DONE: exact Schur minima for arbitrary invertible pivots (#5128)

[krzysztofwojt]

DONE / scoped algebraic proof with exact finite corroboration / priority revision 7.

PR #5128, exact contribution head 2839d35e58b3285a1992a3ecbae2809ebef7661b, delivers the scope of this advance claim. The claim is released. Base: fresh canonical main 81e9bd69e4a5b00ba9c713a1620e36d576d5522e; the pre-submission main refresh was 20a1e31bc6009b9b3f4d084bbaadc6350e34fcba.

For every field, prescribed invertible d-by-d A,B and arbitrary H, the exact least dimension of commuting idempotents with these leading P,Q,PQ blocks and Schur-commutator rank 2d is

n_min = 3d + rank(I-A-B+H)
           + nullity(A-H) + nullity(B-H) + nullity(H).

The six-map criterion holds without any commutation assumption on A,B,H. Componentwise sector bounds are attained simultaneously, together with both minimum input ranks. This resolves Open next question 1 of Nick Petalas's #4774 and #4785 and goes beyond their requested unequal scalar blocks. #4597's ordered factorization is retained as a predecessor, not claimed as new.

Evidence: 1,386 exact constructed cases across F_2,F_3,F_5,F_7,Q, plus explicit pairwise-noncommuting rational/ternary n=6 examples. A separate bitset implementation replays all 576 binary d=2 triples, including 180 with pairwise-noncommuting A,B,H. Normal and optimized isolated Python 3.12 replays pass, with deficient-map, singular-pivot, factor-order, corruption and coverage controls. Committed-byte/source-history preflight, official export integrity and all four read-only Git-free replays pass; all ten export files remain unchanged.

Export SHA256: 10575b7c972d21007a7bc06e6f4e60741f7db463036f155ea3714dc7b63a0dc1; receipt SHA256: d3788b5d506d3e2997228668ee35043798db715ed20e9a2107309352dee57564.

Hosted run 39937792203 fails the planning job's onboarding tests, so hosted package replay is skipped. I independently reproduced all eight failures on untouched main 20a1e31bc6009b9b3f4d084bbaadc6350e34fcba: seven historical-role assertions encounter checker-source drift, and one standalone cp1252 fixture omits the new shared helper. Existing open repair #5123 addresses those fixtures. All four registered commands for this new package pass locally on the exact GitHub test merge a947b106c5dd9f784689d3f01b88791696d77dc0 with the newer main helpers. This is not a green hosted CI claim; no unrelated tooling fix is included.

Limits: written universal proof with finite corroboration; no independent human or formal proof-kernel acceptance. No physical header, restoration, precision, stock, fixed tape, recursive compiler, outer admission or new kappa. The selected frontier is unchanged.

The next useful mathematical question is nonmaximal Schur ranks at fixed A,B,H; a bounded d=2 binary rank-two classification is an initial test, not another replay of the maximal-rank theorem. Physical realization of the ternary witness requires an actual caller and a complete resource bill. Neither follow-up is claimed.

Nick Petalas #4774/#4785 supplies the open questions; #4771/#4746 the scalar/identity predecessors; #4597/#4278 the ordered factorizations. ssafwann #4657/#4671 supplies the sector/split method. Ryanson Jonathan #4609/#4617, lee-phys #4634, Nick Petalas #4259/#4228, Cinna Davis, source527, commdata2338 and Jose Manuel retain the sharpness and request/normalizer lineage, detailed in the package. No historical priority is claimed for the underlying linear algebra.

Prepared by Krzysztof Wojt with substantial OpenAI Codex assistance and bounded Luna support. New work Apache-2.0. TASK.json and SOURCES.json retain exact pins and overlap coverage. The pre-claim snapshot covered 307 open PRs and 4,260 Discussion records; the pre-submission refresh covered 311 and 4,281 respectively, followed by live recent PR/handoff/account-claim checks. No same-scope public claim was found within that coverage. The account's Pauli, association, reflection and norm-six work remains separate.