Scoped audit: cyclic product gaps cannot replace contraction guards

[theon199]

The weighted two-scan inverse needs bounds on individual coefficients and intermediate prefixes. A fixed gap in the complete cycle product preserves invertibility but cannot preserve its inherited precision or magnitude guard: an exact family has closure gap 3/4 at every length, while every scan cut has error at least 2^(k-w)/3. Thus target error 2^-p requires w >= p+k-1. The retained full caller rejects this family; no current acceptance bug is alleged.

Status: mathematical audit / scoped negative result; no new kappa. This adds one source-pinned package, a written all-length/all-cut proof, and a sufficient relaxed-scan certificate that includes both rounded first-pass quantities, the closure denominator, all second-pass prefixes and stored-state height. An in-memory observer checks literal reads, writes, coefficient queries and roundings. No existing source, supplier, selected record, upstream file or frozen artifact is edited.

Validation:

• python3.12 scripts/research_package.py --package research/cyclic-product-gap-audit passes: 16,800 family executions, 240 separately implemented real-block rational solves, 24 valid full-caller executions, nine semantic controls and nine focused test groups.
• 311,880 rounding events are observed. Of the family cases, 9,932 receive a sufficient certificate and 6,868 remain explicitly uncertified; every case satisfies the lattice obstruction.
• Optimized Python 3.14 replay produces the same mathematical receipt. A read-only relocated package, source/receipt corruption controls, an uncharged-read control and make selected-record-check pass.
• The final patch applies to refreshed main e6f9716621f4e01fc190617934cdfaf072bd976f; all four consumed inputs and selected kappa are unchanged. Hosted CI is separate from this local evidence.

Base: 410026dc5513b87db92ae98571d1d5984c69a983. OS assignment seed: 73, adversarial-audit track. No pinned sprint issue existed; the local claim, three candidate tasks, stopping rule and completion are recorded in TASK.json. Source Git blobs and SHA256 pins, exact witness, finite domain and remaining obligations are in the package.

Nearest overlap: #764 at 983d66f0f2e48ee500ec7eee2da9244c001c5064 sharpens the bound under pointwise contraction. This changes that hypothesis and proves why choosing a different cut cannot supply a product-gap-only guard. The final overlap snapshot covered all 812 open PRs and 166 discussion records, with complete reported pagination; it found only that predecessor and the local package.

Credit Chafik Boukhalfa's PR628/PR905 scan and wrapper, Ali Romman's PR621 kernel lineage, and Charles Bethin's PR764 bound. Prepared for Charan (@theon199) with substantial OpenAI Codex assistance in mathematics, code, testing and writing; no subagents, independent human review, formal verification or complete multiplication theorem are claimed. Coefficient preparation, arbitrary dirty storage, tape/physical integration and supplier adoption remain outside this audit. Apache-2.0.

Proof, sources, exact receipt and open next questions.

[Mister-Meeseeks]

Batch review at 9638775e11386fa5db071a94e3a589eb78074cd9.

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

Reviewed scope: All-cut cyclic-scan precision counterexample shows a cycle-product gap alone cannot bound partial-product growth; explicit relaxed sufficient recurrence is checked. The retained caller rejects the expansive family, so no current acceptance failure is claimed.

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.