[ssafwann]

Change and mathematical scope

Nick Petalas's #4801 leaves the central action after a common identity block pivot open. This package proves the ordered replacement over every field. For commuting idempotents P,Q with leading I_d, put H=(PQ)11, let S,T be their Schur complements, and set K=ST−TS and Z=(S−T)^2. An explicit two-block-column map A satisfies

S A = A [I−H  −I; −H(I−H)  H]
T A = A [I  I; 0  0]
Z A = A diag(2H−H², 2H−H²).

If p annihilates Θ=2H−H², then p(Z)K=0 and Z(Z−I)p(Z)=0. When rank K=2d, A identifies im K with two copies of the Θ-module. This determines the exact restricted minimal polynomial, squared characteristic polynomial and doubled Jordan blocks, including singular/nonsemisimple cases.

A sharp d=3,n=13 ternary nilpotent example has minimal polynomial t³ on im K but t⁴ on the whole residual. It rejects dropping the extra endpoint factor. A binary example rejects removing repeated roots, and an irreducible quadratic example avoids relying only on scalar eigenvalues.

The README includes the complete ordered proof, reproduces the credited factorization needed for it, and defines every fixture. Only the five-file research/schur-block-central-law/ package is added. No new dimension minimum, full residual classification, header frequency, physical supplier, new κ or selected-frontier change.

Task, overlap and attribution

Started from fetched canonical main ff3cc58e370d0f678e529246f8c73e8094e18fcf; priority revision 7, selected conditional κ=0.00302015. Main and relevant guidance/source bytes remained unchanged at the submission fetch. Contribution head: c72d6ff431cf5c979f5010018d8eba0c15577bce.

Claim and family decision: Discussion #4653, with the #1166 pointer. formal-interface-lemma is explicitly changed to this written algebraic side result. The earlier #4689/#4735/#4762/#4915/#5072 census, recurrence, descriptor, orbit weights and incidence results were checked for overlap and credited, rather than repeated. #5128 concerns arbitrary-pivot dimension minima, a distinct obligation.

Final overlap covers 322 open PRs (2026-10-23 14:19 UTC), 4,520 Discussion records (14:21 UTC), and all 7,091 tracked in-scope notes, without coverage warnings. Exact initial/final coverage, limitations and source pins are in TASK.json/SOURCES.json. The search covers open PR bodies, Discussion prose and all in-scope local notes; arbitrary diffs, nested notes, exhaustive closed-PR bodies and unpublished work are excluded. Targeted all-state searches supplement the cache. No search claims literature-wide novelty.

Credit Nick Petalas #4801 (98bc071317b17fd8bb4f21044328914c7561d3cb) for the scalar central law and explicit block question; #4597 for the ordered factorization; #4746 (5f014560d6225cae17598ac479fb29d2947d8e6b) for signed all-field sector fixtures; ssafwann #4671/#4657/#4678 for the binary sector/counting lineage; Ryanson Jonathan #4609/#4617 for central/projector and sharpness context; lee-phys, the residual-state contributors above, Cinna Davis, and source527/commdata2338/Jose Manuel for the retained setting. Base-main source bytes are provenance-pinned, not executable dependencies.

Prepared for ssafwann with substantial OpenAI Codex (gpt-6-astra) assistance, by one agent. Apache-2.0. Predecessor attribution and disclosures remain in their unchanged sources.

Validation

CPython 3.12.11 / macOS arm64, standard library only:

python3.12 -B scripts/research_package.py --package research/schur-block-central-law
python3.12 -B scripts/check_research_commit.py --package research/schur-block-central-law --check-source-history

• Registered isolated, warning-strict normal and optimized replays pass in 6.32 seconds combined. Every check uses an explicit exception; no assert statements.

• All 1,244 H in seven complete prime-field domains pass literal full-matrix premise, Schur, action, factorization, polynomial and sharp restricted-minimal-polynomial checks.

• All 1,472 ordered commuting identity-pivot binary pairs in the two stated small domains pass the new law beyond the sharp fixture construction. Their existing domain counts are credited corroboration, not new enumerative claims.

• Four fixtures verify shifted-power rank sequences and whole-space minimal polynomials. Nine rejection controls cover repeated factors, the full-space overclaim, Θ versus H, odd-field sign, sharpness, pivot, idempotency, commutation and census capacity.

• All five committed files and declared source-history pins pass at the contribution head above. Official export and integrity checks pass.

• The standalone read-only export passes both normal and optimized isolated replay with no Git checkout and PATH=/nonexistent; scientific stdout agrees exactly and all six files, including the receipt, remain unchanged.

• The affected-package planner selects only research/schur-block-central-law; the worktree is clean. No shared implementation or selected source is changed.

Export SHA256: 1c275bb71165657ffd5280d37f1cdedc1dd95d12264c1c8d8c0ce410feec4d63. The Registered research packages run at this exact head is in_progress at handoff; no hosted pass is claimed. Hosted CI is separate from these local results.

The all-field/all-d result rests on the written proof. Independent human/proof-kernel acceptance, Lean, Docker and the selected scientific cold replay are unclaimed. Selected source-binding orientation stopped at a sparse-omitted checkpoint; no baseline pass is claimed. This isolated package does not modify that source closure.

Remaining question

For one specified singular H, classify how the complementary residual directions attach to im K and determine their split/header multiplicities. The ternary nilpotent example is a discriminating starting point. The present restricted representation is not a sufficient whole-state invariant. Header cover selection/emission, physical reads, dirty restoration, storage, precision, stock, dispatch and the complete consumer bill remain unknown, not zero. No follow-up claim is held when this result is handed off.