Recover sharp Schur kernel sectors directly from original header data

research/schur-header-kernel-sectors/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-header-kernel-sectors/PROOF.md

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+# Original-header formulas for the sharp Schur kernel sectors
+
+Prepared by Ali Romman with substantial OpenAI Codex assistance. Apache-2.0.
+This is a written all-field result with exact finite corroboration.
+
+## Contract and statement
+
+Let P0,Q0 be commuting idempotents on V over any field F. Their ordered joint
+sectors V00,V10,V01,V11 have dimensions r00,r10,r01,r11. Let U=F^d and take a
+split i:U->V, pi:V->U with pi i=I and pi P0 i=pi Q0 i=I. Thus a header
+[i,B], with B a basis of ker pi, has the common identity pivot I_d. Let S,T
+be the Schur complements, acting on W=ker pi, and put
+
+    H=pi P0 Q0 i,    K=ST-TS,    Z=(S-T)^2.
+
+Assume rank K=2d. This is the same sharpness hypothesis as the predecessors.
+For a,b in {0,1}, let Eab=ker(S-aI) intersect ker(T-bI).
+
+**Theorem.** In the original space V, the four joint residual sectors are
+
+    E00 = (I-i pi)V00,
+    E10 = V10 intersect ker pi,
+    E01 = V01 intersect ker pi,
+    E11 = V11 intersect ker pi.                              (1)
+
+The first displayed map is injective on V00. Consequently
+
+    (dim E00,dim E10,dim E01,dim E11)
+       = (r00,r10-d,r01-d,r11-d).                            (2)
+
+These four spaces give a direct sum decomposition of ker K. No division by
+two is used. H need not be nilpotent, and I-H need not be invertible. Empty
+or rank-deficient 00 factors are allowed.
+
+In particular, the opposite-sector count required by
+[PR #5287](https://github.com/CrocSwap/integer-mult-bounds/pull/5287) is
+
+    a1 = r10+r01-2d
+       = rank P0 + rank Q0 - 2 rank(P0Q0) - 2d.              (3)
+
+Thus that predecessor's complete central-class construction takes H and the
+original joint-sector profile directly. It no longer needs a full residual
+kernel calculation. This statement does not classify the full ordered pair
+(S,T) or determine later-guard behavior.
+
+## 1. Credited sharp-sector criterion
+
+Write E_j for each original joint projector, Y_j=E_j i and X_j=pi|V_j.
+The identity-pivot equations imply
+
+    X00 Y00=H-I,    X10 Y10=X01 Y01=I-H,    X11 Y11=H.        (4)
+
+The sharp-sector criterion of Nick Petalas's
+[PR #4746](https://github.com/CrocSwap/integer-mult-bounds/pull/4746) states
+that rank K=2d is equivalent to each of Y10,Y01,Y11 being injective and each
+of X10,X01,X11 being onto, under these same hypotheses. The specialization
+is retained in ssafwann's
+[PR #5267](https://github.com/CrocSwap/integer-mult-bounds/pull/5267), in
+`vendor/NILPOTENT-SOURCE.md`. The criterion itself does not require nilpotent
+H. This is a credited theorem used here, not a new sharpness criterion.
+The finite verifier separately checks the literal commutator rank.
+
+There is no full-rank requirement on X00 or Y00. In particular, when H has
+an eigenvalue-one part, that sector can differ from the full-rank nilpotent
+models. The proof below keeps it unrestricted.
+
+## 2. Residual maps in the original space
+
+For v in W, the Schur residual maps have the basis-free formulas
+
+    S v = P0 v - P0 i pi P0 v,
+    T v = Q0 v - Q0 i pi Q0 v.                             (5)
+
+For example, in header coordinates P=[I b;c D], the first expression has
+lower coordinates (D-cb)v and zero pivot coordinates. Hence (5) is the
+actual Schur action, not the leading compression or just its central image.
+
+Since T v lies in im Q0, T v=v implies v in im Q0. Conversely if v is in
+im Q0 intersect W, then pi Q0 v=pi v=0, and (5) gives T v=v. The same
+argument applies to S and P0. This proves
+
+    E11 = im P0 intersect im Q0 intersect W = V11 intersect W.
+
+If T v=v and S v=0, put a=pi P0 v. The first equality places v in im Q0;
+the second says P0 v=P0 i a. Its V10 coordinate is 0=Y10 a. Injectivity of
+Y10 gives a=0, so P0 v=0 and v lies in V01. The converse follows at once
+from (5). Thus E01=V01 intersect W. Exchange P0 and Q0 to obtain E10.
+
+It remains to determine E00. If S v=T v=0, put a=pi P0 v and b=pi Q0 v.
+Equation (5) gives P0 v=P0 i a and Q0 v=Q0 i b. Their common V11 component
+gives Y11 a=Y11 b, so a=b. The vector w=v-i a then lies in V00. Since
+pi v=0, pi w=-a, and therefore v=w-i pi w. This proves one inclusion in
+E00=(I-i pi)V00. For w in V00, substitute v=w-i pi w in (5); the two
+terms cancel for both maps. This proves the other inclusion.
+
+If (I-i pi)w=0 with w in V00, then w=i pi w. Its V10 component is
+Y10 pi w=0. Injectivity gives pi w=0, hence w=0. The map on V00 is injective.
+The 00 sector is therefore a projection of V00, not generally V00 intersect W.
+
+## 3. Dimensions, completeness and central class
+
+Surjectivity of X10,X01,X11 gives the last three dimensions in (2).
+Injectivity just proved gives the first. Distinct joint eigenvalue pairs
+give independent subspaces over every field, including characteristic two.
+They lie in ker K, and their dimensions sum to
+
+    n-3d = (n-d)-2d = dim ker K.
+
+They therefore exhaust ker K. This proof also makes each basis constructive:
+project an original V00 basis by I-i pi, and take kernels of the three
+small observation maps X_j for j!=00. `kernel_bases` implements these maps
+in original V coordinates; it does not build S or T to find those kernels.
+
+For commuting idempotents, rank P0=r10+r11, rank Q0=r01+r11 and
+rank(P0Q0)=r11, which proves (3). This is a rank formula, not a trace formula
+that could lose dimension information in positive characteristic.
+
+The unchanged theorem of PR #5287 reconstructs the complete similarity class
+of Z from H, m=n-d and a1 under sharpness. It uses the credited commutator
+image action of ssafwann's
+[PR #5256](https://github.com/CrocSwap/integer-mult-bounds/pull/5256), extends
+its endpoint nilpotent chains, and retains the non-endpoint primary operator.
+Substitute (3) into that theorem. `central_from_header` makes precisely this
+substitution into the unchanged classifier. It handles repeated and nonsplit
+primary factors through that predecessor, rather than through eigenvalue
+samples. The original header profile is a required input or a separately
+computed datum; its acquisition is not assigned zero physical cost.
+
+## 4. Necessary boundaries
+
+Sharpness cannot be removed. Take d=1 and one original coordinate in each
+sector. Choose i and pi entirely in V11. Both pivot blocks are one and H=1,
+but K=0. The residual has one 00, one 10 and one 01 coordinate. Formula (2)
+would incorrectly predict (1,0,0,0). The factor-rank gate rejects this input.
+
+H and residual dimension alone are insufficient. For H=0 over F2, padding
+two original 00 directions or padding one 10 and one 01 direction gives
+the same residual dimension, but changes a1 from zero to two. The complete
+central classes differ. Original profile data supply the missing information.
+
+The formulas concern joint eigensectors on ker K and the complete central
+operator. They do not assert that the four spaces split as invariant direct
+summands of all of W, that Z determines (S,T), or that central classes give
+unconditional header probabilities. No native realization, selection/emission
+cost, later-guard transition, supplier admission or complete kappa change is
+proved. The selected frontier remains unchanged.
+
+## Evidence and credit
+
+The verifier constructs original sector factors, a complete header and both
+full commuting projectors. It checks the original identity pivot, compressed
+H and sharp Schur rank. It then compares each explicit original-space basis
+with separately computed full residual joint kernels and the complete
+central similarity invariants. Cases include empty V00, unequal X00/Y00
+ranks, changes of original sector bases, characteristic two, eigenvalue-one
+chains and repeated nonsplit factors. Finite exact checks support the written
+proof; they are not independent human or proof-kernel acceptance.
+
+Prepared by Ali Romman with substantial OpenAI Codex assistance. Exact source
+and retained-history pins are in SOURCES.json. Credit Nick Petalas for #4746
+and the ordered factor lineage; ssafwann for #5256/#5267 and the source matrix
+helpers; Ali Romman's #5287 for the central classifier; and codyw912, MatShep,
+Ryanson Jonathan and the contributors retained in NOTICE.md. The central
+classifier and independent invariant oracle are unchanged. General projector,
+kernel and Schur-complement facts are classical; the new result is the explicit
+original-header sector formula and its complete central-class binding.

research/schur-header-kernel-sectors/README.md

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+# Sharp Schur kernel sectors from the original header
+
+For a common identity-pivot deletion of size d with maximal residual
+commutator rank 2d, the four joint kernel-sector dimensions are
+
+```
+(r00, r10-d, r01-d, r11-d).
+```
+
+Here the r values are the **original** commuting pair's joint-sector
+dimensions. The [proof](PROOF.md) gives explicit original-space bases:
+project V00 by I-i*pi, and intersect each other original sector with ker pi.
+It works over every field and for arbitrary compressed product H, including
+an eigenvalue-one component and singular or empty 00 factors.
+
+The opposite-sector count is r10+r01-2d. This answers the original-header
+question of [PR #5287](https://github.com/CrocSwap/integer-mult-bounds/pull/5287)
+and supplies its complete central classifier without a full residual kernel
+calculation. `header.py` provides both the constructive kernel bases and
+`central_from_header(h, sizes, q)`. The retained scalar classifier supports
+F2,F3,F5,F7; the written sector theorem is over every field.
+
+Python 3.12 and the standard library are sufficient:
+
+```sh
+python3.12 -B scripts/research_package.py --package research/schur-header-kernel-sectors
+```
+
+Both registered normal and optimized commands end with
+`PASS_SCHUR_HEADER_KERNEL_SECTORS`. `verify.py --write-result` regenerates
+RESULT.json. Normal verification writes no package files. The checks build
+full projectors and headers, verify sharpness independently, and compare all
+four explicit bases with full residual kernels and complete central invariants.
+
+The sharp rank assumption is required. This is a central-state interface,
+not a full ordered-pair classification, header probability, later-guard law
+or physical supplier. The original profile must be supplied or computed;
+its cost is not free. The complete kappa and selected frontier are unchanged.
+
+Prepared by Ali Romman with substantial OpenAI Codex assistance; Apache-2.0.
+Nick Petalas supplies the sharp-sector criterion, ssafwann the block and
+nilpotent Schur extensions, and #5287 the unchanged central classifier.
+`NOTICE.md` and the retained source documents preserve the full lineage.
+`SOURCES.json` pins all copied bytes to an ancestor of this contribution.
+
+## Open next questions
+
+Pin a named consumer and state whether it needs the central class or the full
+ordered-pair extension data. Then specify the header law and complete paid
+selection/emission boundary. The present formula alone does not determine
+unconditional multiplicities, later-guard survival or native physical cost.