Scoped negative result: arbitrary free rectangles retain COPY64 for P112 pair (4,6)

[bbMisk]

Change and mathematical scope

Scoped negative result; no new kappa. This closes the exactly-two projected-product branch left open by Cinna Davis's PR #1373. For the fixed P112 pair (4,6), arbitrary GL(3,C) mixing of two coordinate terms with one nonzero free rectangle retains sharp whole-decoder COPY sum 64 and cannot lower the separate COPY power bill for 0 < t <= 1 at unchanged width and normalization.

The new compensation argument gives |U| <= 2^(r_comp+1)-1. If the compensation rank is at most one, at most three rows can disappear, preserving both other ranks. Otherwise the ranks dominate (7,7,2), whose concave power bill is at least that of (8,8,0). The three-product argument belongs to PR #1373; both new implementations replay its finite premise before combining the branches.

A literal failure control shows why all terms matter: omitting compensation falsely reports 63, while the complete bill is 70. Identity mixing attains 64, including a rank-nine masked completion.

Only research/p112-two-product-compensation/ is added. The written proof states all hypotheses. Other coordinate pairs, zero-factor families, more mixed coordinates, multiple rectangles, shared copies and changed normalization are outside scope. No physical supplier, full paid recurrence, new multiplication bound or formal multiplication theorem is claimed.

Task and overlap

• OS seed 46, structural decoder track; complex-masked-decoder card, scope refined to the open two-product branch. Three candidate tasks and the local claim are in TASK.json. No pinned sprint issue existed.
• Exact base: cd91ac8e2652bfee6bf09da3b8724253bcf78f5f; contribution: 8d53667d9bb09d9b1ffb0f865b5c8dc230eb02f6. Refreshed upstream 410026dc5513b87db92ae98571d1d5984c69a983 preserves every live source pin and the selected record.
• PR Scoped negative result: P112 GL3 COPY floor with three active projections #1373: 3eaef3e4427068b33847dc1147c8a2e5709a970e, the credited three-product predecessor for this same pair. Its seed and pair are not claimed as novelty.
• PR Scoped obstruction: one free P112 rectangle cannot improve one coordinate COPY #1331: f08b698cc2bd74e2425486728a008e531fc35f0e, Arham Tahir's one-coordinate predecessor and survivor-containment method.
• PR Research: COPY floor 64 for two-coordinate P112 mixing with small free rectangles (no new kappa) #1375: 4a533aff2a7a87c9b36e077a31774b332a7f64bd, @commdata2338's all-pair small-rectangle result. This contribution removes the support-size cap for nonzero rectangles on (4,6).
• Concurrent PR Scoped negative result: fixed P112 24-row boundary retains COPY64 #1432: 4bcc49e3f4041018b0607da46a87d8928a6e0fd0, Hanson Sun / bbMisk's fixed 24-row (0,1) result, including zero amplitudes. Its reported (7,8,7) pattern is independently corroborated here; the new contribution is the arbitrary-support compensation-rank argument. It is not a code or theorem dependency.
• The final recorded overlap scan completely paginated 797 open PRs and 149 discussion records, without warnings; nearest heads were rechecked before submission. Source SHA256/Git-blob pins and source-specific notices are retained.

Validation

Python 3.12.13, standard library only:

python3.12 -B scripts/research_package.py --package research/p112-two-product-compensation

Expected: PASS_REGISTERED_RESEARCH_PACKAGE research/p112-two-product-compensation outcome=negative-result.

Both exact engines agree on all five strata, 2,555 character evaluations, 2,800 one-sided ranks and 145,600 credited three-product records. They also check all eight baseline coordinate charges. The primary checker verifies eight Gaussian-rational fixtures (100,352 entries) and eight semantic controls. The separate label-based checker verifies five sharp erasure boundaries and an affine omission control.

The registered replay, optimized Python 3.14.6 checks, three CLI corruption/export test groups, 20 repository workflow tests, selected-record integrity and whitespace checks pass. Receipt SHA256: 8ee4249397f879fecfac63695fe547badddd8492533279689c71079ae943aa1b. VALIDATION.json binds the proof and executed checkers.

The independent implementation covers the finite premises; it does not separately replay the Gaussian-rational fixture implementation. Both implementations have the same AI-assisted author. Full selected-supplier cold replay, native admission, hosted CI and proof-assistant checking were not run locally. The compiler, realization, hosting, precision/tape, analytic and bit-interface obligations remain inherited.

Credit

Credit Cinna Davis / PR1373; Arham Tahir / PR1331; @commdata2338 / PR1375; concurrent Hanson Sun / bbMisk / PR1432; Mihir Srivastava / PR845; DreamingOfClouds / PR717 and PR446; Joel Pulikkan / GamingPuzzled / PR620; and the wider lineage in the retained notices.

Prepared for Hanson Sun (@bbMisk) with substantial OpenAI Codex assistance in mathematics, code, checking and prose. Apache-2.0; no independent human review is claimed.

Open next questions

The README and TASK.json identify a bounded extension to three or more mixed coordinate terms: derive the minimal-active-product normal form and compensation-support bound before synthesis. New projective support classes and simultaneous survivor-rank estimates are the blockers. No gain is predicted.

[Mister-Meeseeks]

100-PR batch review at 8d53667d9bb09d9b1ffb0f865b5c8dc230eb02f6.

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

Reviewed scope: Compensation rank closes the remaining two-product branch for the fixed P112 coordinate pair and arbitrary nonzero free rectangles under GL3 mixing. Two exact premise engines, sharpness and omitted-compensation controls support COPY and concave-power lower bounds; other decoder models 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.