[petalas]

Summary

A scoped algebraic proof gives the exact commutator after deleting a common admitted scalar pivot from two commuting matrices over any field:

P = [[p,r],[c,D]], Q = [[q,s],[e,F]], PQ = QP = [[h,t],[w,J]], p,q != 0
S = D-c r/p, T = F-e s/q
u = c/p-e/q, v = r/p-s/q, x = w-h e/q, y = t-h s/q
[S,T] = x v-u y

This gives a necessary-and-sufficient zero criterion and a complete rank-0/1/2 trichotomy. The rank-two bound is sharp even for two commuting rank-two idempotents in dimension three over F5. It answers the one-step commutativity part of #4259 open question 1; it does not classify the full Schur-pair orbit or count complete headers. No new κ or supplier admission.

Only research/schur-commutator-criterion/ is added. The side theorem changes the scope of the uniform-transfer-audit card to original successor work. The proof is self-contained; predecessor results are cited context, not audit deliverables. This contribution was produced by Nick Petalas.

Evidence

Before, the laminar preservation proof left a nonnested pair's changing Schur state unspecified. After, the explicit vector factorization decides commutativity and its rank defect at that boundary.

• Every admitted commuting pair in five small domains is checked directly: 1,864 arbitrary F2 matrices at m=3; 420 arbitrary F3 pairs at m=2; 253 F2 idempotent pairs at m=3; 2,572 F3 idempotent pairs at m=3; 76 F5 idempotent pairs at m=2. The overlapping domains are not claimed as disjoint counts.
• All 20,160 GL4(F2) headers for one overlapping coordinate pair are enumerated. Both pivots pass in 5,040 headers; 3,696 leave commuting complements and 1,344 give rank-one commutators. These are header counts, not distinct-pair counts.
• A new explicit F5 projector pair attains commutator rank two. All 7,442 declared vector quadruples check the degenerate zero criterion, and four controls detect false preservation, wrong sign, omitted product-pivot correction and missing input commutation.
• Both registered commands pass locally on CPython 3.12.15, macOS 27.0.1 arm64. Final directly timed normal and optimized checks took 1.31 s and 1.27 s, below their 120 s limits. These are check durations, not performance comparisons.
• Package metadata, five-file committed-byte/source-history preflight, official committed export and export integrity pass. Both read-only Git-free modes reproduce exactly the committed normal/optimized receipt, including all five source hashes. Receipt SHA256: fe80df2a5e858cfb377490b2acd739959c25771231ddfa60966e078b08ef15d7.

Reproduce from the repository root, using CPython 3.12. Another CPython 3.12 path may replace this local interpreter:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/schur-commutator-criterion/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -O -B research/schur-commutator-criterion/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/schur-commutator-criterion
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/schur-commutator-criterion --check-source-history

Expected markers are PASS_SCHUR_COMMUTATOR_CRITERION, PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=completed and PASS_RESEARCH_COMMIT_BYTES. No pending predecessor executable, external live input or third-party Python package is needed. The source pins bind historical provenance.

Limitations and unrun obligations

The universal claim is the written field-algebra proof, with finite corroboration. Independent human and formal proof-kernel review are unclaimed. This covers one scalar deletion with two nonzero pivots. Noncommuting inputs, nonfield rings, block pivots, later-step orbit distributions and complete header-count laws remain outside scope. Header selection/emission, an actual producer, dispatch, dirty return, stock and the complete caller bill remain unpriced. No selected cold replay, Lean/Docker build or complete multiplication theorem is supplied.

All registered local checks ran. Ruff is unavailable, so no external static analyzer was run; Python warnings-as-errors and AST syntax/function-size inspection were used. Bounded dynamic allocations and the code constraints are documented in the README.

Open next questions

1. Classify simultaneous Schur-pair orbits and transition weights for nonnested commuting projectors. Start with the two rank-two requests in dimension three; the commutator rank alone does not identify the full orbit. Coordinate with the active corank-one counting work.
2. Extend the factorization to a common invertible block pivot and find its sharp commutator-rank bound. The scalar rearrangements used here do not transfer directly.

Related open work

• Prove the exact full-guard header law for laminar projector families (no new κ) #4259, head 2a582d8f66d1900b22ee2cd0062ee5dca94e8f85: immediate structural predecessor and open question. This adds the exact one-step commutator beyond laminar preservation. No header fraction is computed or numerically compared here.
• Prove the joint full-guard law for an overlapping corank-one pair #4261, head 4728fc766b3084f3ab2f5c8543a5e74e4a12b126: counts overlapping corank-one full guards without classifying Schur pairs. This supplies a structural criterion, with no overlapping aggregate number to compare.
• Prove the exact ternary full-guard law for two corank-one requests #4273, head 53c2d5a3a43c0538c3d8c263bdb65c29e330b068: ternary corank-one counting formula. Different output quantity; no joint fraction is asserted here.
• Prove the full-guard law for one-way coupled corank-one requests #4274, head 497b8862d98ab57e33b262fa2631afc5a559257f: initially noncommuting one-way corank-one requests. Its input premise differs; no shared aggregate number is claimed.
• Prove the full-guard header law for k corank-one requests with orthogonal kernels: binary threshold m ≥ k+1 (no new κ) #4275, head a3ba3670815329ac69c84c0ea0fefcfa16d7cbff: gives the exact joint full-guard fraction over every finite field for k corank-one requests with pairwise orthogonal rank-one kernels. It counts complete passing headers; this package decides whether one Schur deletion preserves a commuting pair. No header fraction is computed here, so no overlapping numerical result is claimed.
• Prove exact intermediate-rank BIT header densities #4228, head bb25dcedb5bbbacd3c35359eb6b5488b087bcde6: existing single-request Schur/rank framework and densities. The sharp F5 example has input ranks 2 and Schur ranks 1, agreeing numerically with its rank-drop lemma. No marginal density is computed here.
• Prove the joint header law for nonorthogonal rank-one requests (no new κ) #4240, head 9f9c89ed75e9fdae6b6762925b080fb308824858: classifies nonorthogonal rank-one header counts. This treats the Schur commutator of a previously commuting pair, without a shared count to compare.
• Prove the joint full-guard header law for orthogonal requests of any rank (no new κ) #4246 / Prove the exact joint header law for orthogonal rank-one requests #4239: orthogonal full-chain and rank-one laws. Those preserved subclasses are context; this adds the defect criterion outside them, with no new guard fraction or overlapping aggregate number.
• Prove exact extreme-rank BIT header densities #4227: extreme-rank marginal laws, a different quantity; no shared marginal count is claimed.
• Propose unconditional κ = 0.00173097911567079247276057 via batched CRT #4100, head 337d04939571a637aeab54a0f58f54fde811663a: Cinna Davis's BIT request/complete-header context. No physical invoice or multiplication claim is compared or admitted.
• Prove same-rho Schur closure for the completed parity-triangular caller #4209: Schur majorant closure for a parity-triangular complex caller, a different matrix contract and quantity. No shared numerical result.

The preceding header packages are by Nick Petalas. Cinna Davis supplies the BIT request context. No predecessor source or credit line is changed. The live three-kernel counting claim is a separate question.

CI notes

Hosted CI is green on e38f8e9477a49689122ed11a83d163595b366460: run 39524663970. plan passed in 2m51s and the package replay passed in 1m23s. The hosted replay log contains both scientific PASS markers, the same receipt digest as the local committed/export replays, and PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=completed. No check failed. All local registered obligations passed; none was deferred solely to CI.

Merge danger

Two-way door. Blast radius: package. This adds one isolated research package and bounded replay commands. The selected frontier is unchanged.