[pinterbanget]

Exhaustive check of petalas's #4597 (head dfeb8ca7), the block Schur commutator identity and rank bound, on every commuting pair in M₃(F₂), M₃(F₃) and M₄(F₂), and every split d. No new κ.

Method

research/pr4597-schur-exhaustive/verify.py uses the standard library only, takes about 80 s and imports nothing from #4597.

• Enumeration. Q runs over the centralizer of P, a nullspace computed here.
• Symmetry reduction. P runs over orbit representatives under GL_d × GL_{n−d} block conjugation, with orbit-size weights.
• Totals. The weights recover all 7,456, 809,433 and 2,526,976 commuting pairs.

Results

• Prove the sharp 2d block Schur commutator bound (no new kappa) #4597 holds everywhere checked. The identity [S,Z] = XV − UY, the bound min(2d, n−d) and the equality criterion all hold for every admitted pair.
• The bound is attained by general commuting pairs wherever it is at least 2. For example, M₄(F₂) at d = 2 has 30,672 pairs of rank 2.
• Finding, for Prove the sharp 2d block Schur commutator bound (no new kappa) #4597's open question 2. Commuting idempotent pairs never exceed rank 1 in these domains: F₂ with n = 3, 4 and F₃ with n = 3. Binary and ternary projectors therefore do not reach 2d at these sizes. A positive control over F₅ exhibits an idempotent pair in M₃ with rank 2 = 2d, consistent with Prove the sharp 2d block Schur commutator bound (no new kappa) #4597's F₅ sharpness family.

Limitations

• Sizes. Coverage is exhaustive only at these small sizes. It corroborates the written proof and does not replace it.
• The idempotent finding is not proved for larger n.

Open question

Do binary commuting idempotents reach 2d at n = 5 or 6? First step: enumerate simultaneously diagonalizable pairs.

Credit: petalas (Nick Petalas) for #4597 and #4278. Ryanson Jonathan, with Claude Code (Claude Opus 5.5) assistance.

Head: aae3cdd

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