims.py
index 757368e..36e8d87 100644
--- a/tests/test_retired_claims.py
+++ b/tests/test_retired_claims.py
@@ -439,9 +439,9 @@ def test_fsben_within_five_of_benfix_in_colorado(posting):
     ) in _read(MANUSCRIPT)
     facts = _read(FACTS)
     assert (
-        f"within $5 of `BENFIX` in {within} of {len(co)} cases ({exact} exactly"
-        in facts
-    )
+        f"within $5 of `BENFIX` in {within} of {len(co)} cases ({exact} exactly "
+        f"equal; {prorated} of the other {len(off)} are prorated allotments"
+    ) in facts
 
 
 # ---------------------------------------------------------------------------

From e72957bda252f5be7f749ae32826de2dabe6cb6b Mon Sep 17 00:00:00 2001
From: Max Ghenis <mghenis@gmail.com>
Date: Sun, 4 Oct 2026 14:11:20 -0400
Subject: [PATCH 9/9] Address review round 8: the lab README quotes the new
 print label

dfc1ddc renamed amterr_replay.py's summary label from engine(original)
to engine(solver inputs); the lab README still quoted the old one. It now
quotes the new label, and a test builds that line from claims_audit.json
and checks the README and the script agree.

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
---
 paper/snapshot/labs/amterr/README.md |  2 +-
 tests/test_retired_claims.py         | 14 ++++++++++++++
 2 files changed, 15 insertions(+), 1 deletion(-)

diff --git a/paper/snapshot/labs/amterr/README.md b/paper/snapshot/labs/amterr/README.md
index 2b5eb0b..0bcd043 100644
--- a/paper/snapshot/labs/amterr/README.md
+++ b/paper/snapshot/labs/amterr/README.md
@@ -135,6 +135,6 @@ cd $WORK/axiom-oracles && \
 cmp $WORK/replay/amterr_replay_results.json $LAB/amterr_replay_results.json
 ```
 
-The script prints `engine(original) vs RAWBEN <=$5: 246/283 (86.9%)` and
+The script prints `engine(solver inputs) vs RAWBEN <=$5: 246/283 (86.9%)` and
 `solver within $5: 246/283; engine agrees (<=$5) on 246 of those`. Without
 `AMTERR_LAB_DIR` it reads and writes next to itself.
diff --git a/tests/test_retired_claims.py b/tests/test_retired_claims.py
index 36e8d87..c6e0521 100644
--- a/tests/test_retired_claims.py
+++ b/tests/test_retired_claims.py
@@ -137,6 +137,20 @@ def test_js_scan_sees_phrases_split_across_concatenated_lines():
     assert "minimodel's full-formula recomputation" in RETIRED
 
 
+def test_lab_readme_quotes_the_replay_printout():
+    """The lab README quotes amterr_replay.py's summary line verbatim."""
+    readme = (LAB / "README.md").read_text(encoding="utf-8")
+    script = (LAB / "amterr_replay.py").read_text(encoding="utf-8")
+    replay = AUDIT["case_level"]["replay"]
+    n, k = replay["n"], replay["reproduced_n"]
+    assert 'print(f"engine(solver inputs) vs RAWBEN {tol}:' in script
+    assert (
+        f"`engine(solver inputs) vs RAWBEN <=$5: {k}/{n} ({100 * k / n:.1f}%)`"
+        in readme
+    )
+    assert "engine(original)" not in readme
+
+
 def test_facts_records_each_withdrawal():
     """The catalog keeps the retired wording only as a superseded record."""
     facts = _read(FACTS)