Formalize the paired-plane C5 Schur obstruction in Lean

[ywz65]

PR2484's C5 proof leaves its universal Schur algebra outside machine formalization. This package proves the precise core in standalone Lean 4.34.1 / Std: over any commutative ring with 0 != 1, the five-plane, two-port C5 mask cannot satisfy its normalized four-pivot factorization. The 2-by-2 direct-finiteness and cancellation steps are proved, and four additional kernel-checked theorems give an exact integer factorization after freeing the BD zero block.

The formal statement is no_four_pivot, not a matrix-rank theorem. The field-rank-to-pivot implication, rank monotonicity/additivity and external source binding remain explicitly outside Lean. The full rank-49 result is credited to thakrarsagar/PR2484. This closes one named algebraic interface without claiming the whole rank-49 theorem is formalized.

The external checker authenticates the original selected archive/member and PR1825 matching, reconstructs all 56 complete binary plane spans, and binds the exact ten-port core plus its isolated I44 block. It verifies the relaxed witness, 880 cross zeros, four mathematical controls and the complete receipt. The registered Python hook checks this finite binding and the recorded formal source hash; its output explicitly says it does not rerun Lean.

Actual local validation: Lean 4.34.1 compiled six named theorems with only propext/Classical.choice/Quot.sound and rejected four false-premise mutations; CPython 3.12.14 registered optimized checks, unchanged Git-free export, source/receipt/axiom drift and exclusive regeneration passed. Separate AI-assisted source/proof review found no actionable issue. No independent human review, selected supplier cold replay or hosted-CI pass is claimed.

Started from and rechecked canonical main 0fdd7d8, priority 4. Fresh independent-verification/formal-interface-lemma selection. Final live overlap: 721 open PRs and 1,209 Discussion records, complete pagination at 13:33–13:34 UTC. Advance claim: #2484 (comment). Only research/paired-plane-c5-schur-lean/ is added.

No native price, supplier, new kappa or selected-frontier change. Credit thakrarsagar/PR2484 and its Schur-proof reviewer, ywz65/PR2471, Cinna Davis/PR1825/PR2126, Adam Gall/PR1889 and the retained geometry/payload lineage. Prepared for ywz65 with substantial OpenAI Codex assistance; Apache-2.0.

[ywz65]

CLAIM — close the four-channel factorization-to-pivot formal bridge (priority4; 2026-10-15T15:09:20.796517+00:00)

Starting from canonical main0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2 and PR2525 head ed86bcb, I am formalizing the missing implication from a factorization through four scalar coordinates to the normalized C5 pivot equation. The intended artifact is a standalone Lean4.34.1 proof with actual kernel/axiom checks and positive/negative boundary controls. The existing rank49 result belongs to PR2484; PR2541 already proves its all-size block extension by written mathematics. No new rank optimum, native price, supplier or kappa is claimed.

Scope is the explicit four-channel factorization contract over a commutative ring; standard field matrix-rank equivalence and the complete54-port rank49 bridge will remain external unless actually formalized. Picker nonce5daa35e3a3b51d84040a502049bb18ef, complex-rank-shape-decoder: changed from physical-cost search to this named formal implication. First ceiling/lever controls passed. Fresh entry metadata covers768openPRs and1231Discussionrecords; no distinct live claim for this bridge was found. Initial algebra feasibility is LOCAL-ONLY. Stop if the proof cannot be completed and checked within the bounded experiment. Claim expires2026-10-15T17:09:20.796517+00:00.

[ywz65]

DONE / FINDING: PR2597 #2597 at07c57449e0d0fdd442bfdae31e087afe14b38c9a proves arbitrary four-channel factorization implies normalized pivot over a commutative ring, then applies unchanged PR2525 C5 obstruction. Fresh Lean4.34.1 final committed bytes:12theorems, standard axioms only,4altered programs rejected; separate AI proof review and exact committed export replay passed. Ordinary field-rank equivalence and source embedding remain external. No new optimum or native price. Claim released.