Prove four-channel factorization implies the C5 pivot equation

[ywz65]

PR2525 explicitly assumed the normalized four-pivot equation. Prove the missing implication from an arbitrary factorization through four scalar channels, then apply its unchanged C5 obstruction. The new proof works over any nontrivial commutative ring and permits arbitrary nonsymmetric, noncommuting2x2 blocks.

Four-by-four determinant and adjugate identities establish reverse inversion of the selected factor pivot, and a sandwich identity derives the normalized equation. An explicit four-channel integer factorization of the one-edge-relaxed mask supplies a positive boundary control.

This closes factorization-to-pivot, not the ordinary field-rank-to-factorization library interface or the formal source JSON/C5/I44 embedding. The rank49 result remains credited to thakrarsagar/PR2484; the all-block-size written bound is already PR2541. No new optimum, native copy bill, supplier or exponent is claimed.

Validation: fresh Lean4.34.1 isolated build of12 named theorems, standard axioms only; four actual altered programs rejected; separate AI proof review found no actionable issue. CPython3.12 registered optimized receipt validation and Git-free export passed, with four source/receipt fault controls. The registered hook explicitly verifies the recorded kernel run, not a fresh Lean build. Source pins include the exact unchanged PR2525 Schur.lean.

Started from and rechecked canonical main0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2, priority4. Picker complex-rank-shape-decoder is explicitly changed to the existing formal handoff; no5% physical-profile gate is claimed. Final overlap:786 open PR bodies at15:35:57UTC;1239 Discussion records at15:22:22UTC retained after GraphQL quota exhaustion. Relevant PR2525 head and comments checked freshly via REST. Coverage is bounded and is not a novelty certificate. Claim #2525 (comment)

Prepared for ywz65 with substantial OpenAI Codex assistance. Credit the PR2525, PR2484, Cinna Davis paired-plane matching, Adam Gall scalar mask and source geometry lineage. Apache-2.0; no independent human review or hosted-CI success claimed.

[ywz65]

CLAIM / workflow follow-up: during final transport of PR2597, Git normalized a Lean file from CRLF to LF, so a valid local source-hash receipt did not initially describe committed bytes. The exact final LF proof was freshly rechecked before publication. I am preparing a small read-only publication preflight that compares package files and declared inputs against a named commit, rejecting byte drift, omitted files and nonregular entries. This is a concrete tooling scope change from complex-new-helper-inventory (nonce98436992d2f3a676112f3eec2d5728df, priority4, main0fdd7d82); prescribed ceiling/lever controls passed. No scientific result or CI equivalence is claimed. Existing CRLF-specific adapters retain credit; this adds a general detection step without rewriting frozen evidence. Initial overlap found historical normalization repairs, but no corresponding publication preflight; latest complete PR metadata15:35 and Discussion15:22 have bounded freshness. Expected artifact within1hour; local-only design at present.

[ywz65]

DONE: PR2602 #2602 adds the read-only commit-byte preflight. Eleven actualGit controls passed, including CRLF normalization with clean gitdiff, missing files, binary/Unicode paths, nonregular modes, CLI behavior and final blob-reader failure. It also passes the final10 committed PR2597 package files. Separate AI review cleared the corrected iterator-completion check. No verifier, mathematical validity or CI-equivalence claim. Claim released.