Scoped negative result: three-mode P112 COPY64 despite four span-loss exceptions

[bbMisk]

Change and mathematical scope

Scoped negative result; no new kappa. Mixing any three P112 coordinate terms with one nonzero free rank-one correction by arbitrary GL(4,C) cannot lower the separate COPY bill below 64 when both correction supports have at least two indices. At least three replacement terms have COPY rank at least eight; the separate power bill cannot improve for 0<q<=1.

The two-mode span lemma from #1473 fails in exactly four representative three-mode cases, each a 24-by-2 rectangle. This package classifies those exceptions and proves the COPY floor using their surviving pairing ranks. Identity mixing attains 64; the explicit cancellation costs 70 after compensation. The proof covers all 56 coordinate triples through verified symmetries.

Only research/p112-three-mode-seed40/ is added. Singleton-sided corrections, more mixed modes, multiple corrections and shared COPY billing remain outside scope. No physical supplier, complete recurrence, compiler/hosting/precision admission or full multiplication theorem is claimed.

Task and overlap

Base: ef5f39403610d231a0c27e8a6b4f08c10209bcd6. OS assignment seed 40, structural decoder research; complex-masked-decoder card, refined scope. TASK/CLAIM record three candidates, the seed-selected triple (2,4,5), exact domain and stopping rule.

Nearest predecessor: #1473 at ea2f6bf0940580dfaeb393ec0786abc8f7bbb7b4, which explicitly leaves three-mode support classification open. The additional evidence is the 39-stratum classification, four counterexamples to extending its span lemma, and the repaired four-term bound. #1432, #1460 and #1468 concern two-mode corrections; #1325 retains fixed readers. No pending theorem is imported.

The complete overlap refresh covered 756 open PRs and 191 discussions with no warnings; the nearest head and latest PRs were rechecked. No pinned coordination issue existed, so the claim is local. Exact Git blobs/SHA256 pins and overlap limits are included. Main and all pinned inputs remained unchanged on refresh.

Validation

Python 3.12.13, standard library only:

python3.12 -B scripts/research_package.py --package research/p112-three-mode-seed40

Expected: PASS_REGISTERED_RESEARCH_PACKAGE research/p112-three-mode-seed40 outcome=negative-result.

Local registered replay passes: 19,926 character-support cuts; six Gaussian-rational fixtures / 75,264 entries; eight controls. A code-disjoint checker independently constructs Euclidean roots and rational flats, checks all 242,424 support/erasure cases, 96 single-row restorations and 702,464 transport pairing entries, and verifies integer compensation.

Six external corruption tests, 20 research workflow tests, selected-record integrity and baseline bindings pass. Both checkers also pass in a read-only relocated export under Python 3.14.6 with -I -O -B; all 13 file hashes remain unchanged. Finite premises support a written quantified proof, not proof-assistant formalization or independent human review. Selected cold replay, new native admission and hosted-CI success are not claimed.

Credit

Hanson Sun / #1473 and #1432; commdata2338 / #1375; Cinna Davis / #1373; Arham Tahir / #1331; Gareth Davidson / #1468; Mihir Srivastava / #845; DreamingOfClouds / #717/446/595; themilkman216 / #968; Joel Pulikkan / GamingPuzzled / #620; and woklerx / #1077. Exact source context is retained in the package.

Prepared for Hanson Sun (@bbMisk) with substantial OpenAI Codex assistance in mathematics, both implementations, verification and writing. The two implementations share the same AI-assisted author. Apache-2.0.

Open next questions

README/TASK identify singleton-sided three-mode corrections and four-mode mixing. Each first requires a new erasure/support classification and a coupled compensation bound; neither follows by rerunning this excluded class. Physical costs remain an additional obligation for any positive construction.

[Mister-Meeseeks]

Batch review at 403158808f075dfafcd80da09f630b9a3fde6baa.

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

Reviewed scope: Complete three-coordinate support flats and exceptional erasures support COPY64 under GL4 mixing with a free correction whose two sides have size at least two. All 56 transports and independent premises pass; singleton sides and different sharing conventions remain open.

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.