Independent verification: A8 simplex COPY71 on all 2,025 P112 frame punctures

[bbMisk]

Change and mathematical scope

PR #1304 left the fixed A8 simplex nine-term decoder's other P112 frame punctures open. This package proves the sharp separate COPY cost is 71 on the remaining 1,080 Fano-frame punctures, with all nine terms active at equality. Exact orbit coverage and a replay of the earlier 945 matching frames extend the statement to all 2,025 orthogonal eight-frame punctures. Thus this fixed family cannot lower the standard P112 copied-information bill of 64.

This is independent corroboration, not a claim of unique discovery. The final overlap refresh found a concurrent investigation of exactly the same orbit, posted after the entry snapshot. At that snapshot it reported the frame census and cost-71 witness, with the lower bound still being checked. The added value here is a completed source-bound all-anchor replay plus a separately implemented Gram-derived verifier. No unpublished concurrent code is imported or assumed correct.

Only research/a8-p112-fano-frame-floor/ is added. The claim covers the fixed nine simplex normals, arbitrary complex row shifts, all-pairs exactness and separate-term binary pairing-rank billing. Other normals, masked identities, non-frame punctures, two unrestricted factors and shared-copy billing remain outside scope. No physical supplier, full moment invoice or new kappa is claimed; the selected record is unchanged.

Task and overlap

• Started from main ebedc0895eb4c06725af020bc1af165722b0a3ce; rebased onto 81e81af67714bdc8d1693b0bf5c29468b92f62ff before committing. Contribution head: 98e1164652922beec5a68c18b982d0076c096426.
• Assigned complex-masked-decoder, priority revision 2, nonce 7b0bdc126cee1998eec8c3dc04fd305c. Scope is declared changed to independent corroboration after discovering the overlap. Main's later revision 3 guidance was reviewed; the original card is historical provenance.
• Exact new representative: [0,85,94,99,105,108,112,113] in PR1304's A8 root order. All 43,435 binary planes at each of nine anchors are checked. The new orbit has 20,019 distinct partner-space tuples and 382,032 compatible choices.
• PR Research: sharp A8 simplex P112 copy floor 71 on matching-frame punctures (no new kappa) #1304, head a052f0da36a3845c2ec785fd44e5b9e50de3b688, covers the 945-frame orbit. PR Research: sharp A8 simplex nine-term decoder copy floor 72 (no new kappa) #1248, head f226f4329a996b0a1bea8794179a91b5d0ed80bf, covers P120. PR Research: nine-term complex E8 decoders are one-sided overcomplete representations (seed 42, no new kappa) #1224, head 00e6ed0810b8ebc7f6133e2c036d0f1c9a3a4162, supplies the normal form and family.
• Final paginated snapshot: 733 open PRs and 244 discussion records, 2026-10-14 03:51 UTC, no coverage warnings. The same-scope claim is explicitly credited; this search is not a novelty certificate.

Validation

python3.12 -B scripts/research_package.py --package research/a8-p112-fano-frame-floor

Passed on macOS 27.0.1 arm64, CPython 3.12.13. The registered hook runs:

• Primary exact replay: 781,830 planes across both representatives and every anchor, 2,025-frame coverage, full all-pairs identities and active cost vector (1,8,8,9,9,9,9,9,9).
• Separate implementation: Gram-derived labels, nullspace partner spaces and Cartesian products with forced-value compatibility; four anchor representatives transported to all 18 by checked stabilizers. It imports neither the primary checker nor the predecessor. Direct pairing-submatrix witness costs agree.
• Six package rejection tests, including modified source bytes, false cost and an omitted anchor. Mathematical controls include coefficient/term/label/puncture mutations and a relaxation counterexample: omitting mutual compatibility gives the false deficiency bound 11 instead of 10.

Both mathematical checkers also pass under isolated, optimized CPython 3.14.6 from a relocated read-only package. Package metadata/source pins, selected-record integrity and whitespace checks pass. Exact source and validation bindings are in TASK.json and SOURCES.json.

Not run: selected construction cold replay, native or Lean builds, and hosted CI before publication. This finite proof does not discharge the inherited all-size, realization, hosting, precision/tape, analytic or bit H1-H5 obligations. Independent human review is unclaimed.

Credit

commdata2338 / #1304 for the matching-frame result, successor question and byte-identical vendored verifier; Philipp / mightyOil / #1248 for deficiency spaces; joaqcoo / #1224 for the simplex family and normal form; DreamingOfClouds / #717, #595, #446 for the decoder model; themilkman216 / #968 and woklerx / #1077, #1180 for the surrounding bounds. The concurrent Hanson / bbMisk session linked above receives same-scope credit.

Hanson Sun (@bbMisk), with substantial OpenAI Codex assistance in research, proof, both new implementations, verification and writing. The separate implementation is by the same assistant and is not independent human or agent review. Apache-2.0 attribution is retained.

Open next questions

The package README and TASK record two scoped follow-ups: another specified nine-normal family, whose kernel ratios must first be derived; and formalizing the plane-coverage/deficiency implication, which is currently a written mathematical bridge to exhaustive Python evidence. No further frame search within this fixed simplex family is needed for this question.

[Mister-Meeseeks]

Batch review at 98e1164652922beec5a68c18b982d0076c096426.

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

Reviewed scope: Independent full two-orbit replay corroborates the fixed-simplex COPY71 result across every orthogonal P112 frame puncture. Primary all-anchor coverage and separate Gram-derived checks pass; parallel contribution and predecessor methods retain credit.

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.