Independent audit: #1560's zero-exclusion Theorem 1 / Corollary 2 and their use in #1616/#1650 (no gap; no new κ)

[JosieElliston]

Separately authored audit of #1560's Theorem 1 (structural zero exclusion) and Corollary 2 (gauge form), and of how #1616 and #1650 use them. Corroborated; no gap found. No new κ, bound or rigidity claim.

#1560 (@an664, with GPT-6 Pro) states no independent human review. #1616 and #1650 apply its criterion at hundreds of pivot frames for the P107, P106 and P105 rigidity results, and cite the theorem without re-deriving it. This package checks that step.

What was checked

1. Written audit of proof.md §2–§4 and §6, step by step, over an arbitrary characteristic-zero field (README table). No gap. One expository omission is filled: §6 says the pivot entries "have the required form" without the calculation. The README gives it, and Corollary 2 holds as stated.

2. Implicit finite facts at every pivot. The proof uses E8 facts relative to F that it never states. prove a frame-incidence zero exclusion criterion with a P107 fixture #1560's checker covers only its own pivot. verify.py was written from the prose alone and checks these facts exactly at all 2025 frames, in about 50 s, standard library only:

   • the expansion 2v_i = Σ G_ik v_k;
   • |T_i| = 4 and G ∈ {0, ±1} off the pivot;
   • the 14 blocks form an S(3,4,8) with 8 lines each;
   • 4+4 orthogonal classes by sign product, each completing to a frame;
   • Hadamard class rows;
   • every exceptional 4+1 configuration (3,628,800) gives w_j = ±1/2;
   • the class-sum dichotomy;
   • Σ_k ε_ik ε_jk = 2G_ij.

   Three negative controls are included.

3. Binding. Both criterion functions were read against §3. They test exactly (D), (G), (W) and nonempty blocks with the full live-frame family. For (D) they use a modular rank lower bound plus the seven kernel vectors, whose membership is identity (6), checked for every frame. binding_crosscheck.py compares Research: rank-8 rigidity on every P105 unit via #1560's criterion, and (G) does not imply (D) for frame sub-families (no new κ) #1650's criterion with a second implementation built on different exact methods: classes by orthogonality, Q-rank of MᵀM by Bareiss elimination, and BFS. All 4354 sampled cases agree (5–10-line deletions, including every live frame of five deletion sets), and so do 1503 thin-family cases (11–40-line deletions). The fixtures include session 13's full-family (G)-without-(D) pivot. The comparison is thin for (G) and (D), because Research: rank-8 rigidity on every P105 unit via #1560's criterion, and (G) does not imply (D) for frame sub-families (no new κ) #1650 stops at its first failing test.

Not covered: #1212 §8 (floor from gauge rigidity) and §10 (stabiliser transport), which #1616/#1650's floors also need. Not covered either: whether (G) implies (D) (session 13's claim in #1502), or a proof-assistant formalization. Part 3 is a same-account check, since #1616/#1650 came from earlier Claude sessions on this account. Parts 1–2 do not use that code.

Task card: formal-interface-lemma (chooser nonce a2a3ecb2c0544e92b8d811b546d670c3), scope refined to this theorem-to-certificate binding (TASK.json). Base main a18b4b01. Claim: #1502 (comment)

Checks run

• python3 -I -B research/zero-exclusion-proof-audit/verify.py on 3.10, 3.12 and 3.14 (also with -O).
• python3 scripts/research_package.py --package research/zero-exclusion-proof-audit --check-only, plus the registered replay.
• The cross-check as stated in the README.
• A Codex read-only adversarial review found no mathematical gap. It raised three low-severity points (one control was really an illustration, the Python version was unstated, one overbroad phrase), all fixed in c29103af.

Credit: Andrey Mas (@an664) with GPT-6 Pro for the theorem, proof and fixture. PR1560 also credits Brett Halligan / @burntout76 / Ninefold (#1212), @DreamingOfClouds (#717) and @themilkman216 (#968). Prepared by Josie Elliston with Claude Opus 5.5 (Anthropic) via Claude Code (substantial AI assistance). Apache-2.0.

🤖 Generated with Claude Code

[Mister-Meeseeks]

Batch review at c29103afcc252aeff3c89e8164b335b25c40096d.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Written audit of PR1560 zero-exclusion argument, explicit pivot-entry gauge calculation and complete E8 facts at all2025 pivots. Code-binding samples are bounded corroboration; PR1212 transport/floor and full unit census remain separate obligations.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.