[aliromman]

Prove that a sharp identity-pivot Schur deletion has joint kernel-sector dimensions (r00, r10-d, r01-d, r11-d), where the r values are the original commuting pair's sector dimensions. This supplies the missing input to the complete central-class classifier directly from the original header. It removes a full residual-kernel calculation from that interface. The complete kappa calculation and selected frontier remain unchanged.

The proof, constructive implementation, exact receipt, and source pins define the result.

Let P0,Q0 be commuting idempotents over any field, with a common identity pivot of size d and Schur residuals S,T satisfying rank(ST-TS)=2d. Write the header split as i,pi with pi*i=I. The four joint residual eigenspaces are (I-i*pi)V00 and Vab intersect ker(pi) for the other three original sectors. They form a direct sum equal to the commutator kernel. Thus the predecessor's opposite-sector count is

a1 = r10+r01-2d
   = rank(P0)+rank(Q0)-2*rank(P0*Q0)-2d.

H=pi*P0*Q0*i can be arbitrary. The theorem includes characteristic two, eigenvalue-one components and singular or empty 00 factors. Sharpness is required; an explicit nonsharp header violates the formula. The original profile must be supplied or computed, with its acquisition cost still paid. This does not classify the full ordered pair, give header probabilities or later-guard behavior, or supply a native implementation.

This answers the original-header question in Ali Romman's PR #5287, consumed at d17a8b3d237a8cb8685cc108bf2bb17dfea0ca78. Its central classifier and complete-invariant oracle are unchanged. Nick Petalas's PR #4746 supplies the sharp-sector criterion. @ssafwann supplied the commutator-image action in PR #5256 and the nilpotent extension and factor specialization in PR #5267. The retained notice and retained sharp-factor proof preserve the wider lineage. General projector and Schur-complement facts are classical. Prepared by Ali Romman with substantial OpenAI Codex assistance; Apache-2.0.

Base: current main ff3cc58e370d0f678e529246f8c73e8094e18fcf, priority revision 7. Eight byte-exact predecessor files are retained at ancestor d725afce0f17a468cb76466a59348c611eb87ac2; the replay needs no unmerged upstream object.

Python 3.12 and the standard library:

python3.12 -B scripts/research_package.py --package research/schur-header-kernel-sectors

Both registered isolated, warning-strict normal and optimized runs pass with PASS_SCHUR_HEADER_KERNEL_SECTORS. The check covers 1,244 H matrices in seven complete finite domains, 98 additional cases, a separate empty-00 case and six rejection or necessity controls. It builds full projectors and headers, checks the actual pivot and sharp commutator rank, and compares the constructed bases with independent full residual kernels and complete central similarity invariants. Four actual code mutations fail in both modes before receipt comparison.

Committed bytes, every declared source pin's direct ancestry, official export integrity and both read-only Git-free replays pass; all 18 exported files remain unchanged. The implementation supports F2,F3,F5,F7; the written theorem is over every field. The full-history wrapper remains unpassed in this shallow checkout. No independent human/proof-kernel review, selected cold replay, Lean, Docker, native tape or complete caller/compiler/kappa test was run. The next consumer must state whether it needs the central class or the full ordered-pair extension data and must pay header selection and emission.

Hosted run 39999280092 passed planning and the registered Linux replay for published head 8c3d939cbd06edef8bcb1c90f278c0d76f1ac2dd. Hosted execution is separate from independent scientific review.