DONE: P112 singleton corroboration and equality detail

[an664]

DONE (2026-10-14) - corroborating finding and equality detail, with the full finding and proof summary and public pinned fork evidence at 340e0c997a41a4279177559bdb574ebeb3831c61. Scope released; no duplicate upstream theorem PR.

Hanson Sun / @bbMisk's PR1613, head a34723742801f877c9636ed72f176a7ffa7bf005, presents a broader written proof for all 56 triples and explicitly credits this earlier parallel claim. Our independent fixed J=(2,4,5), actual |U|>=2, singleton-right/GL4 reconstruction corroborates the common COPY64 and power floor. A separate Pro review gives the common-pure-column equality condition and exact nonisolated COPY65 / power 8*8^q+1 within this fixed scope; no novelty claim relative to the complete PR1613 package is made.

The public package retains the producer proof/checkers, exact source pins, a labeled chat-transferred Pro proof supplement and bounded replay. Local original-checker normal/optimized runs and the registered three-command Python 3.12.14 replay pass. The explicit 672 anchor cases, three equality-boundary fixtures and two false-inference controls pass; a corrupted saved invoice rejects under optimization. The reviewer did not audit the producer programs, and we did not audit PR1613's code or broader transports. No formal/human proof review, physical supplier, kappa or selected-frontier claim.

Credit Hanson Sun / PR1512/1473/1613, Gareth Davidson / @bitplane / PR1468, Arham Tahir / PR1591, and the retained PR1375/1373/1331/845/717/968/620/1077 lineage. Andrey Mas with substantial GPT-6 Pro mathematical/checker work and a separate Pro review through ChatGPT; Codex source coordination, execution audit, local replay and packaging. Apache-2.0.

Original claim and source history

CLAIM: singleton-column correction with three P112 coordinate modes.

Question: fix J=(2,4,5), Y=X[:,J] in the pinned P112 model. For beta among the 112 live indices, v=lambda e_beta (lambda nonzero), arbitrary complex u with actual support U of size at least two and B[U,{beta}]=0, and arbitrary M in GL(4,C), set D=[Y,u]M and T=[Y,v]M^(-T). Keep the other five coordinate terms unchanged. Can the complete separate COPY sum 40+sum_j rank_F2 B[supp D_j,supp T_j] be below 64, or the separate power bill below 8*8^q for 0<q<=1? Every compensation term is charged; transpose is ordinary transpose.

This is the explicit singleton-sided followup of Hanson Sun/@bbMisk PR1512, pinned at 403158808f075dfafcd80da09f630b9a3fde6baa. Its theorem assumes both correction supports have at least two entries; its one-anchor control already shows that the existing span-preservation argument does not extend unchanged. First classify one-anchor losses in the published 39 strata, then settle the coupled four-term bill or report a rigorously bounded partial result.

Scope is refined from the complex-masked-decoder card to this prerequisite. Main checkout base 55a02580ce1b283026e9d3b3510b218f8c93a4de, priority revision 2; a newer main c5108f0 was observed during the direct source check. No changes to selected records. More modes, multiple corrections, shared COPY billing, and the one-entry |U|=|V|=1 domain are excluded.

Overlap: complete 02:38/02:39 UTC helper coverage (719 open PRs, 207 Discussion records; no warnings), all-state targeted searches, then direct PR1512/Discussion1360 reads at 02:44 UTC. PR1512 head is unchanged with no comments/reviews. Discussion1360, Arham Tahir, reserves single-entry completions with all nine factor terms variable; this study deliberately keeps |U|>=2 and mixes only the designated three modes. PR1468, Gareth Davidson/@bitplane, head 5e2d8ec8ebff17c85e813ef1bd6f06312c1fe16a, is the two-mode singleton predecessor. No claim of blanket novelty from search coverage.

Expected artifact: a scoped proof/counterexample with exact source-bound finite premises and meaningful failure controls, or a precise useful handoff. An algebraic identity and support-rank certificate would be feasibility evidence only; any positive supplier would still need preparation, frames, helper/restoration, hosting, precision and complete paid recurrence. No physical supplier or new kappa claimed.

Prepared by Andrey Mas with substantial GPT-6 Pro research through ChatGPT and Codex source coordination/local verification. Credit Hanson Sun/PR1512 and 1473, Gareth Davidson/PR1468, Arham Tahir/PR1331 and 1360, and the retained PR717/845/968/620/1077 lineage. Apache-2.0.

Budget: about 90 minutes study plus 30 minutes verification/handoff. Claim expires 2026-10-14T04:47:32+00:00.

[an664]

Corroborating finding / scope released. Hanson Sun / @bbMisk's PR1613, head a34723742801f877c9636ed72f176a7ffa7bf005, presents a written proof of the broader all-56-triple one-column COPY64 and 8*8^q floor and explicitly credits this earlier claim. Our independently produced fixed-triple investigation reaches the same core conclusion. We will not open a duplicate theorem PR.

Public proof and checkers: pinned fork-only evidence package, commit 340e0c997a41a4279177559bdb574ebeb3831c61; exact Pro review Sections 5-7 with provenance. From that pinned fork checkout:

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

The registered three-command Python 3.12.14 replay passed, as did final metadata/source validation. No upstream duplicate PR was opened.

A separate GPT-6 Pro review gives the following equality description within exactly this claim's scope: J=(2,4,5), actual |supp(u)|>=2, free singleton right correction v=lambda e_beta, nonzero lambda, arbitrary complex M in GL4, five unchanged coordinate terms and separate billing. Write the columns of M and M^(-T) as (a_j,c_j) and (b_j,d_j), with a_j,b_j in C^3.

• Equality at COPY64, or at power 8*8^q for any 0<q<=1, holds exactly when some common column has a_p=b_p=0 and the other three prices are each eight. After common column permutation, the transformation has an arbitrary invertible three-by-three block and an isolated correction block; equality does not force individually unmixed modes.
• When there is no common zero column, the exact minima for each fixed admissible correction are COPY65 and power 8*8^q+1. The explicit M=I+E_(0,3), N=I-E_(3,0) attains ranks (8,8,8,1). The reviewed lower bound uses the four-case componentwise floors and a chord comparison, not a rank-sum-only argument. No claim of novelty for this refinement relative to the complete PR1613 package is made.

The nonisolated lower envelopes in the Pro proof are (7,7,7,7), (8,8,8,1) and (7,7,7,6), up to permutation. With z=8^q>1, the chord bound x^q >= (8-x)/7 + (x-1)z/7 for 1<=x<=8 gives the first envelope at least 1+3z+3(z-1)/7 and the last at least 1+3z+2(z-1)/7; the middle is exactly 1+3z. Adding the five unchanged terms gives the 8z+1 floor. The stated unipotent matrix attains it. This summary is transcribed from the separate Pro review, whose exact Sections 5-7 and finite-example checker are retained with the evidence.

Local CPython 3.12.14 replay of our original two same-session checkers passed normal and optimized modes: all 39 support strata, 170,352 anchor pairings, 19,926 nonempty functional cuts, 4,368 single deletions, five exact fixtures and nine primary rejection controls. The separate Pro reviewer reconstructed the finite premises from the theorem transcription but did not audit those original programs.

A further local replay of the review's explicit examples passed normal and optimized modes with identical JSON: 672 coordinate/anchor/in-or-out support ranks all equal eight, plus a nonmonomial isolated equality fixture (8,8,8,0), an isolated nonoptimal fixture (9,9,9,0) and the nonisolated attaining fixture (8,8,8,1). The two false inferences “equality forces monomial mixing” and “isolation alone guarantees equality” are rejected. These checks do not enumerate arbitrary complex amplitudes or the real exponent interval; the universal conclusions remain the Pro-authored written argument.

This corroborates the common fixed-triple result, not PR1613's implementation, all 56 transports, zero/single-entry extensions or physical admission. PR1591 remains distinct completed single-entry/all-nine-factor work. The proof, immutable producer files, original local receipts, transported review text, new mechanical checks and overlap record are retained in a reproducible evidence bundle. Review text was recovered through chat and is not authenticated as byte-identical to the original downloadable reviewer file; reviewer-code replay is not claimed.

Credit Hanson Sun / PR1512 and PR1613, Gareth Davidson / @bitplane / PR1468, Hanson Sun / PR1473, Arham Tahir / PR1591, and the PR1375/1373/1331/845/717/968/620/1077 lineage. Andrey Mas with substantial GPT-6 Pro mathematical/checker work and separate Pro review through ChatGPT; Codex source coordination, execution audit, local replay and packaging. Apache-2.0. No formal or human proof review, new supplier, kappa or selected-frontier change.

Final overlap: 741 open PRs and 267 Discussion body/comment/reply records at 04:26-04:28 UTC, complete pagination/no warnings; direct PR1613 head/body unchanged and this claim had no comments. Nearby PR1325 concerns fixed-reader sparse-lift equality in a different model. These source comparisons do not establish novelty or validate the neighboring proofs.