Flag the replay's weakly identified matches and reset-off misses in the paper

[MaxGhenis]

Stacked on #95 (base retire-qc-claims, final at e72957b), with #96 merged in. The replay paragraph this edits exists only on #95's branch, and the counts it quotes come from #96's claims_audit.json. Merge order: #96 → #95 → this PR, which GitHub retargets to main once #95 merges. Until then, the diff against #95 also shows #96's lab files. The paper changes are commits 1e2c45d, 773e170 and a945cde. a945cde applies #96's review-r1 fixes and this PR's review-r1 suggestions (~/reviews/snap-qc-sim-pr97/review-r1.md, which approved the content).

What it changes in the paper

Replay paragraph (#sec-decompose)

• Weak identification, after the threshold rates: "In 60 of the 230 moved matches, 31 of them above the threshold, the issued benefit lies within $5 of a stretch where the solver's benefit formula is flat in the moved input (the maximum allotment in 53, the minimum benefit in 4 and rent past the shelter-deduction cap in 3). By that formula every amount of the input on the stretch reproduces the issued benefit, so the match bounds the input on one side at most and is weakly identified. In 34 of the 53 the issued benefit is the maximum allotment and the solver lowered an income; the solver pays the maximum whenever net income is zero or less, so every value of that income up to the point where net income reaches zero yields it."

  • This is the caveat review round 3 of Correct retired SNAP QC claims in the paper, README, FACTS and simulator #95 called "the natural caveat on the 78.4% and 91.4%" (R8).
  • The brief named the 34. An adversarial pass (8 independent solver ports) showed the same argument covers all 60, so the paper gives both.

• Accounting of the 37 misses. The solver's-limits sentence now covers each one exactly once:

  • 17 where nothing moved, which the paper already said;
  • "In 6 more, its $3 steps reached within $3 of the issued benefit before the utility reset moved the input off that match";
  • 9 where a stop rule ended the steps short;
  • 5 household-size misses.

  This is round 3's R7. It goes beyond the brief, which asked for the weak-identification caveat only.

• "Of the 10 [computational misses], 2 are among the 6 the utility reset moved off a match."

• "a stepped utility amount is then reset" now reads "the utility amount is then reset". Two rows were reset without a step.

Footnote and FACTS D4. The stop rules now say "at or above the maximum allotment" (202311-40369 has RAWBEN above it). They also add the mirror rule: at or below a one- or two-person unit's minimum benefit, income-raising steps run until the uncapped benefit falls below zero.

New FACTS rows

• D5: weak identification (60 = 53 + 4 + 3; 34 income cuts at the maximum; the at_max flag's 54).
• D6: what the solver did in the 20 moved misses, the 3 moves that end farther from RAWBEN than FSBEN, and the audit's two checks of D4.

Abstract

The abstract is unchanged. Its claim, that the engine "reproduces the issued benefit within $5 for 78.4% … and 91.4% … which is consistent with correct arithmetic on a wrong input", still holds for every weakly identified match. The caveat qualifies how much those matches show; it doesn't make the sentence false. So this PR needs no decision of its own. Its merge waits only on #95, which is Max's call (d933).

Rendering and revision

• Re-rendered with cd paper && quarto render index.qmd (Quarto 1.9.36). paper/out/index.html and index.pdf are copied to app/public/paper/web/.
• The HTML diff is the edited paragraphs, the footnote and the date. The PDF is still 27 pages.
• Revision 10 is not yet published (production serves revision 9 until d934), so it keeps its number. Its date moves to 2026-10-04 in index.qmd and the wrapper, and the wrapper's cache key moves to r10-20261004.

Tests

• tests/test_retired_claims.py:
  • replay_quotes locks the new sentences against claims_audit.json (case_level.solver_outcomes);
  • the Hypothesis count strategy draws the new counts;
  • test_replay_counts_partition checks that the stretches sum to 60, that 34 ≤ 53, and that 14 + 3 + 6 + 9 + 5 = 37 exactly;
  • test_facts_quotes_the_solver_outcomes locks D5 and D6.
• Mutation checks. Each of these fails a lock:
  • 60 → 61 in the paragraph;
  • an altered miss-accounting sentence;
  • 4 → 5 in D5.
• Result: tests/test_retired_claims.py, test_paper_embed.py, test_amterr_lab.py, test_asset_versioning.py and test_cause_shares.py give 73 passed, none skipped.

axiom: n/a: manuscript text and fact catalog; no policy encoding changes

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