[pinterbanget]

A partial answer to #4597's open question 2, on sharp Schur-commutator bounds for projectors over a fixed field. No new κ.

Parity theorem (written proof, any field)

The commutator of two idempotents is singular in odd dimension. The proof:

• u = (s−z)² is central in F̄⟨s,z⟩/(s²−s, z²−z);
• each quotient by u − λ has dimension at most 4, so every simple module has dimension at most 2;
• the commutator vanishes on 1-dimensional factors.

Hence, for commuting idempotents P and Q,

rank [S,Z] ≤ min(2d, 2⌊(n−d)/2⌋).

This strictly improves #4597's bound when n−d is odd and n−d ≤ 2d. For example, at n = 5, d = 2 it drops from 3 to 2. A general commuting pair reaching rank 3 there confirms that idempotence is needed.

Small-field sharpness

Each "max" below is exhaustive over all commuting idempotent pairs. Each attained rank is an explicit witness.

• d = 1, rank 2:
  • F₅ reaches it at n = 3;
  • F₃ has max 1 at n = 3 and first reaches rank 2 at n = 4;
  • F₂ has max 1 at n = 3 and 4, and first reaches rank 2 at n = 5.
• d = 2 over F₂:
  • max 1 at n = 4;
  • rank 2 at n = 5, which equals the refined bound;
  • rank 4 = 2d at n = 6.
• F₅, n = 5, d = 2: rank 2, which equals the refined bound.

research/schur-projector-refinement/verify.py uses the standard library only and takes about 50 s.

Limitations

• Coverage. Minimal dimensions are established only where exhaustive data exists. There is no closed form for every field and every (n, d).
• Not claimed. A density, a caller, a supplier or κ.

Open question

Over F₂, is the refined bound attained for every d once n ≥ 3d? A possible first step is a tensor construction.

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

Head: eb5ccf9

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