[lee-phys]

PR #4617 gives binary Schur-commutator sharpness in dimension n=5d and leaves the minimum dimension open. This contribution proves

For every d >= 2, the minimum is n=3d.

There are commuting idempotents P,Q over F2 with invertible leading d-by-d blocks and Schur-complement commutator rank 2d. Both projectors have rank 2d. The necessary lower bound is immediate from the commutator's (n-d)-by-(n-d) size; the new construction attains it.

The requested first case d=3,n=9 is given as literal matrices. The general construction starts with T and I+T both invertible, constructs P,Q in three d-by-d blocks, and computes the Schur complements explicitly. Their commutator K satisfies
K^2 = diag(R^2(R+I)^2, R^2(R+I)^2), where R=T(I+T)^(-1) and R+I=(I+T)^(-1), so K is invertible. Fixed 3-by-3 and 2-by-2 seed matrices, followed by direct sums, cover every odd d>=3 and every even d>=2.

Scope and credit. The genuinely new dimension cases are odd d>=3. The even cases already follow by direct sums from Ryanson Jonathan's d=2,n=6 example in #4609; the new formula also realizes them. Nick Petalas's #4597 supplies the preceding block-Schur rank bound and #4278 the precursor setting; Ryanson's #4617 supplies the uniform n=5d construction and open question. Exact inspected heads are recorded in the README and TASK.json. The new theorem requires invertible pivots, not identity pivots; its equal pivots are generally nonidentity. The d=1 classification is credited to #4609 rather than re-proved here.

Only the five-file package research/binary-schur-minimal-dimension is added. README Sections 1–4 contain the complete proof. The standard-library verifier imports no predecessor code and has no external mathematical inputs.

Validation on Python 3.12.13 / macOS:

• The literal 9-by-9 witness passes direct full-matrix idempotence, commutation, pivot-invertibility and rank checks.
• The formula passes at d=2..12 and with larger-ambient zero padding.
• All 16 binary 2-by-2 and 512 binary 3-by-3 seeds are enumerated: 2 and 48 have both T and I+T invertible, and every admissible seed passes.
• Six semantic controls reject singular pivots, zero and nonzero singular T, identical projectors, broken idempotence, and noncommuting projectors.
• Registered normal and optimized Python replays, metadata/pins, committed-byte checks, export integrity and isolated standalone replay from a Git-free export pass. The export's six files, including its receipt, remain unchanged.
• The affected-package planner selects only this package; staged whitespace checks pass.

Hosted CI run 39786020403 passed on exact head 9f053f7, including the registered package replay.

Reproduce:

python3.12 -B scripts/research_package.py --package research/binary-schur-minimal-dimension

The universal statement is the written algebraic proof, not an inference from finite samples. This is author-checked work with substantial OpenAI Codex assistance; independent human and formal proof-kernel review are unclaimed. It supplies no header density, physical caller, supplier admission or new kappa. The selected scientific replay, Lean, Docker and the optional full-history preflight were not run.

Base and submission refresh: canonical main fbc9ea0. This follows the donor's explicit mathematical assignment; the formal-interface-lemma card is marked changed to a written constructive theorem. The final overlap search refreshed 408 open PRs and 3,577 Discussion records through 2026-10-22 13:18 UTC, finding the credited predecessors and this scoped claim. Local sparse coverage omits 5,824 historical note paths; no exhaustive or global novelty claim is made.

Prepared by lee-phys with substantial OpenAI Codex assistance. Apache-2.0.