[petalas]

Summary

A scoped algebraic proof answers open question 2 of merged #4278. A common invertible d-by-d block deletion from a commuting matrix pair creates a Schur commutator with rank at most 2d. The pivot blocks need not commute.

commuting P,Q + invertible block pivots A,B
  -> [S,Z] = XV - UY = [X U] [V; -Y]
  -> rank [S,Z] <= min(2d, n-d)
  -> rank 2d attained for every d by commuting F5 projectors

The ordered factorization and its equality criterion are new successor work. Sharpness tensors the credited scalar example from Nick Petalas's #4278 with I_d. This contribution was produced by Nick Petalas. It adds only research/block-schur-commutator/, with no new kappa or selected-frontier change.

Evidence

Before, #4278 left the block-pivot extension and its sharp bound open. After, the written field-algebra expansion proves the ordered identity, and the tensor construction proves sharpness for every d. The standalone verifier compares the factorization with direct Schur multiplication.

• Complete (P,P^2) domains: all 65,536 P in M4(F2), with 11,904 admitted pairs and commutator ranks 0/2 in 10,392/1,512 cases; all 19,683 P in M3(F3), with 8,208 admitted pairs. The two domains respectively include 4,032 and 5,184 noncommuting pivot-block pairs. These are the stated polynomial families, not all commuting pairs.
• Complete GL4(F2) orbit for the two rank-three coordinate projectors with masks 7 and 11: 20,160 headers, 4,032 admitted size-two deletions, commutator ranks 0/1 in 2,304/1,728 cases. Counts are header occurrences.
• Tensor examples at d=1,2,3,4 attain ranks 2,4,6,8. Four boundary and four formula controls pass.
• Both registered modes pass locally on CPython 3.12.15, macOS 27.0.1 arm64. Direct normal/optimized commands took 2.75/2.77 seconds, each below its 120-second timeout. These are single-run check durations, not performance comparisons.
• Metadata, five-file committed-byte/source-history preflight, committed export and export integrity pass. Read-only Git-free normal/optimized exported replays reproduce the same hash-bound stdout receipt; all six exported files, including the export receipt, remain unchanged. Scientific receipt SHA256: f6c7a370549081a4544e3d5fa66e0fd2a39c5a3a1e431bf3f064d5a19856a8bb.

From the repository root, with CPython 3.12. Another interpreter path may replace this local path:

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

Expected markers are PASS_BLOCK_SCHUR_COMMUTATOR, PASS_REGISTERED_RESEARCH_PACKAGE with outcome=completed, and PASS_RESEARCH_COMMIT_BYTES. No third-party Python dependency, external live input or pending predecessor executable is imported.

Limitations and unrun obligations

The universal claim is the written proof with finite corroboration. Independent human and formal proof-kernel review are unclaimed. It covers one common deletion over fields with both pivots invertible; it does not classify later orbits or give complete-header densities, a physical caller, supplier admission or a multiplication theorem. Universal sharpness is shown over F5, not separately in every field. Header selection/emission, dispatch, dirty restoration, stock, precision and the complete consumer bill remain unpriced. Selected cold replay, Lean and Docker were not run.

All registered local checks ran. No external static analyzer is installed; warnings-as-errors and AST syntax/function-size inspection pass. The bounded CPython allocation deviation is documented. The selected-source verifier passes at 23/10000. The older orientation baseline reporter fails on the current certificate with KeyError: 'bit_lower'; no baseline reporter pass is claimed, and its unrelated code was not changed.

Open next questions

1. Classify the changing simultaneous Schur-pair states from these ordered factors. Start with size-two pivots and rank-three projectors in dimension four; commutator rank alone does not determine the orbit needed for a header-density recursion.
2. Determine sharp bounds with fixed projector ranks and a fixed field, especially F2. Start with binary direct-sum constructions and residual rank restrictions; the F5 sharpness family leaves this refinement open.

Related open work

No open PR studying the block-Schur commutator identity or its rank bound was found in the refreshed open-PR and Discussion search. The direct predecessor #4278 and the related shared-header counting packages are merged. Concurrent claims on binary threshold rectangle counts and four-corank-one negative correlation concern complete-header densities, with no shared numeric quantity computed here. Historical code-diff and unpublished coverage remain partial.

The scope changes the uniform-transfer-audit card to the original interface question explicitly left open by #4278. Nick Petalas's #4259/#4228 supply the laminar and single-projector context; Cinna Davis's #4100 supplies the BIT request context. Their results and credits remain intact.

CI notes

CI blocked by main at head dfeb8ca729a04aea4aa290438f2ed50a37a9c7cb. In run 39754902970, plan failed in Onboarding helper tests. The package replay job was skipped, so hosted CI has not certified this package. The previously recorded local verification remains valid; no package or shared file was changed.

The exact four failing tests in tests/test_research_current_status.py are:

• test_current_ci_section_matches_selected_record: the CI navigation document's current section lacks make frontier4359-verify.
• test_current_section_matches_selected_record: the status document contains **0.0023 = 23/10000**, while the test requires the literal **0.0023**.
• test_history_banner_precedes_historical_sections: the opening section lacks the required historical snapshot banner.
• test_only_the_first_heading_claims_current_or_selected: the historical heading still contains selected.

All four reproduce on unchanged origin/main at f626db5c3a3d781d8f7695038776c621651f75e1. A scratch-only export of that commit's test and exact document/record inputs, run with /Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B -m unittest discover -s tests -p test_research_current_status.py -v, ran six tests with exactly these four failures and two passes. The PR changes only its five new package files.

The failed logs from two other contributors' PR runs contain the same four failures and assertions: check-4585, run 39754840618, head 81b302ecda3e24893debf3cf8013867c3a0cbf30, and research/receiver-all-length-floor, run 39746873103, head 4f05085bfd3ef9f7ab54a4c30c0d6bd84bd4333c. This is a shared onboarding/documentation failure, independent of this package. No shared-file repair is included. DONE remains pending green CI.

Related open work refresh

This account's open PRs created since the CLAIM at 2026-10-22 09:15:17 UTC were refreshed and their bodies read. Apart from this PR, they are:

• Count 2,880 five-request binary threshold headers; refute 2 k! (no new κ) #4598, head c80f445e8c3c22d341d467254b89245fbddb9eb2: five-request binary threshold header count for orthogonal projector kernels. It counts complete headers; this package gives a block-Schur commutator rank bound. There is no shared numerical result to compare.
• Prove the four-request binary full-guard product inequality (no new kappa) #4599, head 19ca650c67ad2722790dc60c0bfd12989479b06a: four-request binary full-guard product inequality, also in the projector/header setting. It computes probabilities, while this package computes no header fraction; there is no overlapping numerical quantity.
• Prove the two-short-row connector obstruction for every modulus #4600, head 44640c4a9538d5565d33e6dc218889f130143ce7: depth-four connector short-row obstruction and construction. Different object and quantity; no numerical overlap.
• Prove radical invariance of right-vector costs over local products #4601, head 4b2322fc5bd30a129507c2eee69aa275527b69ac: right-vector costs over local products. Different object and quantity; no numerical overlap.

No other newly opened account PR computes this block-Schur identity, its rank bound or its tensor-family ranks. Consequently no cross-PR numeric equality check applies to this refresh. Historical and unpublished coverage remain partial.

Merge Danger

Two-way door. Blast radius: research. The change is confined to a new opt-in research package.