[petalas]

Result

Claim type: scoped all-field proof (lower bound and interval structure) plus an exact finite certificate (d=2 over F2). No new κ.

This answers Open next question 1 of Krzysztof Wojt's #5128 (research/arbitrary-pivot-schur-minimum): which Schur-commutator ranks below 2d are feasible at each ambient dimension, given prescribed invertible pivots A,B and product block H.

With T00=I-A-B+H, T10=A-H, T01=B-H, T11=H, n0 = Σ rank T_j, rho0 = rank(T M T) and nmax from #5128:

1. Over every field: no pair exists below n0, and the rank is forced to rho0 at n0. For n >= n0 the feasible ranks form the interval [rho0, rhomax(n)], and
   n_min(rho) >= max(n0 + rho - rho0, nmax - 2(2d - rho)).
   The second term uses a new deficiency identity: rank R = 2d - a and rank L = 2d - b, where a and b are the sector-image and sector-kernel deficiencies.
2. For d=2 over F2, equality holds for all 576 triples (288 with AB != BA). rho0 is 0, 2 or 4 for 408, 144 and 24 triples, with 16 profiles. Rank two is feasible at n exactly when rho0 <= 2 and n >= max(n0+2-rho0, nmax-4).

Proof: research/schur-intermediate-ranks/PROOF.md.

Evidence and reproduction

verify.py uses only the standard library. For every triple it enumerates every admissible block-subspace pair and computes the exact least dimension for each rank. On every pair it checks the rank R/L identities and the sandwich bound, and it checks the closed law. It then builds 2,496 literal commuting idempotent pairs, one for each triple and feasible rank at the claimed least n, and recomputes S, Z and the commutator rank from the matrix entries. Four seeded controls are rejected.

~/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/schur-intermediate-ranks/verify.py
~/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/schur-intermediate-ranks/verify.py
~/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/schur-intermediate-ranks
~/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/schur-intermediate-ranks --check-source-history

All four pass locally (CPython 3.12, macOS). Each registered check takes about 2.7 s. Expected rows_sha256 e18d15092793582f8d16939086fb512563ecbffc95e0f537ddece6873232dc81.

Limitations

• Equality is proved only for d=2 over F2. For other d and fields the formula is a lower bound.
• One common deletion with invertible pivots. No header count, physical producer or consumer, restoration, precision, stock or outer transfer.
• Prove exact Schur minima for arbitrary invertible block pivots (no new κ) #5128's factorization and rank-2d minimum are cited, not rechecked. No independent human or formal proof-kernel review.

Open next questions

1. Does equality hold for every d and field? First step: run the same reduction at d=2 over F3 and d=3 over F2, then try a two-phase step chain that keeps im R ∩ ker L = 0.
2. Is there a closed form for rho0? The ranks of the T_j alone do not determine it. Start from the proved sandwich 2d - 2Σ nu_j <= rho0 <= 2d - Σ nu_j.

Related open work

• No open PR studies nonmaximal Schur-commutator ranks or this prescribed-pivot model. Prove exact Schur minima for arbitrary invertible block pivots (no new κ) #5128 (merged) is the direct predecessor.
• This account's PRs opened since the claim (Certify that balanced depth-three chain multiplicities break every unit symmetry for m ≤ 24 (no new κ) #6037, Prove the conjugate-pair target needs four calls in every group (no new κ) #6038, Classify mixed Q8 call-space shear identities; no pair of shear images separates (no new κ) #6039) study chain multiplicities and call counts, not this object; no shared quantity.

CI notes

All hosted checks passed on head acc000f4edfa0b0c4a90b505c6e02a4c3572f74c: plan and replay (research/schur-intermediate-ranks) in run https://github.com/CrocSwap/integer-mult-bounds/actions/runs/40607943406.

Credit: produced by Nick Petalas. Claim: #1166 (comment)