Scoped negative result: two-anchor span preservation for P112 decoders

[bbMisk]

Change and mathematical scope

Scoped proof and independent structural certificate; no new kappa. A free P112 rectangle with at least two rows and two columns cannot erase a binary direction from any support of a nonzero combination of two coordinate modes. Explicit signed coordinate symmetries certify this for all 28 mode pairs.

Consequently, arbitrary GL(3,C) mixing of those two modes and one such correction has at least two replacement terms of COPY rank eight or more. With the other six modes unchanged, the sharp separate COPY sum remains 64, and the power bill remains at least 8*8^q for 0<q<=1. Identity mixing attains it. The statement covers the exactly-two and three active projected-product branches. Singleton-sided rectangles, zero-factor cases, more mixed modes and shared COPY billing are outside scope.

Only research/p112-multiline-free-rectangles-seed42/ is added. The proof uses finite geometry plus a written argument over arbitrary complex coefficients. No physical supplier, complete recurrence, compiler/hosting/precision admission or multiplication theorem is claimed.

Task and overlap

• Base: 410026dc5513b87db92ae98571d1d5984c69a983; refreshed upstream 4040a0f46d67b41c36a472a7cd4af791fd4c36cc preserves the inputs and accepts the additive patch.
• OS assignment seed 42, structural decoder track; card complex-masked-decoder, refined scope. TASK.json records three candidates, the single overlap-driven pivot, exact sources and stopping rule.
• The initial one-column idea was released after finding Discussion1434. Its completed PR1468, head 5e2d8ec8ebff17c85e813ef1bd6f06312c1fe16a, covers the complementary singleton-sided case.
• PR1460, head 8d53667d9bb09d9b1ffb0f865b5c8dc230eb02f6, already proves COPY64 for arbitrary nonzero rectangles at pair (4,6). This package adds the stronger per-active-term span lemma, explicit all-pair transport, and a different maximal-erasure check. It does not claim first priority for that COPY64 subcase.
• PR1375 supplies small-rectangle bounds; PR1373 identifies the two-product boundary; PR1331 and PR1432 supply preceding handoffs and fixed-boundary evidence. No pending theorem is assumed. Exact heads are in TASK.json.
• The full overlap refresh covered 828 open PRs and 162 discussion records with complete pagination and no warnings; the latest 25 PRs and nearest heads were rechecked before submission. Claim.

Validation

Python 3.12.2, standard library only:

python3.12 -B scripts/research_package.py --package research/p112-multiline-free-rectangles-seed42

Expected: PASS_REGISTERED_RESEARCH_PACKAGE research/p112-multiline-free-rectangles-seed42 outcome=negative-result.

• Primary: all 2,553 nonempty character cuts, 28 symmetry maps, seven Gaussian-rational fixtures / 87,808 entries, 12 controls.
• Separate implementation: all 6,216 two-anchor deletions / 31,080 span identities, 351,232 symmetry pairing entries, one 12,544-entry integer identity, six controls. It imports neither primary nor predecessor code.
• Five external source/receipt corruptions rejected. Both optimized Python 3.14.6 checkers pass in a relocated read-only export with identical file hashes before/after. All 20 research workflow tests pass under Python 3.12.2, as do selected-record integrity, metadata and whitespace checks.

The singleton control disproves a one-anchor extension of the span lemma, not a singleton COPY64 theorem. Omitting compensation changes a constrained entry. No selected cold replay, native admission, proof assistant, independent human review or hosted-CI success is claimed. VALIDATION.json binds the checked proof and implementation bytes.

Credit

Credit commdata2338/PR1375, Cinna Davis/PR1373, Arham Tahir/PR1331, Hanson Sun/PR1432/1460 and Gareth Davidson/PR1468; Mihir Srivastava/PR845, DreamingOfClouds/PR717/446/595, themilkman216/PR968, GamingPuzzled/PR620 and woklerx/PR1077 for the geometry, model and orientation lineage.

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

Open next questions

README/TASK identify two bounded directions: classify three-coordinate support strata before testing GL(4,C), or supply an explicit legal shared-COPY endpoint contract. The former needs new geometry; the latter needs a complete gathering/frame/restoration invoice.

[Mister-Meeseeks]

Batch review at ea2f6bf0940580dfaeb393ec0786abc8f7bbb7b4.

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

Reviewed scope: Two-anchor span-preservation proof and exact symmetries cover every coordinate pair for free rectangles with at least two rows and columns. At least two active terms retain COPY rank eight; arbitrary singleton sides and different cost interfaces are excluded.

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.