[aliromman]

Recovers the complete similarity class of Z=(S-T)^2 after a sharp identity-block Schur deletion, including characteristic two, non-split components and repeated roots. For commuting idempotents P,Q with common pivot I_d and rank(ST-TS)=2d, the class is determined by H=(PQ)_11, residual dimension m, and one opposite-sector kernel count. This enabling algebraic result determines the missing central state data. It does not change the complete kappa calculation or selected frontier.

Proof, exact receipt, classifier, and source bindings.

On each central primary space at 0 or 1, extend every Jordan block of K=ST-TS restricted to im K by one, then square the chain. The remaining length-one K blocks are determined by ker K. Away from 0 and 1, K is invertible and the whole space already lies in im K. The one extra count is

a1 = dim(ker S intersect ker(T-I))
     + dim(ker(S-I) intersect ker T).

It is necessary in general: two admitted pairs can have the same H and residual dimension but different a1 and different complete characteristic polynomials. The result classifies Z, not the full ordered pair (S,T) or its header weights. The sharp rank assumption is required.

This answers the explicit characteristic-two handoff in @ssafwann's PR #5267 and extends the restricted block action in PR #5256. For nilpotent H in characteristic two, each block k gives two central blocks of length ceil((k+1)/2) and two of length floor((k+1)/2), plus the specified scalar summands. The padded binary H=J3 fixture has blocks (2,2,2,2,1,1); the ternary fixture retains the predecessor's (4,3,1,1,1). The predecessor's nilpotent ordered-pair rigidity, odd-field partition and fixed-H weights remain credited results.

The retained block proof, nilpotent proof and handoff, and notice preserve the exact attribution to ssafwann, Nick Petalas, Ryanson Jonathan and the wider Schur/normalizer lineage. Generic primary decomposition, nilpotent-image chains and Faddeev-LeVerrier are classical. No global novelty is claimed. Prepared by Ali Romman with substantial OpenAI Codex assistance; Apache-2.0.

Base: canonical main ff3cc58, priority revision 7. The exact source helper and proof/credit files are retained from upstream heads c72d6ff and 8ab280c. Their identical bytes are pinned to ancestor 60bd962 in this publication history. No unmerged upstream Git object is required by the registered replay.

With CPython 3.12 and the standard library:

python3.12 -B scripts/research_package.py --package research/schur-whole-central-class

Each mode ends with:

PASS_SCHUR_WHOLE_CENTRAL

The package runner then prints:

PASS_REGISTERED_RESEARCH_PACKAGE research/schur-whole-central-class outcome=completed

Both registered modes pass all 1,244 H in seven complete small prime-field domains, 26 additional mixed/padded/Jordan examples and nine control groups. The full-matrix oracle checks original premises, sharp rank, integer-lift characteristic polynomials and every primary power-rank partition through a separate column-elimination routine. A non-split repeated central factor is explicitly checked. Four code mutations fail in both modes before receipt checking: omitted chain extension, ignored opposite-sector count, wrong commutator and wrong non-endpoint action.

Source pins, all declared source ancestries, committed bytes and official export integrity pass. Both read-only Git-free replays pass with all 13 exported files unchanged. Main and scoped overlap were refreshed before submission. Hosted CI passed both planning and the focused replay at this exact head. The code implements F2, F3, F5, F7; the written theorem is over every field. The full-history wrapper was not run in the shallow checkout. No independent human/proof-kernel acceptance, selected cold replay, Lean, Docker, nonsharp classification, general ordered-pair classification, header probability or physical supplier is claimed.