Research: a stronger P112 two-call COPY floor for arbitrary cross forms

[commdata2338]

Change and mathematical scope

For two independent P112 root-code decoders with arbitrary directed binary cross forms, prove that c01+c10 >= min(8,r01,r10) forces the exact separate-total COPY minimum 128. Here r denotes global cross-form rank and c the minimum rank on allowed support hyperplanes. A term touching both calls costs at least 16-min(8,r01,r10) by a block Schur complement; combining this with PR2294's coefficient-count argument yields the new criterion.

The resulting rank-only test certifies 93 of 100 ordered rank pairs in 0..9. The explicit rank-(4,4) example has support minima (3,3): PR2294's earlier c01+c10>=8 test is inconclusive, while this theorem proves minimum 128. Sixteen coordinate terms attain it. Seven low-rank pairs remain undecided by the rank-only test.

This is a scoped written proof with exact finite premises and three Lean-checked natural-number interface theorems. The binary/complex matrix arguments and P112 support implications are not fully formalized. No physical supplier, rank-power moment, kappa improvement, selected-frontier change or independent human review is claimed.

Task and overlap

The claim is in Discussion #2372. Package: research/p112-cross-form-rank-tradeoff/. Contribution commit: 361de309e5fc05f83862f0b6c611210eda9837a1; main was rechecked unchanged at 07:24 UTC before submission.

Exact base: 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2, priority revision 4; final canonical-main refresh found no changed files.

Predecessors:

• PR2294, 18d6ac4f864af2b71aa8cf5f4541fcfc4350e2f0: model, original count argument, sufficient tests and open arbitrary-cross-form question. Its proof is pinned verbatim. No predecessor code is imported.
• PR2338, 82700676e1d42acf98dc1921cb6ef82551620008: context only. Its single rank-nine label table differs from the arbitrary forms on V direct-sum V considered here. The README and notice are pinned for scope and credit; its SAT results are not assumed.
• Main-branch geometry research/p112-single-lift/reference-geometry.json: blob d862663d3c0ffe3c72df510e89997beb85cef640, SHA256 4f44af368e046002553d632ee90c6470990827cea3fae41f0f54018522d7a080.

Task card:

Task ID / scope key: formal-interface-lemma / 636f26dec98e26b296e8;
  P112/two-call/arbitrary-cross-forms/Schur-price-to-COPY-count.
Base / exact dependencies: the base and predecessor heads above.
Concrete object: any finite complex root-code factorization A C^T=I16;
  arbitrary binary cross forms; unchanged within-call P112 pairings.
Scope status: refined to PR2294's explicit open question and a small
  formal count interface, with the matrix hypotheses stated explicitly.
Changed restriction: replace isometric or large-rank coupling assumptions
  by a sufficient condition using the smaller directed cross-form rank.
Method / domain: written Schur and complex-rank proof; Lean 4.21.0 core
  count theorem; exact P112 premises, rank pairs 0..9 and finite controls.
Nearest overlap: PR2294's weaker c01+c10>=8; PR2338's single-table model;
  different from current single-call masked-completion/correction work.
Full cost target: prove a separate-total COPY floor128, attained by the
  standard 16 terms; no physical bill or reduced recursive moment inferred.
Success / falsification / cap: kernel count proof, source-pinned support
  census, exact fixtures and rejected false variants; stop at the sufficient
  theorem, with residual low ranks and physical admission open;60min budget.
Credit / artifact / command: retained attribution below;
  research/p112-cross-form-rank-tradeoff/;
  python3 -I -B scripts/research_package.py --package research/p112-cross-form-rank-tradeoff.

The final shared snapshot covered 587 open PRs and 1,116 Discussion records, with complete pagination (PRs 2026-10-15 07:07:38 UTC; Discussions 07:09:50 UTC). Searches included the task, cross forms, cross-form, Schur and rank tradeoff, plus source and current claim reads. This is metadata coverage, not proof of novelty. No competing same-object claim was found beyond the canonical claim above.

Validation

CPython 3.12.3, standard library; Lean 4.21.0 core, commit 6741444a63ee.

python3 -I -B scripts/research_package.py --package research/p112-cross-form-rank-tradeoff --check-only
python3 -I -B scripts/research_package.py --package research/p112-cross-form-rank-tradeoff
python3 -I -O -B research/p112-cross-form-rank-tradeoff/verify.py
python3 -I -B research/p112-cross-form-rank-tradeoff/test_integrity.py
lean +leanprover/lean4:v4.21.0 -j1 research/p112-cross-form-rank-tradeoff/CountFloor.lean
python3 -I -B research/p112-cross-form-rank-tradeoff/check_lean.py --lean /path/to/lean

Expected finite marker: PASS_P112_CROSS_FORM_RANK_TRADEOFF ranks=93/100 blocks=9216 decoders=505 controls=8 abstentions=1.

The checker independently reconstructs the Hamming-shell roots, checks 12,544 label pairings, both 511-functional support censuses, all 64 within-call support ranks, all 9,216 invertible-diagonal 2-by-2 block cases and 505 exact decoder fixtures. Five 16-dimensional coefficient identities imply the full 224-by-224 scalar demands using the checked left inverse. One deliberately degenerate rank-(1,1) form makes the sufficient test abstain.

Normal, optimized and registered replays pass. Five integrity tests cover Git-free relocation, live source and predecessor drift, a false receipt and integer-to-Boolean receipt corruption. Lean checks three conditional arithmetic theorems and four concrete counterexamples; four false proofs reject. The theorem axiom audits contain only propext and Quot.sound. The Python registration does not build Lean; the separate command does.

Not checked: the seven residual rank classes in full, matrix premises in a proof assistant, all cross-form matrices by enumeration, physical embedding/hosting/precision, rank-power moments, full selected cold replay, hosted CI or independent human review. These checks are not required to claim the scoped COPY-floor theorem. The selected record is byte-for-byte unchanged.

The registered replay and the Lean check (Lean 4.21.0: three theorems that rely only on propext and Quot.sound, four rejected proofs) were run again on the final package bytes before submission, under an 8 GiB memory cap; both passed in a few seconds and left the worktree unchanged. The worked example checks by hand: ranks (4,4) with support minima (3,3) satisfy 3 + 3 = 6, at least min(8,4,4) = 4, so the new test applies, while PR #2294's test needs a sum of at least 8.

Credit

Sagar Thakrar (@thakrarsagar) supplied PR2294's model, count argument, prior sufficient conditions and open question. Alephnan supplied PR1714's batch model; Mihir Srivastava and Adam Gall supplied the P112 support-lemma lineage. The main P112 single-lift geometry is retained unchanged. DreamingOfClouds supplied PR2338's related single-table results and scope comparison. Copied predecessor text preserves its attribution and Apache-2.0 notice; new work uses the repository's Apache-2.0 license.

Prepared for commdata2338 (@commdata2338), with substantial OpenAI Codex assistance.

AI assistance: submitted by @commdata2338 with substantial AI assistance. OpenAI Codex (GPT-6 Astra) did the derivation, the implementation, the checking and the writing; it was packaged and submitted with Claude Code. No independent human review is claimed.

Open next questions

The package README and TASK.json identify two bounded follow-ups: classify the remaining small-rank forms when the actual support-minimum criterion fails, starting with rank-one annihilator hyperplanes; and formalize the binary/complex matrix-to-count premises, starting with the Schur bound. Finite fixture costs do not settle the residual minima. Physical realization remains a separate question.

Workflow feedback (optional)

The current board's explicit PR2294 open question and exact source pins enabled a focused continuation. A claim index keyed by object and changed hypothesis, across task-card names, would reduce simultaneous semantic collisions more effectively than filename reservations alone.

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https://claude.ai/code/session_01X86sCSXdoDd61SDFfC1PKG