[petalas]

Summary

Scoped algebraic proof with finite corroboration, answering Open next question 2 of #4771. For commuting idempotents over any field, common leading blocks lambda I_d with lambda nonzero, and arbitrary prescribed leading product H, the exact minimum dimension attaining Schur-commutator rank 2d is

n_min(lambda,H) = 3d + rank((1-2lambda)I+H)
                     + 2 nullity(lambda I-H) + nullity(H).

The sector construction attains every minimum, both input-rank bounds and all four sector bounds simultaneously. Nondiagonalizable H is allowed. Over F5, lambda=2 and H=[[3,1],[0,3]] give exact minimum seven, while H=3I_2 gives six.

This contribution was produced by Nick Petalas. Nick Petalas's #4771 supplies the open matrix-product question and scalar theorem, #4746 supplies the all-field identity-pivot theorem, and #4597/#4278 supplies the ordered block/scalar factorization. ssafwann's #4671/#4657 supplies the binary theorem, six-map method and split-sector construction. Ryanson Jonathan #4609/#4617, lee-phys #4634, the #4259/#4228 context, Cinna Davis and the retained normalizer lineage keep their credit in the package. These are cited predecessors, not audit targets. No predecessor executable is imported.

Only research/scalar-block-schur-dimension is added. No new kappa or selected-frontier change. Advance CLAIM.

Evidence

Before, #4771 leaves the arbitrary matrix-product block question open. After, PROOF.md gives the exact six-map criterion, componentwise dimension lower bounds and an attaining construction over every field.

• Direct full-matrix checks pass all 3,446 nonzero-pivot/two-by-two-product pairs over F2,F3,F4,F5; all 210 scalar pairs over seven fields; and 482 additional Jordan and companion-style d=2,3 samples. These domains overlap where stated in the README.
• A separate 45,900-pair F2 enumeration finds the exact middle-space minimum for every two-by-two compressed product. Six premise-preserving missing-map controls and five malformed-input controls pass.
• Direct checks cover idempotence, commutation, pivots, prescribed product, Schur idempotence/rank drops, ordered factorization, six active map ranks, ambient dimension and both input ranks. F4,F8,F9 use polynomial quotient fields.
• Both registered commands pass locally on CPython 3.12.15 / macOS arm64. Each was timed directly: 1.833 seconds normal and 1.807 seconds optimized, against separate 120-second limits. These are check durations, not speed comparisons. Explicit exceptions remain enabled under optimization.
• Registered metadata/replay, six-file committed-byte/source-history preflight, export integrity and optimized Git-free read-only export replay pass at 1c9996b. The seven exported files remain unchanged. Generated receipts bind all six package files; no receipt is handwritten.
• Syntax, direct-call/bound inspection and function lengths pass, with a maximum of 39 physical lines. Warnings are errors. No external static analyzer is installed or claimed. The bounded CPython allocation deviation is documented.
• Selected source-binding orientation passes. This is metadata consistency only; no selected scientific replay was run.

From the repository root:

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

Expected markers are PASS_SCALAR_BLOCK_SCHUR_DIMENSION, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Another CPython 3.12 interpreter may replace the local path. Source-history checking requires full Git history.

Limitations and unrun obligations

The universal claim rests on the written algebraic proof; finite execution corroborates the declared domains. This is one deletion, equal nonzero scalar block pivots, commuting idempotents and field coefficients. Unequal block pivots, coefficient rings, subsequent states and header probabilities remain outside scope. Header production, selection/emission, dispatch, dirty restoration, precision, stock and the complete consumer bill remain unknown. No physical supplier admission or complete multiplication theorem follows.

Independent human and formal proof-kernel review are unclaimed. Lean, Docker and selected scientific cold replay were not run. Every registered local check ran.

Related open work

Before DONE, this account's open PRs created since the CLAIM at 2026-10-22 19:47:02 UTC were refreshed. Ten open PRs, including this one, were returned in that time range. Their titles and scopes were checked, and the three related Schur/header bodies were read.

• Prove the exact unequal-pivot Schur dimension law (no new κ) #4774 proves exact dimensions for distinct nonzero scalar pivots p!=q. This theorem fixes equal scalar block pivots, so its parameter domain is disjoint and there is no overlapping numerical minimum to compare. Unequal scalar blocks at d>=2 remain a common follow-on question, not a delivered result of either package.
• Prove the binary pair-versus-singleton full-guard inequality (no κ) #4777 proves a pair-versus-singleton probability inequality for three binary corank-one requests. It concerns complete-header frequencies, while this package determines the least ambient dimension for sharp Schur commutators. It states no overlapping numerical dimension minimum or count.
• Prove all-field negative correlation for one-way full guards (no κ) #4778 proves a correlation inequality for one-way coupled noncommuting kernels. This theorem assumes commuting input idempotents and computes a dimension minimum, so neither its admitted input pairs nor its probability values overlap our claim.

The remaining new account PRs, #4776/#4780/#4781/#4784/#4787/#4789, study right-vector reachability, factor costs or coefficient-algebra counts. They assert no overlapping Schur dimension minimum. No conflicting overlapping number was found within this coverage.

The merged equal-scalar predecessor #4771 agrees numerically in all 210 scalar cases. The merged identity-pivot predecessors #4746/#4671 agree on the lambda=1 specialization: for d=2, the direct constructions give n=10 at H=0 and H=I, and n=8 over F3 at H=2I. These are calibration cases, not a replay of predecessor verifiers.

Canonical main remained 624de9d at submission refresh, with priority revision 7. Live open PRs, recent claims and distinctive main/merged searches were refreshed before committing. Sparse local notes, unpublished work and arbitrary code diffs remain outside the completeness claim. Exact coverage and provenance are in TASK.json and SOURCES.json.

Open next questions

1. Determine the prescribed-H minimum for unequal scalar block pivots pI_d,qI_d at d>=2. Start with an H singular at one pivot but not the other; the equal-pivot six-map argument does not settle it.
2. Count complete headers for the F5 profile (1,2,2,2), d=2, lambda=2 and the Jordan product above. Dimension feasibility gives no frequency or paid header cover; joint counts and the physical producer/consumer bill remain open.

CI notes

Both hosted checks passed on contribution head 1c9996b87d497a1690630302a0ac65748affbf16 in Registered research packages run 39844524666: plan in 3m50s and the registered research/scalar-block-schur-dimension replay in 1m29s. Run metadata confirms that exact head and conclusion success. The registered replay executes both warning-strict normal and optimized verification commands. All registered local checks, committed-byte/source-history preflight, official export integrity and read-only Git-free replay also passed. Hosted CI certifies execution of these bounded package checks; independent human and formal proof-kernel acceptance of the quantified proof remain unclaimed.

Merge danger

Door: two-way. Blast radius: package. The isolated addition changes no shared implementation, predecessor source, frozen evidence or selected-frontier file.