[ssafwann]

Change and mathematical scope

PR #4657 leaves the identity-pivot Schur minimum open when H+H² is singular, beginning with d=3,n=11. This independently replayable package closes that gap and proves the exact answer for every prescribed leading block H of PQ:

n_min(H) = 4d + dim ker(H+H²) over F2, for commuting idempotents with common leading blocks I_d and Schur-commutator rank 2d.

Thus the unrestricted identity-pivot minimum is 4d for every d>=2, and 5 for d=1 (the scalar result was already known from #4609). In particular d=3,n=11 is impossible and n=12 is attained. Singular H+H² is permitted, with the exact extra dimension given above.

The new proof shows that sharpness is equivalent to injectivity of the inclusion and surjectivity of the projection on each of the three active joint eigenspaces. Their dimensions must be at least (d-b,d+b,d+b,d+a), with a=nullity(H), b=nullity(I+H). A split factorization construction attains all four bounds for every H, including non-semisimple H. The README contains the complete proof and a literal 13-by-13 nilpotent-product witness.

Only research/identity-pivot-sector-bound/ is added: five files, no shared implementation, frozen source, or selected-frontier changes. No header density, physical caller, supplier admission or new κ. Universal claims rest on the written proof, not finite extrapolation.

Task and overlap

New branch from fetched canonical main b58ad9ef4d4e861b0eaa5dba41742bdc072dca51; priority revision 7 and selected conditional κ=0.00302015 are unchanged. Publication head: 8d2e3bc.

Handoff-first scope, claimed in the existing canonical Discussion #4653. The formal-interface-lemma card is explicitly marked changed to this algebraic side theorem. The family decision is narrow and stop the dimension question; any subsequent complete-header count needs its own ensemble and caller contract. All physical costs remain unknown, not zero.

The final overlap refresh covered 439 open PR bodies (2026-10-22T15:02:24.135490+00:00), 3,595 Discussion records (2026-10-22T15:04:47.720156+00:00), and all 5,860 tracked in-scope notes on main, with no coverage warnings. It found the credited predecessors and this scoped claim, with no duplicate sector-dimension result located. Coverage does not include arbitrary diffs, closed remote PRs or unpublished work.

All five predecessor heads were rechecked unchanged at 15:03 UTC. Exact head, path, Git blob and SHA256 citations appear in TASK.json. Pending sources are attributed citations, not replay-time Git dependencies.

Validation

Python 3.12.11 on macOS arm64; standard library only:

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

• Registered ordinary and optimized (-O) replays pass; explicit exceptions keep checks active in both modes.
• All 530 H of sizes 1,2,3 pass prescribed-product, direct full-matrix idempotence/commutation, identity-pivot, commutator-rank and minimal-sector checks.
• The d=2..12 optimal family, literal 13-by-13 singular non-semisimple witness and padding pass.
• All 1,472 ordered commuting-idempotent pairs in the stated small domains pass the factor identities, six-map criterion and sector bounds.
• Ten rejection controls pass. Six preserve all matrix premises while removing exactly one of the six full-rank map conditions, losing sharpness; four address domain and overclaim errors.
• All five committed files and declared source-history pins pass. Official export/integrity checks pass. The read-only, Git-free export passes -I -W error -O -B with PATH=/nonexistent; all six exported files, including the receipt, remain unchanged. The affected-package planner selects only this package, and the working tree is clean.

No independent human or proof-kernel acceptance is claimed. Lean, Docker and the selected scientific cold replay were not run for this isolated package. The Registered research packages run on exact head 8d2e3bc76aa7d43e5c629daaf1e72d4dd5a81ed9 reports action_required: fork workflow approval is pending. No hosted replay pass is claimed.

Credit

Nick Petalas (#4597 and #4278/#4259/#4228) supplies the ordered factorization and precursor setting. Ryanson Jonathan (#4609/#4617, with disclosed Claude Code assistance) supplies the known scalar minimum and 5d family. lee-phys (#4634, with OpenAI Codex assistance) supplies the unrestricted 3d optimum, reused seeds, nonidentity control formula and original identity-pivot question. #4657 supplies the identity specialization, prior lower bound and 4d family, and adapted matrix helpers. Cinna Davis's #4100 supplies the earlier BIT/header setting.

Prepared for ssafwann with substantial OpenAI Codex (gpt-6-astra) assistance, by one agent. Apache-2.0.

Open next questions

The prescribed-H dimension question is closed by this theorem. Successor PR #4678 now supplies the exact one-block complete-header count for every fixed joint-sector profile, retaining shared-H dependence and both unconditional/identity-pivot-conditioned densities. The next residual-state question for later full guards is recorded there and in Discussion #4653. Physical costs and supplier admission remain open; this PR retains its dimension proof and verification artifact unchanged.