[Arham-Tahir64]

Changing one odd entry of the P112 decoder creates a rank-nine completion and allows all nine factor terms to change. This package proves that every exact nine-term complex factorization still has separate COPY cost at least 64, with sharp power floor sum rho_k^q >= 8*8^q for 0<q<=1. It adds only research/p112-singleton-rank9/.

The result covers any of the 5,824 ordered odd locations and any nonzero complex amplitude, with ordinary transpose and the retained separate label-pairing cost convention. The proof classifies reduced supports using all 511 binary hyperplanes after each singleton deletion, then uses the forced factor spaces and cross-coefficient identities to price the compensation. A nontrivial attaining profile (7,8,8,8,8,8,8,8,1) illustrates why a cost-seven replacement cannot be accepted without its extra term.

Evidence and reproduction

python3.12 -B scripts/research_package.py --package research/p112-singleton-rank9

Local registered Python 3.12.14 replay passed: 57,232 hyperplane cases, 1,008 exact reduced-normal kernels, 14,400 root-hyperplane pairings, all odd singleton locations, five exact Q(i) fixtures / 62,720 matrix entries, and 13 tests including 12 failure controls. Optimized read-only relocation also passed. Modular rank is used only with explicit rational null vectors or a full-rank minor to justify complex ranks. The quantified argument is written in PROOF.md; finite tests alone are not claimed to quantify all coefficients. Exact source pins, environments, coverage and unrun obligations are in the package metadata. No hosted-CI success is claimed here.

Task, overlap and credit

Started from current main bce54fb3f895cbe5cd4c0eb477c3afeed100190f, chooser nonce 8546cf43704f750d58dc187afac16177, task complex-masked-decoder, refined to one changed entry with all nine factors free. Discussion1360 records the claim. The final complete overlap snapshot covered 726 open PRs and 231 Discussion records without warnings; it does not establish novelty.

PR1331 allows arbitrary rectangles but one mixed coordinate mode; PR1375/1432/1473 and Gareth Davidson's PR1468 cover broader correction supports with two mixed coordinate modes. This package removes the restriction on factor changes for a singleton correction and supplies a different whole-factor-space argument. Exact heads and predecessor attribution are retained in TASK.json. Credit DreamingOfClouds PR717/595, Mihir Srivastava PR845, themilkman216 PR968, GamingPuzzled / Joel Pulikkan PR620, woklerx PR1077, Arham Tahir PR1331, commdata2338 PR1375, and Hanson Sun PR1432/1473. PR1224's nine-term all-pairs result is contextual, not assumed.

Limits and next questions

This is a scoped negative result, not a new supplier or kappa. Multiple changed odd entries, larger rectangles, ten or more terms, other punctures and shared-copy pricing remain outside the theorem. No whole supplier invoice, physical gathering, frames, restoration, hosting, precision/tape, full cold replay or formal multiplication theorem is supplied. The selected frontier and all frozen inputs remain unchanged. The README gives concrete next steps for two changed entries and ten-term factorizations.

Prepared by Arham Tahir with substantial OpenAI Codex assistance in the research, proof, checker, tests and documentation. Apache-2.0. Independent human review of the universal proof remains welcome.

[Mister-Meeseeks]

Batch review at 264b8dcfc56584797ebf36d69ee35de18313fd27.

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

Reviewed scope: Complete support-kernel premises plus the written eight/nine mixed-term argument establish the sharp separate COPY64 and power-moment floor for exactly nine factors and one changed odd entry. Arbitrary rectangles, additional factors 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.