CLAIM: sharp copy floor for all rank-nine factorizations of a single odd-entry P112 completion

[Arham-Tahir64]

DONE — PR #1591, head 264b8dcfc56584797ebf36d69ee35de18313fd27, on starting main bce54fb3f895cbe5cd4c0eb477c3afeed100190f. Claim released.

Established a sharp scoped theorem: every exact nine-term complex factorization of a nonzero single-odd-entry P112 completion has total separate COPY cost at least 64 and power bill at least 8*8^q for 0<q<=1. All factor terms may change. The completion has rank nine, forcing the factor spaces; the proof combines the complete reduced-normal classification with a coupled compensation argument.

Evidence: 57,232 hyperplane/deletion cases; 1,008 exact reduced-normal kernels; 14,400 root-hyperplane pairings; all 5,824 ordered odd locations; five exact Q(i) fixtures / 62,720 entries. Thirteen tests including twelve failure controls, registered Python 3.12.14 replay, optimized read-only relocation and selected-record integrity pass. The finite facts support a written universal proof; no independent human review or formal multiplication theorem is claimed.

The nontrivial profile (7,8,8,8,8,8,8,8,1) attains the sum floor and shows why discarding the compensation is invalid. No whole supplier invoice, physical realization or new kappa is supplied. All predecessor files and the selected record remain unchanged. Larger corrections, ten or more terms and shared-copy pricing remain outside scope.

Reproduce: python3.12 -B scripts/research_package.py --package research/p112-singleton-rank9. Open next questions and exact pins are in the package. Final complete overlap refresh covered 726 open PRs and 231 Discussion records without warnings. PR1331/1375/1432/1473/1468 are acknowledged: this result restricts the correction to one entry but frees all nine factor terms. Credit DreamingOfClouds, Mihir Srivastava, themilkman216, GamingPuzzled / Joel Pulikkan, woklerx, commdata2338, Hanson Sun and Gareth Davidson; exact heads are retained.

Prepared by Arham Tahir with substantial OpenAI Codex assistance; Apache-2.0.

--- Original claim retained ---

CLAIM

Question: for P112, does every exact nine-term complex factorization of K = 2G + lambda e_alpha e_beta^T, with alpha,beta live odd-paired root lines and lambda nonzero, have total binary label-pairing copy cost at least 64? All nine factor terms may change. The completion has rank nine, so both factor spaces are forced by K.

Base: bce54fb. Task complex-masked-decoder, chooser nonce 8546cf43704f750d58dc187afac16177, scope refined to one-entry completions, arbitrary GL(9,C). Predecessor geometry and cost convention: DreamingOfClouds PR717, Mihir Srivastava PR845, themilkman216 PR968, GamingPuzzled PR620.

Method/domain: enumerate the 511 binary label hyperplanes after deleting each of the 112 possible singleton rows. Exact root-kernel certificates classify every complex reduced-support normal; a coefficient-rank argument then handles all nine-term factorizations. Include exact rank-nine attaining fixtures, dense factor-basis changes and meaningful failure controls. Target is a sharp scoped obstruction, not a physical supplier or new kappa.

Overlap: PR1331 (f08b698, Arham Tahir) handles arbitrary free rectangles but one mixed coordinate term. The active GL(3,C) singleton follow-up in Discussion1316 is deliberately acknowledged: this is an independent, larger factor-space proof for all eight coordinate modes and every nine-term factorization of the one-entry completion; it uses a different 511-hyperplane classification rather than two-coordinate cancellation strata. PR1180/1191/1212/1213/1224/1236 are contextual leads, not assumed theorem dependencies. Larger free rectangles and multiple changed odd entries remain outside scope.

Expected artifact: research/p112-singleton-rank9/, source pins, a written quantified proof, standard-library finite checker and controls. About 60 minutes, final quarter reserved for checking. Claim expires 2026-10-14T01:25:00Z.

Prepared for Arham Tahir (@Arham-Tahir64) with substantial OpenAI Codex assistance; Apache-2.0.