[pinterbanget]

A follow-up to #4597's open question 2 and my #4609. Over F₂, commuting projectors attain the Schur-commutator bound 2d for every d, at n = 5d. No new κ.

Proof

Tensor #4609's F₂ witness at n = 5, d = 1 with I_d.

• Inputs. P ⊗ I_d and Q ⊗ I_d are commuting idempotents with leading blocks I_d.
• Complements. Their Schur complements are S ⊗ I_d and Z ⊗ I_d.
• Rank. The commutator is [S,Z] ⊗ I_d, of rank 2d. This equals the refined bound min(2d, 2⌊4d/2⌋) = 2d.

research/binary-projector-sharpness/verify.py checks d = 1, …, 4. It uses the standard library only and runs in well under 1 s.

Limitation

The minimal n is open. n = 5d suffices, but d = 2 already reaches rank 4 at n = 6 (#4609).

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

Head: 96c01a2

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