[ssafwann]

Change and mathematical scope

PR #4634 leaves open what changes when both leading blocks must be identities. This adds one self-contained package, research/identity-pivot-schur/, proving that for commuting binary idempotents with leading blocks I_d and Schur-commutator rank 2d:

• n >= L(d) = min_{t=0,...,d} max(3d+2t, 4d-3t) >= ceil(17d/5). In particular the unrestricted-pivot optimum n=3d is impossible.
• A new explicit construction attains n=4d for every d>=2, improving Binary commuting projectors attain the Schur-commutator bound 2d for every d at n=5d (follow-up to #4597 OQ2 and #4609; no new κ) #4617's identity-pivot 5d family.
• For d=2, the exact minimum is n=8. For general d, 4d is optimal within the class where H+H^2 is invertible, with H the leading block of PQ.

The proof specializes Nick Petalas's ordered factorization to K=AB and derives BA=[[J,0],[J,J]], where J=H+H^2. Rank-nullity and the four joint-sector projectors give the lower bounds. A four-block change of basis supplies the upper bound. README Sections 1–3 contain the complete argument and a literal 8-by-8 witness.

No new κ, header density, physical caller or supplier admission. At this package's first publication the unrestricted identity-pivot minimum lay between L(d) and 4d. Successor PR #4671 now closes that gap with the exact minimum n(H)=4d+nullity(H+H²) for every prescribed leading block H of PQ, hence 4d for all d>=2. Universal claims rest on the written proof, not extrapolation from samples.

Task and overlap

Canonical base and submission refresh: b58ad9ef4d4e861b0eaa5dba41742bdc072dca51, priority revision 7, selected conditional κ=0.00302015. Head: b79f90bc8d839a4998a40b10bcb437e87baa8397.

Handoff-first scope, coordinated in Discussion #4653 and the #1166 reply. formal-interface-lemma is explicitly marked changed to a written algebraic side result, rather than a Lean formalization or selected-caller audit.

Nearest predecessors: #4597 (dfeb8ca7, factorization), #4609 (eb5ccf92, scalar witness), #4617 (96c01a21, 5d family), #4634 (9f053f70, unrestricted 3d optimum and this open question). Full hashes and source paths are in TASK.json. All four heads were unchanged on the final live check.

The final overlap refresh covered 426 open PR bodies (14:27 UTC), 3,588 Discussion records (14:29 UTC), and every one of 5,862 in-scope local notes, with no coverage warnings. It located the credited open question and this claim, not a duplicate result. This is not a search of every diff, closed PR or unpublished result.

Validation

Python 3.12.11 / macOS arm64, standard library only:

python3.12 -B scripts/research_package.py --package research/identity-pivot-schur
python3.12 -B scripts/check_research_commit.py --package research/identity-pivot-schur --check-source-history

• Registered ordinary and optimized Python replays pass; checks use explicit exceptions.
• Direct full-matrix checks cover the literal witness, d=2..12, zero padding, all 2/48 admissible binary seeds of sizes 2/3, and a mixed singular-J fixture.
• Exhaustive identity-pivot commuting-idempotent domains contain 253 ordered pairs at (n,d)=(3,1) and 1,219 at (4,2); both factorizations and joint-sector identities pass.
• Eleven controls reject invalid premises, factor omissions, false sharpness and the invalid inference that a sharp rectangular product forces J invertible. The last control uses the credited Binary commuting projectors attain the Schur-commutator bound 2d for every d at n=5d (follow-up to #4597 OQ2 and #4609; no new κ) #4617 scalar witness.
• All five committed files and source-history pins pass. Official export and integrity checks pass. The exported verifier passes with -I -O -B, no Git checkout and PATH=/nonexistent; all six export files, including its receipt, remain unchanged.
• The affected-package planner selects only research/identity-pivot-schur; the working tree is clean.

Pending predecessors are citations, not replay-time Git dependencies. The two attributed scalar control matrices are contained in the checker. No predecessor executable is imported.

No independent human or formal-kernel review is claimed. Lean, Docker and the selected scientific cold replay were not run. The separate selected baseline attempt stopped at an omitted sparse-checkout checkpoint; no baseline pass is claimed. Hosted CI is separate from these local results. The Registered research packages run on exact head b79f90bc8d839a4998a40b10bcb437e87baa8397 reports action_required: a maintainer must approve the fork workflow before it can run. No hosted replay pass is claimed.

Credit

Nick Petalas supplies the block-Schur factorization and #4278/#4259/#4228 lineage; Ryanson Jonathan supplies the binary scalar witness and 5d tensor family; lee-phys supplies the unrestricted 3d construction, reused binary seeds and explicit identity-pivot question. Cinna Davis supplies the earlier BIT request/header setting. Predecessor AI disclosures and Apache-2.0 attribution are retained.

Prepared for ssafwann with substantial OpenAI Codex (gpt-6-astra) assistance, by one agent. No shared code, frozen evidence or selected frontier changes.

Open next questions

The original singular-J dimension question is answered by successor PR #4671: sharpness requires n>=4d+nullity(J), and the bound is attained for every H. The remaining complete-header counting question and its scope are recorded there and in Discussion #4653. This PR retains the first proof, bounds, construction and verification artifact unchanged.