Research: sharp COPY71 on the remaining A8 simplex P112 frame orbit (no new kappa)

[bbMisk]

Change and mathematical scope

PR #1304 leaves the fixed A8 simplex decoder on nonmatching P112 frame punctures open. This proves the sharp separate COPY cost 71 for the complementary 1,080 frames, over arbitrary complex per-row kernel shifts. An all-nine-active exact witness attains the bound. The frame census partitions all 2,025 E8 eight-frames into the predecessor's 945 matching frames and these 1,080 frames, each consisting of one pair root and a Fano plane on the remaining coordinates.

This is a scoped proof with exhaustive finite lemmas, not a supplier or κ improvement. The nine simplex normals are fixed, exactness is on all live pairs with ordinary transpose, and terms are billed separately. The COPY power bill also exceeds the eight rank-eight coordinate terms for 0<=a<1, when width, normalization and multiplicity are fixed. No full supplier invoice is claimed. Only research/a8-one-pair-frame-floor/ is added; no pending theorem is assumed for its bound.

Task and overlap

• Task complex-masked-decoder, scope refined to PR Research: sharp A8 simplex P112 copy floor 71 on matching-frame punctures (no new kappa) #1304's explicit other-frame question; task and overlap record.
• Research base: ebedc0895eb4c06725af020bc1af165722b0a3ce (priority revision 2); submission rebased onto 81e81af67714bdc8d1693b0bf5c29468b92f62ff. Revision 3 was read and this completed COPY obstruction retained; it is not a new packing/κ claim.
• Contribution head: da6308454686c77372e82a60e196ccb828285360.
• Predecessors: Research: sharp A8 simplex P112 copy floor 71 on matching-frame punctures (no new kappa) #1304 a052f0da36a3845c2ec785fd44e5b9e50de3b688; Research: sharp A8 simplex nine-term decoder copy floor 72 (no new kappa) #1248 f226f4329a996b0a1bea8794179a91b5d0ed80bf; Research: nine-term complex E8 decoders are one-sided overcomplete representations (seed 42, no new kappa) #1224 00e6ed0810b8ebc7f6133e2c036d0f1c9a3a4162. Their heads were rechecked. Source file blobs and hashes are pinned.
• Scoped claim. Submission overlap covered 733 open PRs and 244 discussion records with complete pagination at 03:52 UTC on October 14. No competing result for this orbit was found within that coverage; no novelty guarantee.

Validation

python3.12 -B scripts/research_package.py --package research/a8-one-pair-frame-floor

Passed on macOS arm64 / CPython 3.12.13. The primary checks all nine anchors: 390,915 binary planes, 20,019 partner-space tuples and 382,032 compatible choices. Two sharp witnesses have costs (0,8,9,9,9,9,9,9,9) and (1,8,8,9,9,9,9,9,9); both all-pairs identities are checked exactly. The written proof explains why the finite compatibility cover bounds every complex shift.

The second arithmetic implementation uses Gram-derived labels, nullspaces, Cartesian products and verified stabilizer transport. A separate pair/Fano construction agrees with the full frame census. Both implementations are explicitly credited adaptations of #1304; this is not independent peer review.

Ten semantic controls and all five package test groups pass, including false/missing/malformed receipts, incomplete anchor coverage, changed frame coverage, optimized execution and read-only relocation. Both mathematical checkers also pass under optimized CPython 3.14.6. The prescribed predecessor positive control, selected-result integrity check and metadata/source pins pass.

Not checked: physical gathering, frames, restoration, hosting, precision/tape, all-size/compiler, common-label/mutable-source, analytic and bit H1–H5 obligations. No selected cold replay, Lean, hosted-CI success, human review or maintainer acceptance is claimed.

Credit

commdata2338 (#1304) supplies the matching-frame result, both adapted implementations and the open question; Philipp/mightyOil (#1248) the deficiency method; joaqcoo (#1224) the normal form and A8 family. Credit the retained DreamingOfClouds (#717/#595/#446), Joel Pulikkan/GamingPuzzled (#620), themilkman216 (#968) and woklerx (#1077) lineage. Exact original file pins and adaptation notices are retained. Apache-2.0.

Prepared for Hanson Sun (@bbMisk), with substantial OpenAI Codex assistance in the mathematics, implementations, verification and writing.

Open next questions

Structured questions and limits: a different specified nine-normal family, starting with its kernel-shift and column-rank calculation; or a physical shared-copy contract with a complete bill. The fixed simplex and separate-term assumptions block transferring this theorem to those cases.

[Mister-Meeseeks]

Batch review at da6308454686c77372e82a60e196ccb828285360.

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

Reviewed scope: All-anchor deficiency enumeration and two complete frame orbits prove sharp COPY71 for the complementary A8 simplex orbit, including an all-active witness and a separate moment obstruction. Arbitrary normal families, masks and shared-copy pricing remain outside scope.

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.