Audit (no new κ): side-condition coverage of the pending T74 outer at #2044 — 554 written conditions, none load-bearing false; or-True and g² review items resolved

[thakrarsagar]

Status: adversarial audit and research note. No counterexample was found. There is no new κ and no change to the frontier, and this does not claim the audited arguments are proved.

Question

The pending T74 family claims conditional κ = 0.0016797549561255445…, which is +0.43% over the selected record. The latest head is #2044 322949a5. Its final parameter point puts β and ε within 10⁻³⁰⁰¹⁰ of 1, but many inherited PR1673 notes were written at β ≈ 0.09.

The ledger only checks what it encodes. So the audit asks: is every asymptotic side condition written in the 28 notes encoded and true at that point, true there for an elementary reason, or explicitly superseded? "Side condition" covers exponent inequalities, "o(p)", "fits the record" and "eventually". Maintainer reviews of #1673, #1758, #1876 and #1928 leave these parameter-inequality interfaces open.

Result

class	count
COVERED (ledger check or recorded identity)	262
AUTO_TRUE (elementary)	53
OBSOLETE (explicitly superseded, file:line cited)	113
STALE_TEXT_ONLY	64
NON_EXPONENT (listed, not assessed)	58
FALSE_AT_POINT (all non-load-bearing)	3
UNCOVERED_TRUE	1

No load-bearing side condition is false at the #2044 point.

• Every condition that fails there was explicitly superseded by a later note: the linear gauge, 2β<1, β<1−ε, the g-factor union row, union density and θ ≥ 1/(4d). Each of these held at PR1673's own point.
• The smallest margins are 10⁻³⁰⁰²² to 6·10⁻³⁰⁰¹⁰. They hold only eventually, which suffices for an asymptotic exponent but is not a cutoff.

Defects found. None changes κ.

1. FINE-PADDING-COMPOSITION.md:37 claims all non-main margins are ≥ 10⁻³⁰⁰¹⁰. This is false: the router row is 5.03·10⁻³⁰⁰¹¹.
2. FINE-PADDING-COMPOSITION.md:38 has a false aside about ξ being free.
3. HALO-THEOREM.md:205 asserts K₀² = o(p), which is false at the point. The theorem's explicit-metadata branch holds, but no final note records that it is the branch used.
4. The PR1786 p¹⁰ umbrella exponent is true, but no ledger row ties it to THETA_POWER.
5. Stale digits: ξ = 0.90792965… appears where the pinned value is 0.90792975…, and 62 other stale numbers remain.

Other observations:

• The offline fine-θ precision term a/(4θ) = Θ(p² log p) is handled only in PR1786 §3, not by any ledger row.
• ζ is unused by every ledger formula.
• The removals of β<1−ε, g<√B and union sparsity are load-bearing.

Open review items, both resolved.

• Conditional kappa: verified 400-digit refinement over PR1875 (+6.06686e-301) #1876/Conditional kappa refinement: 500-digit enclosures and complete native-proof reuse #1897 or True guard. Line 23 has both sign errors and would fail if made live. Nothing depends on it: the proof's shifts use live assertions, and neither the ledger nor run_outer.py uses it. A second vacuous or True sits at outer/struct_check.py:130.
• GAUGEFREE §7, "no other g² per record". No active step charges g² to one record. Every hit is superseded and rejected by a ledger control, apart from the HALO option in defect 3.

Relation to concurrent work

philippeVV published closely related checks on the same head. They are #1897 and #2044 comments with fork commits a3c40a6f, e6bc8393, bc4ed55b, 76af2562 and d1e01310, posted 2026-10-14 19:40 UTC to 2026-10-15 00:46 UTC. They include:

• the broad-shift / or True reconciliation and the 3b₀θB correction;
• an exact knockout showing the lift rests on PR1673's removal of the Vd row;
• a PR1673 fusion cost census, including the same HALO finding;
• the H-call and layout-contract analyses;
• the CRLF and stage-E issues.

That work has priority on those points, and this PR independently corroborates them. The overlap helper does not index PR comments, so this audit's overlap check missed them. This PR adds:

• a machine-checkable inventory of all written side conditions in all 28 notes (outer, PR1758 and PR1786 notes included), mapped to the 62 ledger checks;
• a registered exact verifier;
• findings 1, 2, 4 and 5, the offline-precision and ζ observations, struct_check.py:130, and the g² sweep table.

It does not repeat the knockout, the census or the release-path findings.

Verification

• Agents:
  • Four independent extractors cover the 28 notes.
  • An exact ledger re-evaluation covers 62 checks and 18 controls.
  • Every flagged item was checked by two agents, one trying to refute it and one re-deriving it.
  • A separate agent independently re-checked a seeded sample of 28 non-flagged classifications. It agreed on 27; the mistranslation it found, together with a second instance of the same error, is corrected and does not change any truth value.
  • All 38 verdicts are recorded in INVENTORY.json.
• verify.py:
  • rebuilds the final point from one pinned 30,000-digit rational plus closed forms;
  • evaluates the 133 keys at 4 points with an AST-whitelisted exact evaluator;
  • enforces the classification rules;
  • recomputes 49 ledger checks and 7 controls;
  • runs 14 negative controls (16 with --source-root).
• review_items.py: exact Laurent-polynomial certificates and final-point exponent checks. It runs 88 checks by default; with --source-root it runs 122, re-checking 108 citations.

Reproduce

From the repository root, Python 3.10+ standard library only, about 35 s:

python3 -I -B research/t74-side-condition-coverage-audit/verify.py
python3 -I -B research/t74-side-condition-coverage-audit/review_items.py

The last lines should begin PASS_T74_SIDE_CONDITION_COVERAGE conditions=554 keys=133 covered=262 … load_bearing_false=0 and PASS_T74_REVIEW_ITEMS checks=88 mode=default ….

To check against the audited sources, export #2044 head 322949a5 and pass --source-root DIR to both scripts. This re-hashes 32 files and locates every quote.

Checks run locally:

• The default and source-root modes of both scripts pass on CPython 3.14.4.
• research_package.py --check-only gives PASS_PACKAGE_METADATA_AND_PINS.
• The 65 test_research_*.py tests pass.
• The registered Python 3.12 replay was not run (3.12 is not installed); the scripts parse under the 3.10 grammar.

Scope

• Checked: the written side conditions and their exact truth at the point.
• Not checked:
  • the arguments themselves (accretive Toeplitz theorem, SHALLOW-TOEPLITZ, union-schedule correctness, halo/locality, Trudgian step);
  • the all-size, hosting and precision/tape interfaces;
  • supplier admission;
  • code-only assumptions.
• Extraction limits: extraction is exhaustive over the prose, but it is model reading checked by verifiers, so a condition no extractor recognised is not covered.
• Head: the inventory is pinned to 322949a5. Conditional κ = 0.00167975495612554452133567099392070257699831143850258029546032752016286454826032563882630565613284: tighter free epsilon over PR2044 #2161 and Conditional κ = 0.0016797549561255445213356709939207025769983114385025802954603275201628645482603256388263...: fixed-grid ceiling over PR2161 #2180 would need their final point re-substituted.

Provenance and credit

• Base and refresh: 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2, priority revision 4. Canonical main was unchanged at submission (2026-10-15 03:30 UTC).
• Track and claim: track 60–74, OS seed 72. No sprint coordination issue exists, so the claim was recorded locally.
• Credit:
  • Vlado Vojdanovski (PR1673);
  • Jeremy Kothe and the PR1758/1839/2044 contributors;
  • Chafik Boukhalfa (PR1786);
  • Rohan Arun (PR1876) and M6LI (PR1897);
  • LeverRank;
  • Philippe Van Velzen (concurrent audit, above).

Prepared for Sagar Thakrar with substantial AI assistance (Claude Code, Anthropic). Subagents did the extraction, matching, verification, sampling and verifier construction. The main session checked GAUGEFREE Lemmas 2.1, 3.1–3.3 and Theorem 4.1(a)(b) by hand and reran every command. Apache-2.0.

🤖 Generated with Claude Code