Certify PR288's restricted sign-gauge obstruction with four constraints (no new κ)

[erebos-labs]

Change and mathematical scope

PR288 §1.4(d) already reports the restricted sign-gauge obstruction for the Braverman–He PG(2,9) instance. This contribution supplies an explicit four-constraint witness, three inspectable five-edge paths and a separate exact verifier. Two session signs are forced to be both equal and opposite. No new multiplication bound, κ, exponent or practical speedup is claimed.

The model has fixed additive 5×5 rational vertex matrices, rank-at-most-one physical edge differences and independently signed session targets equal to one common invertible matrix. The result does not cover nonadditive or varying frames, unrelated Levi targets, higher-rank edges or larger dimensions. The matrix-to-sign implication is an ordinary written proof, not a Lean proof, independent human peer review or verification of the full multiplication theorem. Unprovided SAT/MITM/Gröbner searches are outside this contribution.

The standalone package contains the normalized instance, certificate, proof, independent checker, deterministic producer and 24 corruption/positive-control tests. Its checker imports neither the producer nor upstream checker. A new Make target and split CI job check the package; existing algorithms and witnesses are unchanged. Scope: ten research files, five Makefile lines and nineteen workflow lines.

Sources are pinned to upstream main 12a4934d15006e8c10f9b2264f13f8f341be802b and PR288's head 475a088b5862f75f0b10596cfd6842d19531a5ec in eumemic/integer-mult-bounds. The original checker already excludes R's trailing paper-page-number 13. The normalized 175 R, 182 π, nine C and nine J entries matched the primary paper and pinned companion data in the preparation audit. Immutable references and input hashes are in SOURCES.md.

Validation

Local environment: macOS arm64, CPython 3.14.7; isolated SymPy 1.14.0 environment for host gates. The new certificate package itself uses only the standard library.

• make pr288-sign-cycle-verify: all five stages passed, including 24 tests normally, the same 24 under -O, byte-identical deterministic regeneration and verification of the regenerated certificate. The preparation audit also passed isolated replay, independent positive/negative CLI controls and regeneration with three hash seeds.
• Baseline and candidate make gen4-bank-verify, invoked within their full runs: both completed all nine fresh stages, 112 immutable inputs and nine omitted-stage controls. Inputs remained unchanged and mathematical certificates were byte-identical. This preserves the host result; it is not a new result of this sign-cycle contribution.
• Full make verify: incomplete in both checkouts under separate 1,200-second wall limits. Baseline stopped in scripts/complex_network.py; candidate stopped in scripts/paired_network.py, both within verify-certificates. Later groups are not cleared. No mathematical check failure was observed before termination. A redundant standalone gen4 repeat was stopped after 213 seconds; its second repeat was not started. The gen4 successes above refer to the completed embedded runs.
• Focused make joint-dual-verify: fails in pristine baseline and candidate, exit 2, AssertionError: Makefile at joint_dual_compose.py:70, before mathematical replay. Historical SOURCE.json expects Makefile SHA-256 768131bd4ce18433f6abccdc9812a38c92e11220e3a1019c6d95334b42f529d1; pristine baseline is cd0fb59dc078d1fdf54a163ea13bb3660d03c8fe69c1a7f99fc8e5f476a3d5c2, candidate is bfe01daff2349b7260d6580ee2b1ce149fddf98ac1a59aca5bbca102299f0958.
• PR318 proposes repairing the historical bindings and was still open at the final freshness check. Its fix was not applied here. The candidate's intentional Makefile change also requires reviewed binding regeneration before that gate can pass.
• Local YAML validation and patch whitespace/application checks passed. The configured Linux/Python 3.11, 3.13 and 3.14 CI matrix had not run at submission; local tests do not substitute for it.

This is a draft for review, not a claim of merge readiness. Completing repository verification, reviewing the historical binding repair, running CI and reviewing the ordinary proof remain required before merging.

Braverman and He receive instance credit; eumemic/PR288 receives priority for the known obstruction and original checking work. The explicit witness and separate checking implementation are the addition. Apache-2.0 attribution and substantial OpenAI ChatGPT/Codex assistance are disclosed in NOTICE; PR288's Anthropic assistance disclosure is retained. Companion source code and the full paper are not redistributed. No worldwide novelty or exclusive-ownership claim is made.