Research: COPY floor 64 for two-coordinate P112 mixing with small free rectangles (no new kappa)

[commdata2338]

Change and mathematical scope

Coherently mixing two P112 coordinate terms with a free rectangular correction cannot lower their COPY bill in the certified small-support domain. At least two replacement terms keep pairing rank at least eight. So the whole decoder keeps COPY cost 64, and its separate COPY power bill cannot improve. This answers part of the two-coordinate handoff left by PR #1331.

Domain: arbitrary GL(3,C) and all 28 coordinate pairs, with one of:

• a nonzero free correction with at most 23 rows and 23 columns;
• supports of at most 11 per side, including zero factors.

Identity mixing attains the floor with a rank-nine masked completion.

Status: a scoped proof with exact finite premises, a negative result for this route. No physical supplier, improved κ or full multiplication theorem is claimed.

Precise delta: adds only research/p112-two-coordinate-small-rectangle/. It holds:

• a written quantified proof and a deterministic exact certificate;
• two differently implemented finite-premise checks, with failure controls;
• source, task and replay metadata, and a validation receipt.

Selected and frozen sources are unchanged.

The certificate:

• All 140 projective support strata are classified over C.
• The 71,540 nonzero functional evaluations give ordinary erasure distances 24/24/12/12/24 and free-compatible distances 24/24/34/34/40.
• All 700 ordered support pairings have rank at least eight (448 rank 8, 252 rank 9).
• A coefficient-rank identity forces at least two active products, which preserves the two-child power bill and not just the sum of ranks.

Witnesses: a 12-row span-loss witness has no common odd neighbor, and a compatible 24-row witness has neighbor {0}. These mark the edges of the span-preservation premises. They are not the sharp decoder threshold, and neither is a gain.

Remaining obligations.

• Not covered: larger nonzero supports, zero-factor cases above eleven, more mixed coordinates, other punctures, and shared-copy accounting.
• Unchecked: physical gathering, frame and owner legality, restoration, hosting, precision/tape, the all-size compiler, common-label/mutable-source conditions, the analytic transfer and bit H1-H5.
• Not claimed: a selected cold replay, a proof-assistant check, independent human review, hosted CI or maintainer acceptance.

Task and overlap

• Task / scope: complex-masked-decoder / 2a6f2c726b78f9ae7d26, refined to p112-two-coordinate-small-rectangle/GL3C/all28-pairs/nonzero-support23-or-support11. Chooser nonce 44ef721f92930cbeabe2df5c07bc184b, track complex-mechanisms, priority revision 2.

• Base and exact heads:

  • base bce54fb3f895cbe5cd4c0eb477c3afeed100190f; contribution 4a533aff2a7a87c9b36e077a31774b332a7f64bd;
  • Arham Tahir's PR Scoped obstruction: one free P112 rectangle cannot improve one coordinate COPY #1331, head f08b698cc2bd74e2425486728a008e531fc35f0e, fetched and rechecked; it is context, not a code or theorem dependency;
  • the live PR Scoped negative result: one added P112 total cannot reduce fixed-reader copy cost #845/Format limits part 2 (no κ claim): erratum to #595, exact invoice prices, frontier-word censuses, decoder floors, spec requirements, centre and fragmentation walls #717 geometry and the PR Research: frame-pair decoders and the exact 8-term complex copy-cost floor (no new kappa) #968 model proof, pinned by Git blob and SHA256 in SOURCES.json;
  • selected PR Conditional kappa 0.001672660197076918: rare-pattern T16 exchanges #1094, head 0fc329d0bec93ca7cd288eaea1295172e8789810, accounting context only.

• Question: can a small free correction, mixed with two original coordinate terms, reduce the complete separate COPY bill?

• Scope status: refined. The claim began as the singleton follow-up to the discussion CLAIM: one free odd rectangle mixed with one P112 coordinate term #1316 handoff. The same exact erasure calculation then extended it to the two support domains above. It changes the masked/overcomplete assumptions rather than repeating the eight-term all-pairs theorem.

• Method and domain: a complete five-stratum classification for every coordinate pair, an exact F2 census of minimum-support functionals and pairings, and a quantified coefficient-rank proof for every complex invertible factor change. No random search and no floating-point acceptance.

• Optimistic budget: six unchanged rank-eight COPY terms plus three replacements. At least two replacements individually have rank at least 8, giving at least 2*8^(1-a) for 0<=a<1, with the same width, normalization and multiplicity. All added gathering, frame and restoration costs are optimistically zero for this rejection screen.

• Nearest work:

  • PR Scoped obstruction: one free P112 rectangle cannot improve one coordinate COPY #1331 treats one mixed coordinate with arbitrary rectangles.
  • Discussion CLAIM: sharp copy floor for all rank-nine factorizations of a single odd-entry P112 completion #1360 independently considers every nine-term factorization of a singleton completion and explicitly leaves larger free rectangles out of scope. This package treats larger rectangles with two original coordinates mixed, so the two are complementary.
  • PR Research: rank-8 complex decoders are gauge-rigid; E-graph eight-term floor (no new kappa) #1212/Research: classify the P112 masked decoder coordinate component (no new kappa) #1213 concern rank-eight masked rigidity, PR Research: nine-term complex E8 decoders are one-sided overcomplete representations (seed 42, no new kappa) #1224 nine-term all-pairs normal forms, and this account's PR Research: sharp A8 simplex P112 copy floor 71 on matching-frame punctures (no new kappa) #1304 fixed A8 simplex normals.

  None supplies an assumed pending theorem.

• Claim and final check: the claim is in discussion #1316. A live overlap check just before submission (847 open PRs and every discussion record at 23:45 UTC, complete pagination) found no competing two-coordinate result.

Validation

Python 3.12, standard library only:

python3 -B scripts/research_package.py --package research/p112-two-coordinate-small-rectangle
python3 -I -O -B research/p112-two-coordinate-small-rectangle/verify.py
python3 -I -O -B research/p112-two-coordinate-small-rectangle/independent_check.py

The registered replay checks the metadata, provenance and live pins, then runs both checkers in about four seconds. It ends with PASS_REGISTERED_RESEARCH_PACKAGE research/p112-two-coordinate-small-rectangle outcome=negative-result.

• Fixtures: nine exact Gaussian-rational fixtures check 112,896 scalar entries in all. They include nonreal coefficients, ordinary transpose, cancellation, a 23-by-one rectangle and zero-factor cases.
• Controls: nine primary and nine separate-checker mutation controls pass.
• Separate checker: it builds Euclidean E8 roots and a binary dual evaluation code without importing the primary checker or the predecessor geometry. Both optimized-mode checks pass.
• Orientation: the card's first working example also passed (PASS_COMPLEX_DECODER_FIELD_EXTENSION_S44).

The registered replay and the metadata/pin check were run again on the final commit before submission, and both passed.

PASS_P112_TWO_COORDINATE_SMALL_RECTANGLE pairs=28 strata=140 characters=71540 pairings=700 erasures=11 nonzero_free_erasures=23 fixtures=9 controls=9 copy_floor=64
PASS_P112_SMALL_RECTANGLE_INDEPENDENT strata=140 characters=71540 pairings=700 boundaries=12,24 controls=9

Credit

• Arham Tahir (PR Scoped obstruction: one free P112 rectangle cannot improve one coordinate COPY #1331) for the one-coordinate predecessor and the two-coordinate question.
• DreamingOfClouds (PR Format limits part 2 (no κ claim): erratum to #595, exact invoice prices, frontier-word censuses, decoder floors, spec requirements, centre and fragmentation walls #717) for the masked B14 model, and PR Format obstructions (no new κ): exact bounds on window width, copy cost and label dimension #446/Format limits (no κ claim): certified ceiling κ ≤ 5.1595e-3, spec census, structural lemmas, helper bounds and open problems #595 for the separate-total lineage.
• Joel Pulikkan / GamingPuzzled (PR Research notes (no κ claim): E8 decoder floors are tight; extension-ring, triple and Lorentzian units lose after helpers; cold-end, dispatch and fusion-search negatives #620) for the coordinate COPY64 decoder.
• Mihir Srivastava (PR Scoped negative result: one added P112 total cannot reduce fixed-reader copy cost #845) for the exact P112 geometry.
• themilkman216 (PR Research: frame-pair decoders and the exact 8-term complex copy-cost floor (no new kappa) #968) for the pairing-cost model.
• woklerx (PR Scoped structural result: arbitrary complex coefficients retain the exact eight-term E8 decoder floor #1077) for the task's complex-coefficient positive control.

Apache-2.0; source attribution is retained.

AI assistance: submitted by @commdata2338 with substantial AI assistance. OpenAI Codex (GPT-6 Astra) did the mathematics, both implementations, the verification and the writing; it was packaged and submitted with Claude Code. Both implementations come from the same AI-assisted contributor, and no independent human review is claimed.

Open next questions

From the package README:

• Can a nonzero free rectangle with at least 24 indices on one side exploit a span-losing coordinate support to reduce the two-coordinate COPY moment? This is the first size the proof leaves open. First step: take the recorded 24-row free_boundary witness and its common odd neighbor {0}, then solve the actual GL(3,C) cancellation constraints while pricing every term. Blocker: a free support rectangle with span loss does not by itself satisfy the coupled inverse-factor identities or establish any gain. The 12-row unrestricted witness has no common odd neighbor and should not be used for a nonzero correction.
• Zero-factor cases above eleven per side are a separate open branch of this proof. Larger rectangles, three or more mixed coordinates, other punctures and shared-copy objectives remain outside scope.

Workflow feedback (optional)

• Main detour: the first twelve-erasure boundary had no common odd neighbor. Conditioning the erasure distance on an actual free rectangle produced the useful 23-per-side bound and a compatible 24-row witness.
• Suggestion: decoder-support task cards could suggest checking the common odd-neighbor set before proposing a span-loss witness as a free-correction candidate. That separates a linear-algebra boundary from a realizable masked support at once.

[Mister-Meeseeks]

100-PR batch review at 4a533aff2a7a87c9b36e077a31774b332a7f64bd.

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

Reviewed scope: All-complex factor-change proof plus independent complete finite geometry: a nonzero free rectangle of at most 23 rows/columns cannot improve the two-coordinate P112 COPY bill. Zero-factor cases use the smaller 11-erasure domain; larger rectangles 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.