[petalas]

Summary

Scoped algebraic proof with finite corroboration. For commuting field idempotents with the same nonzero scalar leading entry lambda and prescribed leading product entry h, this gives the exact least ambient dimension attaining Schur-commutator rank two:

                    4 + [1-2lambda != 0],  h=0,
N(lambda,h) =       5 + [lambda != 1],     h=lambda,
                    3 + [1-2lambda+h != 0], otherwise.

Every minimum is attained. In particular, a nonidentity pivot with h=lambda requires exactly six dimensions. Over F3, lambda=h=2 requires six, while the same pivot with h=0 or h=1 requires four.

This answers the scalar case of Open next question 1 in #4746, head 5f014560d6225cae17598ac479fb29d2947d8e6b. That identity-pivot theorem, ssafwann's #4671/#4657, and Ryanson Jonathan's known unrestricted small-field minima retain their credit. The new contribution is the exact dependence on both prescribed scalars over every field, including the replacement sharpness criterion. No predecessor executable is imported or audited. No new kappa or selected-frontier change.

Only research/scalar-pivot-schur-dimension/ is added. The result was produced by Nick Petalas. Nick Petalas's #4278/#4597 supplies the scalar/block factorization; Ryanson Jonathan #4609/#4617 and lee-phys #4634 supply earlier sharpness constructions. The #4259/#4228, Cinna Davis and normalizer lineage credit is retained in the package.

Advance claim. The formal-interface-lemma card is explicitly changed to this algebraic side question. The initial claim names intake checkpoint b58ad9ef4d4e861b0eaa5dba41742bdc072dca51; canonical main advanced before branch creation, so the actual package base is 1f87c3578ae546c8f903f7f4d83dc2e2fb99650b. Task/priorities/selected guidance is unchanged. This correction is recorded in TASK.json.

Evidence

Before: #4746 explicitly leaves nonidentity pivots open. After: PROOF.md gives the four compressed sector products and the exact two-factor rank criterion. Zero-product sectors require two dimensions precisely in the endpoint cases. A concrete sector construction attains each bound.

• All 210 prescribed parameter pairs over F2,F3,F4,F5,F7,F8,F9 are constructed as actual full matrices. Direct checks cover idempotence, commutation, pivots, leading product, Schur multiplication, commutator rank and the ordered scalar factor identity.
• A separate exact optimization enumerates 14,291 scalar vector pairs and 1,212 four-sector patterns; every minimum agrees. This includes all possible scalar zero/nonzero map types, each realizable in dimension at most two. It is not an ambient-matrix word-length truncation.
• Seven controls detect both omitted zero-product sectors, a zero pivot, an incorrect sector sign, a mutated projector, a singular basis and a nonfield order.
• Both registered commands pass on CPython 3.12.15 / macOS arm64. Each was timed locally: 0.043 s ordinary and 0.041 s optimized, against a 60-second timeout. These are check durations, not speed comparisons. Explicit checks remain active under -O.
• Package metadata, six-file committed-byte/source-history preflight, official export integrity and a read-only optimized Git-free export replay pass on 22cb0bda57237d39d0e6944b232447d3a2809311. The export produces the same generated six-file hash receipt and leaves all seven exported files unchanged, including the export receipt.
• AST syntax, function-capacity and dynamic-code inspections pass; the longest function has 35 lines. Warnings are errors. No external static analyzer is installed or claimed. The bounded CPython allocation deviation is documented.
• Selected source-binding orientation passes after materializing its declared inputs. This checks metadata consistency only; no selected construction replay was run.

From the repository root:

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

Expected markers: PASS_SCALAR_PIVOT_SCHUR_DIMENSION, PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=completed, PASS_RESEARCH_COMMIT_BYTES. Another CPython 3.12 path may replace the local path. The direct verifier writes no files; source-history checking needs full Git history.

Limitations and unrun obligations

The universal claim rests on the written algebraic proof, with finite corroboration. It assumes equal nonzero scalar pivots, field coefficients and commuting idempotents. Unequal pivots, larger prescribed matrix products, coefficient rings, later-state transitions and header probabilities are outside scope. Header production, selection/emission, dispatch, 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. All registered local checks ran.

Related open work

• Prove the exact identity-pivot Schur minimum over every field (no κ) #4746 proves the all-field identity-pivot minimum. Our overlapping lambda=1 specialization agrees numerically: F2,h=0/1 gives n=5; F3,h=2 gives n=4, all with commutator rank two. The new result classifies nonidentity scalar pivots.
• Prove exact identity-pivot Schur dimension for every prescribed H (no κ) #4671 supplies the binary prescribed-H theorem. At d=1,H=0/1 the same two n=5/rank-two values agree. Its larger-d matrix theorem is credited rather than replayed.
• Prove identity-pivot Schur dimension bounds; attain 4d (no κ) #4657 supplies the binary identity-block precursor. Its d=2,n=8 result concerns a larger deleted block; there is no common d=2 minimum in this scalar theorem.
• Prove optimal binary Schur sharpness dimension n = 3d for d >= 2 #4634 permits arbitrary invertible block pivots and obtains n=3d for d>=2 over F2. Its block minima are outside the scalar domain; no overlapping d>=2 minimum is asserted.
• Count complete headers with sharp identity-pivot Schur blocks (no κ) #4678 counts binary sharp identity-pivot headers. Our exact scalar constructions have sectors (1,1,1,2) at h=0 and (0,2,2,1) at h=1, both n=5/rank two, agreeing with its initial profiles. No header-count number is recomputed here.
• Classify seven residual Schur-pair states and full-guard counts (no new κ) #4689 classifies subsequent binary residual pairs. Its initial (1,1,1,2),n=5,rank-two profile agrees with the direct h=0 construction. Its continuation counts are different quantities and are not replayed.
• Derive invariant Schur guard transitions without GL4/GL5 enumeration (no new κ) #4735 derives invariant binary residual transitions. The same initial n=5/rank-two profile agrees; no transition probability or orbit count is claimed here.
• Prove a complete factor invariant for ordered idempotents (no new kappa) #4762 gives a complete factor invariant for general ordered idempotents. It does not optimize ambient dimension under prescribed equal scalar pivots and product. Its orbit counts and active-dimension formula are different quantities, with no overlapping numerical minimum asserted here.

The current titles and bodies of these related PRs were read. Live claims and open PRs were refreshed immediately before committing; distinctive searches of every research README on canonical main and merged PRs found the credited factorization, not this dimension classification, within the stated search coverage. Sparse local-note and unpublished/code-diff coverage remain partial. Exact coverage is recorded in TASK.json.

Open next questions

1. Classify unequal nonzero scalar pivots p,q with prescribed h, starting with sector products (1-p-q+h,p-h,q-h,h).
2. Determine the minimum for common nonidentity scalar blocks lambda I_d with prescribed matrix H, beginning with one nondiagonal H.
3. Count complete headers for one fixed nonidentity scalar profile after specifying the ensemble. A paid producer and consumer remain separate obligations.

CI notes

CI pending for 22cb0bda57237d39d0e6944b232447d3a2809311. No hosted certification is claimed yet. All registered checks passed locally. Hosted completion will be recorded here after the orchestrator resumes the contribution.

Merge Danger

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