Recover the whole sharp Schur central class and characteristic-two partition

research/schur-whole-central-class/NOTICE.md

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+# Attribution and source availability
+
+New central-class proof, classifier and independent invariant checks: Ali
+Romman, with substantial OpenAI Codex assistance. Apache-2.0.
+
+The frozen files in vendor are byte-exact copies of ssafwann's
+[PR #5256](https://github.com/CrocSwap/integer-mult-bounds/pull/5256), head
+c72d6ff431cf5c979f5010018d8eba0c15577bce, and
+[PR #5267](https://github.com/CrocSwap/integer-mult-bounds/pull/5267), head
+8ab280c8a3ef0544f13ff23a1c69df4757cd4900. They are prepared for ssafwann
+with substantial OpenAI Codex assistance, under Apache-2.0. The first source
+supplies the exact fixture constructor, matrix helpers and ordered block action;
+the second supplies the nilpotent representative, existing odd-field result
+and open characteristic-two question. Both original proof/credit documents
+are retained because the new proof uses their contracts.
+
+SOURCES.json records original commits, paths, blobs and SHA256, and binds the
+same bytes to an ancestor in this publication history. A clean registered
+replay does not require an unmerged upstream object. No predecessor is edited.
+
+Retain Nick Petalas's #4597/#4746/#4801/#4915, MatShep's #4762, codyw912's
+#4735, ssafwann's #4657/#4671/#4678, Ryanson Jonathan's #4609/#4617, lee-phys,
+Mihir Srivastava, JoyboyBrian, Krzysztof Wojt, Cinna Davis and the source527 /
+commdata2338 / Jose Manuel lineage exactly as documented in the retained
+sources. These results and the generic factor-extension/counting methods are
+not our new work. Classical primary decomposition, nilpotent Jordan chains
+and Faddeev-LeVerrier retain their standard attribution.

research/schur-whole-central-class/PROOF.md

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+# Reconstruct the whole central operator from the sharp image
+
+## Contract and statement
+
+Work over any field F. Let P,Q be commuting idempotents with a common leading
+identity block I_d. Let S,T be their Schur complements on a residual space V
+of dimension m. Put H=(PQ)_11, K=ST-TS, Z=(S-T)^2, and require rank K=2d.
+This is the sharp-image hypothesis. It is not optional.
+
+The complete similarity class of Z is determined by H, m and the one integer
+
+    a_1 = dim(ker S intersect ker(T-I))
+          + dim(ker(S-I) intersect ker T).
+
+The result concerns the central operator Z. It does not classify the ordered
+pair (S,T), count headers or give a probability law. It covers nonnilpotent H,
+non-split primary components, repeated roots and characteristic two.
+
+The credited block action in ssafwann's [PR #5256](https://github.com/CrocSwap/integer-mult-bounds/pull/5256)
+identifies M=im K with F^(2d). Set J=H(I-H) and Theta=2H-H^2. In this basis,
+
+    D = Z|M = diag(Theta,Theta),
+    B = K|M = [ J   2(I-H); -J  -J ].                 (1)
+
+The second equation follows by taking the commutator of the source's ordered
+matrices U_H and V_H. Thus B and D commute and B^2=D(D-I). The unchanged
+source proof and fixture construction are retained in vendor, with exact
+original and retained-history pins. Their proof and attribution remain intact.
+
+For lambda=0,1, let M_lambda=ker(D-lambda I)^(2d). Compute the nilpotent Jordan
+block sizes j_(lambda,1),...,j_(lambda,c_lambda) of B on M_lambda. Set
+
+    a_0 = m-2d-a_1.
+
+Then the lambda-primary Jordan blocks of Z are:
+
+- for each j_(lambda,i), one block of length ceil((j_(lambda,i)+1)/2)
+  and one of length floor((j_(lambda,i)+1)/2);
+- a_lambda-c_lambda additional blocks of length one.
+
+The remaining primary part is exactly D restricted to
+im((D(D-I))^(2d)). This is an invariant complement to M_0+M_1 and gives a
+representative over F itself. No splitting field is needed for this part.
+For an admitted source pair, both extra multiplicities are nonnegative.
+
+## Proof
+
+For any two idempotents S,T, direct expansion gives
+
+    K^2=Z(Z-I),    ZS=SZ,    ZT=TZ,
+    KS=(I-S)K,    KT=(I-T)K.                           (2)
+
+These are credited projector identities from the retained Schur lineage.
+In particular ker K is invariant under both S and T. Their restrictions
+commute on ker K. The four joint eigensectors therefore split ker K over
+every field, including characteristic two. On the equal sectors 00 and11,
+Z is zero. On the opposite sectors 01 and10, Z is identity. Rank-nullity gives
+
+    dim ker K = m-2d = a_0+a_1.                       (3)
+
+Decompose V into its Z-primary spaces at0, at1, and away from these endpoints.
+On the last part, Z(Z-I) is invertible. Equation (2) makes K invertible there,
+so that entire part is contained in im K. Its Z action is therefore exactly
+the non-endpoint restriction of D in (1).
+
+On V_lambda, lambda=0 or1, K is nilpotent: its square is the product of a
+nilpotent operator and a commuting invertible operator. Moreover
+M intersect V_lambda is im(K|V_lambda). On ker K within V_lambda, equation
+(2) forces Z=lambda I, so the kernel dimension of K|V_lambda is a_lambda.
+
+A nilpotent Jordan block of K of length ell>=2 induces on its image a block
+of length ell-1. A block of length one has zero image. Thus every block j of
+B|M_lambda comes from exactly one block j+1 of K|V_lambda. The remaining
+blocks of K have length one, and their number is a_lambda-c_lambda.
+
+The factor of Z(Z-I) other than Z-lambda I is invertible on V_lambda and
+commutes with K. Therefore, for every positive integer r,
+
+    rank((Z-lambda I)^r | V_lambda)
+       = rank(K^(2r) | V_lambda).                    (4)
+
+These ranks determine the nilpotent Jordan partition. Squaring a nilpotent
+block of length ell splits its chain into the odd and even positions, with
+lengths ceil(ell/2) and floor(ell/2); omit a zero-length chain. Apply this to
+the reconstructed K blocks. The formula in the statement follows.
+
+This is a complete similarity classification of Z under the stated contract.
+It does not require a choice of how complementary vectors attach to M.
+The integer a_1 is also recoverable from the full class, since
+
+    dim V_1 = dim M_1 + a_1.
+
+It cannot in general be omitted. Append one scalar00 sector or one scalar01
+sector to the same admitted pair. Both operations preserve H, sharpness and
+the new residual dimension. The first appends a zero eigenvalue to Z; the
+second appends an eigenvalue one. Their characteristic polynomials differ.
+The verifier retains this same-H, same-dimension control.
+
+## The open characteristic-two nilpotent case
+
+Suppose char F=2 and H is nilpotent with block sizes k_1,...,k_a, where
+sum k_i=d. Then Theta=H^2, and (1) becomes
+
+    B = diag(H(I-H),H(I-H)) [I 0; I I].
+
+The second factor is invertible and commutes with H. Since I-H is invertible,
+rank B^r = 2 rank H^r for every r. Thus B has two copies of every nilpotent H
+block. The complete central partition is consequently
+
+    sum_i [2 J_ceil((k_i+1)/2)(0) + 2 J_floor((k_i+1)/2)(0)]
+    + (m-2d-a_1-2a) J_1(0) + a_1 J_1(1).            (5)
+
+This answers the explicitly open characteristic-two question in ssafwann's
+[PR #5267](https://github.com/CrocSwap/integer-mult-bounds/pull/5267).
+For its nilpotent profile
+
+    (r00,r10,r01,r11)=(d+s00,d+s10,d+s01,d+a+s11),
+
+that predecessor's all-field representative gives a_1=s10+s01 and
+m=3d+a+s00+s10+s01+s11. The extra zero-block count in (5) is therefore
+`d-a+s00+s11`, and the extra one-block count is `s10+s01`.
+The predecessor already proves the whole ordered-pair rigidity, its odd-field
+k/(k+1) law and the fixed-H weights. None is claimed again here.
+
+For the padded H=J3 fixture, (5) gives `(2,2,2,2,1,1)`. Its whole central
+minimal polynomial is t^2. In odd characteristic the credited partition is
+`(4,3,1,1,1)`, with minimal polynomial t^4. Division by two would give the
+wrong characteristic-two answer.
+
+## Computation, evidence and trust
+
+[central.py](central.py) computes the two primary subspaces in dimension2d,
+the nilpotent B rank sequences, and an explicit non-endpoint restriction.
+Its current scalar implementation supports F2,F3,F5,F7. The written theorem
+applies to every field. The function requires the stated source contract;
+H,m,a_1 alone are not a proof that a proposed original pair is admissible.
+
+The verifier constructs full source pairs with the frozen credited helper,
+then independently forms their Schur complements and checks the original
+premises and sharp rank. It computes full characteristic polynomials through
+integer-lift Faddeev-LeVerrier, factors them over the small prime field, and
+checks all primary power-rank partitions using a separate column-elimination
+routine. These full-matrix invariants are compared with the proposed complete
+representative. Non-split repeated factors and mixed primary examples prevent
+scalar-eigenvalue sampling from serving as the acceptance test.
+
+The generic nilpotent-image and squared-Jordan arguments are classical linear
+algebra, not claims of global novelty. The new result is their complete
+application to the sharp block-Schur central contract and the explicit
+characteristic-two partition. Written proof, exact scalar arithmetic, Python
+and source transcription remain in the trust base. Finite corroboration is
+not independent human or proof-kernel acceptance. Nonsharp pairs, full ordered-
+pair classification for general H, header weights, native realization and
+all physical consumer costs remain outside scope. No kappa or frontier change.
+
+Prepared by Ali Romman with substantial OpenAI Codex assistance; Apache-2.0.

research/schur-whole-central-class/README.md

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+# The whole central class after sharp Schur deletion
+
+For commuting idempotents P,Q with common identity pivot I_d, let S,T be the
+Schur residuals, K=ST-TS, Z=(S-T)^2 and H=(PQ)_11. If rank K=2d, the complete
+similarity class of Z is determined by H, residual dimension m, and one count:
+
+    a1 = dim(ker S intersect ker(T-I))
+         + dim(ker(S-I) intersect ker T).
+
+[PROOF.md](PROOF.md) gives the all-field reconstruction. On the central
+primary spaces at0 and1, first extend each Jordan block of K restricted to
+im K by one, then square that chain. The remaining scalar blocks come from
+ker K. Away from0 and1, the entire space is already in im K. This recovers
+non-split and repeated primary factors as well as scalar eigenvalues.
+
+This answers the characteristic-two handoff from
+[PR #5267](https://github.com/CrocSwap/integer-mult-bounds/pull/5267) and extends
+the restricted central action in
+[PR #5256](https://github.com/CrocSwap/integer-mult-bounds/pull/5256).
+For nilpotent H in characteristic two, each block k gives two 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 case has blocks `(2,2,2,2,1,1)`; its ternary
+counterpart retains the predecessor's `(4,3,1,1,1)`.
+
+The result classifies the central operator, not the whole ordered residual
+pair or its header weights. The sharp rank assumption is required. The complete
+kappa calculation and selected frontier are unchanged.
+
+## Reproduce and evidence
+
+Requires CPython 3.12 and the standard library:
+
+```sh
+python3.12 -B scripts/research_package.py --package research/schur-whole-central-class
+```
+
+Expected: `PASS_SCHUR_WHOLE_CENTRAL` in normal and optimized modes, followed
+by `PASS_REGISTERED_RESEARCH_PACKAGE`. [RESULT.json](RESULT.json) records the
+exact result digest, examples and file hashes. All1,244 H in seven complete
+small prime-field domains are checked. Additional examples cover Jordan
+blocks through size5 over F2,F3,F5,F7, scalar padding, mixed primary spaces
+and a non-split repeated factor. Nine control groups include two admitted
+pairs with the same H and residual dimension but different central classes.
+
+The full-matrix oracle uses integer-lift characteristic polynomials and
+primary power-rank invariants with a separately transcribed column-elimination
+routine. It does not accept scalar eigenvalue samples or the image restriction
+as a substitute for the complete class. The classifier currently implements
+F2,F3,F5,F7; the written theorem is over any field. Finite corroboration does
+not establish independent human or proof-kernel acceptance.
+
+The exact source helpers and proof/attribution documents are retained from
+pending predecessor heads in our own publication history. [SOURCES.json](SOURCES.json)
+records original and retained pins. Every declared source commit is an
+ancestor of this publication; no hidden local Git object is needed. Base:
+ff3cc58e370d0f678e529246f8c73e8094e18fcf, priority revision7. The full-history
+wrapper, selected cold replay, Lean and Docker were not run.
+
+## Open next questions
+
+Can a1 be recovered from cheaper original header data for general H? Compare
+it with the four original joint-sector dimensions when H has a nontrivial
+eigenvalue-one primary component. This could avoid large residual kernel
+calculations. No general original-header formula or physical consumer is
+proved here. General nonsharp central reconstruction and full ordered-pair
+classification remain outside scope.
+
+## Credit
+
+Prepared by Ali Romman with substantial OpenAI Codex assistance; Apache-2.0.
+ssafwann supplies the restricted action, nilpotent model and open question;
+Nick Petalas, Ryanson Jonathan and the wider Schur lineage retain the results
+listed in [NOTICE.md](NOTICE.md) and the unchanged source documents. The
+nilpotent ordered-pair rigidity, odd-field partition and fixed-H weights are
+predecessor results. Classical Jordan-image, primary-decomposition and
+Faddeev-LeVerrier methods are not claimed as new.