[petalas]

Summary

Scoped algebraic proof with finite corroboration. This extends ssafwann's #4671 fixed-H identity-pivot theorem to every field:

- F2: n_min(H) = 4d + nullity(H+H²)                 (#4671)
+ any field: n_min(H) = 4d + nullity(H(I-H))        (this contribution)

The inputs are commuting idempotents with common leading blocks I_d, prescribed leading product block H, and Schur-commutator rank 2d. The signed sector proof attains the exact minimum for every H, including singular and nonsemisimple H. Consequently the unrestricted identity-pivot minimum is 4d for every d over every field with more than two elements. The new contribution is the arbitrary-characteristic extension; the binary theorem, six-map argument and split-factor construction retain #4671's credit. No new κ.

Adds only research/all-field-identity-schur/, five files. Produced by Nick Petalas. Credit also remains with ssafwann's #4657, Nick Petalas's #4597/#4278/#4259/#4228, Ryanson Jonathan's #4609/#4617, lee-phys's #4634, and the retained BIT/normalizer lineage. Pending sources are exact-head citations, not runtime dependencies.

Evidence

• Before: Prove exact identity-pivot Schur dimension for every prescribed H (no κ) #4671 states its prescribed-H dimension theorem over F2. The new proof changes the compressed 00 sector to H-I and retains the correct signs in both ambient factors.
• After: the independently written standard-library verifier constructs full matrices for every H of sizes 1 and 2 over F2,F3,F4,F5: 18+84+260+630 = 992 cases. Direct idempotence, commutation, pivots, product, Schur ranks, sector minima and ordered factors pass. It also checks 18 deficient-map controls, 12 nonbinary 4d witnesses, 12 nondiagonal size-three blocks, three invalid-input rejections and two unsigned-sector rejections.

Both registered commands passed locally in 0.67 s / 0.70 s, including optimized Python with explicit checks. CPython 3.12.15, macOS arm64. Committed package replay, exact committed-byte/source-history preflight, official export integrity and a read-only Git-free export replay pass. Export stdout is identical with PATH=/nonexistent; all six exported files remain unchanged. The generated stdout receipt binds the package files by SHA256. Internal proof review found no blocking issue; independent human and formal proof-kernel review are unclaimed.

Exact reproduction from the repository root:

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

Markers: PASS_ALL_FIELD_IDENTITY_SCHUR, PASS_REGISTERED_RESEARCH_PACKAGE, PASS_RESEARCH_COMMIT_BYTES.

Limitations

The universal claim is the written field-algebra proof, with finite corroboration. Nonidentity pivots, coefficient rings, later orbit transitions and header counts are outside the claim. Header production, reads, restoration, stock, precision and the full consumer bill remain unresolved. No selected construction, Lean or Docker replay was run. The historical card baseline attempt stopped at omitted sparse input research/fresh-helper-boundary/ARITHMETIC-RESULT.json; no baseline pass is claimed.

Canonical main remained b58ad9ef4d4e861b0eaa5dba41742bdc072dca51 at the pre-commit refresh. Live overlap, all-main README searches, merged searches, scope change and exact source heads are recorded in TASK.json. Advance claim.

Open next questions

• Determine the prescribed-H minimum for common nonidentity scalar pivots. First derive the sector products and replacement full-rank criterion.
• Extend Count complete headers with sharp identity-pivot Schur blocks (no κ) #4678's shared-H complete-header count to one odd-field scalar profile. A paid physical header cover remains separate.

Related open work

• Prove exact identity-pivot Schur dimension for every prescribed H (no κ) #4671 supplies the binary fixed-H theorem. Numerical overlap agrees: F2, d=1, H=0 or 1 gives n=5 and commutator rank 2; d=2, H=[[0,1],[1,1]] gives n=8 and rank 4.
• Prove identity-pivot Schur dimension bounds; attain 4d (no κ) #4657 supplies the identity-pivot precursor. Its d=2 minimum 8 agrees with that direct n=8/rank-4 construction.
• Prove the sharp 2d block Schur commutator bound (no new kappa) #4597 supplies the ordered block factorization. Its 2d upper bound is attained here, including F3,d=1,n=4,rank=2 and F2,d=2,n=8,rank=4.
• Refine #4597's Schur commutator bound for projectors: odd-codimension parity theorem and minimal binary/ternary sharpness witnesses (no new κ) #4609 supplies field-specific sharpness and the prior ternary scalar minimum. Our F3,d=1,H=2 construction has n=4 and rank 2, agreeing numerically. Its unrestricted-pivot statement is credited, not audited.
• Binary commuting projectors attain the Schur-commutator bound 2d for every d at n=5d (follow-up to #4597 OQ2 and #4609; no new κ) #4617 supplies a binary identity-pivot 5d family. Our H=0,d=2 construction has n=10 and commutator rank 4, agreeing numerically with that upper bound.
• Prove optimal binary Schur sharpness dimension n = 3d for d >= 2 #4634 attains unrestricted binary pivots at 3d. This contribution imposes identity pivots; the ambient minima concern different contracts. Both attain rank 4 at d=2, checked here at n=8 for the identity contract; its n=6 construction is not replayed.
• Count complete headers with sharp identity-pivot Schur blocks (no κ) #4678 counts binary sharp headers. Our scalar H=0/1 constructions have sectors (1,1,1,2)/(0,2,2,1), n=5 and rank 2; the d=2 witness has sectors (2,2,2,2), n=8 and rank 4. These structural numbers agree; no header-count number is recomputed.
• Classify seven residual Schur-pair states and full-guard counts (no new κ) #4689 studies subsequent binary residual states. Its sharp initial sector (1,1,1,2) agrees with our F2,H=0,n=5,rank-2 construction. Subsequent orbit/header counts are outside this result.
• Derive invariant Schur guard transitions without GL4/GL5 enumeration (no new κ) #4735 derives a binary state recurrence and invariants. This extension changes the field of the initial dimension theorem; there is no common numerical recurrence count.
• Exhaustively check #4597's block Schur commutator bound on all commuting pairs in M3(F2), M3(F3), M4(F2); binary/ternary idempotent pairs stay at rank <= 1 (no new κ) #4608 is a finite small-domain audit of Prove the sharp 2d block Schur commutator bound (no new kappa) #4597. This contribution extends a structural theorem to arbitrary fields and dimensions; none of its census totals is claimed or rerun.

Related open work refresh

This account's open PRs created since the CLAIM at 2026-10-22 18:55:31 UTC were refreshed before DONE, and their current titles and bodies were read:

• Prove exact square-zero radical reachability and binary count (no new κ) #4745 classifies square-zero radical right-vector reachability. It concerns a different algebraic target and cost; no Schur-dimension quantity overlaps.
• Prove the corank-one pair header law over every finite field (no new κ) #4747 gives a corank-one pair full-guard header law over finite fields. It counts complete headers, while this contribution determines the minimum ambient dimension for a prescribed compressed product. No numerical header probability is claimed here, so there is no overlapping number to equate.
• Classify all ten shortest binary depth-six chain schedules (no new κ) #4748 classifies shortest depth-six binary chain schedules. Its schedule counts are unrelated to the prescribed-H Schur dimension.
• Prove the semilocal right-vector cost formula (no new κ) #4749 proves an invertible semilocal right-vector cost formula. Its factor costs and matrix-ring target counts are different quantities, with no numerical overlap.

No newly opened account PR duplicates this all-field fixed-H minimum. The earlier related Schur work and numerical comparisons remain listed above.

CI notes

All 2 checks passed on 5f014560d6225cae17598ac479fb29d2947d8e6b in Registered research packages run 39836723974. plan passed in 2m53s and the registered research/all-field-identity-schur replay passed in 1m10s. The hosted replay certifies the package checks at this exact commit within their stated finite scope; it does not supply human or proof-kernel acceptance of the universal theorem. All local obligations above also passed.

Merge danger

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