[apoursh]

Change and mathematical scope

Resolves both open questions in #6031: the sharp minimum ambient dimension at every Schur-commutator rank holds over every field and every positive block size, and the forced minimum rank has a closed formula using only d-by-d matrices. The eight-file package is research/schur-ranks-all-fields/; predecessors and shared files are unchanged.

For invertible A,B and arbitrary H, put F=A-H, J=B-H and T00=I-A-B+H. For commuting idempotents with prescribed leading blocks A,B,(PQ)11=H, and Schur complements S,Z:

n0   = rank T00 + rank F + rank J + rank H
nmax = 3d + rank T00 + nullity F + nullity J + nullity H
rho0 = rank(J A^-1 F) + rank(F B^-1 J) + 2 rank H - 2d
n_min(rho) = max(n0 + rho - rho0, nmax - 2(2d-rho))

Every integer rho in [rho0,2d] occurs exactly at dimensions n >= n_min(rho). Here nmax is the least dimension attaining rank 2d, not a maximum ambient dimension. The proof identifies two independent cross-sector pairing blocks; sector-preserving rank cores and radicals attain the lower bound over any field. A ternary example has rho0=1 and dimension profile 5,6,7,9, showing why the binary d=2 evenness pattern does not generalize.

This is a written algebraic proof with bounded computational corroboration, relying explicitly on the credited predecessor factorization and sector-realization results. It is not proof-kernel or independent human acceptance. No new kappa, physical supplier, signed invoice, or selected-record assumption discharge is claimed. Production, consumption, restoration, precision, stock, compiler and outer costs remain unpriced.

Task and overlap

Canonical base: 721c03822c421b7f7fcbb2598993b947c4d88419, priority revision 8, selected conditional kappa=0.003125804. The original formal-interface-lemma card is retained verbatim in TASK.json with scope explicitly changed to this released algebraic side question, not formalization or a selected-assumption proof. Exact rank/dimension feasibility is useful before pricing a concrete caller.

Handoff-first intake followed the released #6031 questions in discussion #1166. Canonical progress update. Targeted all-state Schur/#6031 searches and recent board claims found no later extension; coverage is not a historical census. Immediate follow-ups to our #5672 were already covered by #5683/#5689/#5692/#5712. README records the five-point family-expansion decision and stop rule.

Exact predecessors:

• Nick Petalas Prove nonmaximal Schur-commutator ranks form an interval with a sharp binary dimension law (no new κ) #6031 at acc000f4edfa0b0c4a90b505c6e02a4c3572f74c: lower bound, sector-subspace cost reduction and equality at d=2 over F2. This submission proves equality in all dimensions over every field and supplies the requested closed rho0 formula.
• Krzysztof Wojt Prove exact Schur minima for arbitrary invertible block pivots (no new κ) #5128 at 2839d35e58b3285a1992a3ecbae2809ebef7661b: arbitrary-pivot maximal-rank theorem and realization.

SOURCES.json binds exact predecessor bytes as provenance; VERIFY.json has no external live inputs.

Validation

CPython 3.12.14, macOS arm64, standard library only. All package checks passed at commit 12385ca23315181e3d15eb23a2d9ffec65d9a909:

python3.12 -I -W error -B research/schur-ranks-all-fields/verify.py
python3.12 -I -O -W error -B research/schur-ranks-all-fields/verify.py
python3.12 -B scripts/research_package.py --package research/schur-ranks-all-fields
python3.12 -B scripts/check_research_commit.py --package research/schur-ranks-all-fields --check-source-history
python3.12 -B scripts/export_research_package.py export --package research/schur-ranks-all-fields --ref HEAD --output /tmp/schur-ranks-all-fields.tar
python3.12 -B scripts/export_research_package.py check --check-inputs /tmp/schur-ranks-all-fields.tar

The official export also passed normal/optimized replay in a read-only extraction without a Git checkout or Git on PATH; payload bytes and modes remained unchanged. The selected-baseline source-binding check passed; it is not a scientific frontier replay.

Coverage: 886 triples, 303,878 admissible subspace pairs, 3,516 literal witnesses. Exhaustive d=1 over F2/F3/F5 and all 576 binary d=2 triples; fixed samples of 192 ternary d=2 and 24 binary d=3 triples. The direct subspace oracle does not use the proposed minimum formula; it shares only exact arithmetic with the formula checker. Literal witnesses recheck idempotence, commutation, prescribed blocks and Schur rank. Eight controls reject omitted/equated cross blocks, forced evenness, a wrong minimum, corrupted witness, wrong product block, slope-one-only law and reordered products. Coverage inventories and expected result digest are checked:

c38f9ced06db3ecf1bc4c7f8802df7f1b3228f3435946700e7340b115d3156db

Known base CI issue: the unchanged base fails two of six test_research_current_status.py tests: docs/ci.md still identifies the earlier frontier command, and the current-status decimal is bolded together with its rational expansion. #6050 already addresses both. This package does not duplicate that shared-document fix. No full selected-construction replay, Lean build or Docker replay was run; none is implicated by these isolated files. Hosted CI is reported separately.

Credit

New proof and verifier by apoursh, with substantial OpenAI Codex assistance in derivation, implementation, independent subagent checking and verification. Apache-2.0. Retains Nick Petalas #6031/#4597, Krzysztof Wojt #5128, ssafwann #4657/#4671 and Nick Petalas #4746/#4771/#4774/#4785 lineage. The literal realization follows the credited predecessors. Two independent AI-assisted derivations are not independent human review.

Open next questions

README and TASK.json ask whether an actual decoder/header caller can use this sharp frontier with a favorable complete physical bill. First specify exact A,B,H, target rank, allowed sector dimensions and all charged production/consumption/restoration operations. The missing caller and complete cost contract are the blockers; no further algebraic construction round is justified without them.