Prove exact Schur minima for arbitrary invertible block pivots (no new κ)

research/arbitrary-pivot-schur-minimum/PROOF.md

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+# Prescribed arbitrary pivots: the exact Schur dimension
+
+Krzysztof Wojt, with substantial OpenAI Codex assistance. Apache-2.0.
+
+This is a scoped all-field algebraic proof. The ordered Schur factorization
+is Nick Petalas's #4597. The joint-sector and split-factor methods come from
+ssafwann #4657/#4671 and Nick Petalas #4746/#4771/#4774/#4785. The new step is
+the six-map criterion for arbitrary invertible, possibly noncommuting block
+pivots, and its exact prescribed-boundary minimum. No historical priority
+is claimed for the underlying linear algebra.
+
+## 1. Statement
+
+Let K be any field, d >= 1, and let A,B in GL_d(K) and H in M_d(K) be
+prescribed. There is no commutation assumption on A,B,H. Among all commuting
+idempotents P,Q in M_n(K) satisfying
+
+```
+P11=A, Q11=B, (PQ)11=H,
+S=P22-P21 A^-1 P12, Z=Q22-Q21 B^-1 Q12,
+rank(SZ-ZS)=2d,
+```
+
+the exact least dimension is
+
+```
+n_min = 3d + rank(I-A-B+H)
+           + nullity(A-H) + nullity(B-H) + nullity(H).       (1)
+```
+
+For the simultaneous eigenspaces indexed by 00,10,01,11, every such pair
+satisfies the componentwise bounds
+
+```
+r00 >= rank(I-A-B+H),
+r10 >= d+nullity(A-H),
+r01 >= d+nullity(B-H),
+r11 >= d+nullity(H).                                      (2)
+```
+
+All four equalities are attained together. The same construction attains
+both least input ranks
+
+```
+rank P = 2d+nullity(A-H)+nullity(H),
+rank Q = 2d+nullity(B-H)+nullity(H).                        (3)
+```
+
+Consequently it also minimizes both residual ranks, obtained by subtracting d.
+One may add any number of unused 00-sector dimensions, so every n >= n_min
+is feasible. Formula (1), not the finite verification, is the universal claim.
+
+## 2. Ordered ambient factors
+
+Let U=K^d, V=K^n, i:U -> V be inclusion of the first d coordinates and
+pi:V -> U the corresponding projection. Thus pi i=I and
+pi P i=A, pi Q i=B, pi PQ i=H. Define maps L:U^2 -> V and R:V -> U^2 by
+
+```
+L = [PQ i-Q i B^-1 H,  P i A^-1-Q i B^-1],
+R = [A^-1 pi P-B^-1 pi Q;
+     -pi PQ+H B^-1 pi Q].                                (4)
+```
+
+The order of all factors is part of (4). Direct substitution gives
+pi L=0 and R i=0. The operators
+
+```
+Ptilde=P-P i A^-1 pi P, Qtilde=Q-Q i B^-1 pi Q
+```
+
+have block forms diag(0,S), diag(0,Z). Multiplying (4) and cancelling the
+two identical terms Q i B^-1 H B^-1 pi Q gives
+
+```
+L R = Ptilde Qtilde-Qtilde Ptilde = diag(0,SZ-ZS).          (5)
+```
+
+This is the ambient form of #4597's ordered identity. It neither moves H
+past B^-1 nor moves any pivot past another block. In particular
+rank(SZ-ZS)<=2d. Equality holds exactly when L is injective and R is
+surjective: a surjection onto U^2 followed by an injection from U^2 has
+rank 2d, and rank 2d of the product forces both conditions.
+
+## 3. The six-map criterion without scalar pivots
+
+Commuting idempotents split V as the direct sum of their four simultaneous
+eigenspaces, in every characteristic. Write
+
+```
+E00=(I-P)(I-Q), E10=P(I-Q), E01=(I-P)Q, E11=PQ,
+Yj=Ej i:U -> Vj, Xj=pi|Vj:Vj -> U.
+```
+
+Their compressed products are
+
+```
+X00 Y00=I-A-B+H, X10 Y10=A-H,
+X01 Y01=B-H,     X11 Y11=H.                              (6)
+```
+
+In these coordinates the left factor is
+
+```
+L(x,y) = (0,
+          Y10 A^-1 y,
+         -Y01 B^-1(Hx+y),
+          Y11((I-B^-1 H)x+(A^-1-B^-1)y)).                (7)
+```
+
+**Criterion.** rank(SZ-ZS)=2d if and only if each of Y10,Y01,Y11 is
+injective and each of X10,X01,X11 is surjective.
+
+First prove that L is injective exactly when all three active Y maps are
+injective. For necessity:
+
+* If 0!=z is in ker Y10, then (A-H)z=0 by (6). The nonzero pair
+  (x,y)=(-z,Az) is killed by (7): its first active component is Y10 z,
+  its second uses -Hz+Az=0, and its last argument is
+  -z+B^-1 Hz+z-B^-1 Az=0.
+* If 0!=z is in ker Y01, then Hz=Bz. The pair (z,0) is killed by (7),
+  since B^-1 Hz=z and (I-B^-1 H)z=0.
+* If 0!=z is in ker Y11, then Hz=0. Again (z,0) is killed by (7).
+
+Conversely, if all three Y maps are injective and L(x,y)=0, its 10 component
+first gives y=0 because A is invertible. Its 01 component then gives Hx=0
+because B is invertible. Its 11 component finally gives x=0.
+
+Necessity of surjectivity of the X maps follows by applying this argument to
+P^t,Q^t. Their leading blocks are A^t,B^t, their product block is H^t
+(because PQ=QP), and their Schur commutator is -(SZ-ZS)^t. For each sector,
+rank(Ej^t i)=rank(pi Ej)=rank Xj. Hence equality of commutator rank forces
+all three X maps onto. This argument uses ranks, not an assumption that
+the given coordinate splitting is orthogonal to the joint eigenspaces.
+
+For sufficiency, the preceding argument gives injectivity of L. If a pair
+of row covectors (u,v) annihilates R, its restrictions to sectors 10,01,11
+are respectively
+
+```
+u A^-1 X10,
+(-u+vH) B^-1 X01,
+(u(A^-1-B^-1)-v+vH B^-1) X11.                           (8)
+```
+
+Surjectivity of X10 implies u=0. Surjectivity of X01 then implies vH=0.
+The 11 expression reduces to -v X11, so v=0. Thus R is onto, proving the
+criterion by (5). This also shows explicitly why noncommutation of A,B,H
+does not obstruct the argument.
+
+## 4. Lower bound and simultaneous attainment
+
+If X:K^r -> K^d is onto and Y:K^d -> K^r is injective, the rank inequality
+rank(XY)>=rank X+rank Y-r gives r>=d+nullity(XY). Apply it in each active
+sector of (6). The 00 sector needs only r00>=rank(X00Y00). These are (2),
+whose sum is (1). Adding the active sector dimensions of P and Q gives (3).
+
+Here is an explicit construction attaining them for every prescribed A,B,H.
+For any T in M_d(K), choose a matrix N_T whose columns complement im T
+in K^d and put
+
+```
+X_T=[T N_T], Y_T=[I;0].
+```
+
+The middle space has dimension d+nullity(T); X_T is onto, Y_T is injective,
+and X_T Y_T=T. Use these factors for T=A-H, B-H, H in sectors 10,01,11.
+For T00=I-A-B+H choose an ordinary rank factorization X00Y00=T00 through
+dimension rank T00. A zero-dimensional 00 sector is allowed.
+
+Let i be the vertical concatenation of the four Y maps and pi the horizontal
+concatenation of the four X maps. By (6), pi i=I. On the sector direct sum,
+define
+
+```
+P0=diag(0,I,0,I), Q0=diag(0,0,I,I).
+```
+
+They commute and are idempotent. Compression gives pi P0 i=A,
+pi Q0 i=B, pi P0 Q0 i=H. Choose a basis N of ker pi. Since pi i=I,
+M=[i N] is invertible. Set
+
+```
+P=M^-1 P0 M, Q=M^-1 Q0 M.
+```
+
+In the resulting coordinates, i is the first-block inclusion and pi its
+projection. Thus P,Q have exactly the prescribed three leading blocks.
+All six active maps have full rank, so their Schur commutator has rank 2d.
+Their sector dimensions and input ranks attain (2) and (3) simultaneously.
+This completes the proof; no diagonalization of H or compatibility condition
+between A,B,H is needed.
+
+## 5. Specializations and a noncommuting sharp example
+
+If A=B=lambda I, (1) is precisely the common scalar-block formula of #4785.
+If d=1 and A=p, B=q are distinct nonzero scalars, the three nullities sum
+to [h in {0,p,q}], recovering #4774. A=B=I specializes to #4746/#4671.
+These are consistency comparisons, not new results or predecessor audits.
+
+Over Q take
+
+```
+A=[[2,1],[0,2]], B=[[2,0],[1,2]], H=[[3,1],[1,3]].
+```
+
+All three pairwise commutators of A,B,H are nonzero. Yet H=A+B-I, and
+H, A-H=I-B, B-H=I-A are invertible. Formula (1) therefore gives n_min=6,
+with sectors (0,2,2,2), rank P=rank Q=4 and Schur-commutator rank 4.
+The same matrices reduced modulo three still give these ranks. These
+examples lie outside the scalar-block predecessors. The verifier constructs
+their complete matrices and checks their literal Schur products exactly.
+
+## 6. Boundary of the conclusion
+
+The theorem assumes both pivots invertible and P,Q commuting idempotents.
+It concerns one common deletion over a field. It does not classify arbitrary
+nonidempotent commuting inputs, singular-pivot generalized complements,
+coefficient rings, nonmaximal commutator ranks or complete-header frequencies.
+The attaining algebraic representation does not construct or price a physical
+header producer/consumer, arbitrary dirty-owner restoration, precision, stock,
+fixed tapes, recursive compilation or outer transfer. No new kappa follows.

research/arbitrary-pivot-schur-minimum/README.md

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+# Exact Schur dimension for arbitrary invertible block pivots
+
+Krzysztof Wojt, with substantial OpenAI Codex assistance and a bounded Luna
+subagent for the separate binary checker. New work: Apache-2.0.
+
+For any field K, any d >= 1, arbitrary invertible d-by-d matrices A,B and
+arbitrary H, the least dimension of commuting idempotents P,Q with
+`P11=A`, `Q11=B`, `(PQ)11=H` whose Schur complements have commutator rank
+`2d` is exactly
+
+```
+n_min = 3d + rank(I-A-B+H)
+           + nullity(A-H) + nullity(B-H) + nullity(H).
+```
+
+The matrices A,B,H need not commute. [PROOF.md](PROOF.md) proves the lower
+bound and constructs a pair attaining it over every field. It also gives
+the simultaneous minimum input ranks
+
+```
+rank P = 2d + nullity(A-H) + nullity(H),
+rank Q = 2d + nullity(B-H) + nullity(H).
+```
+
+The new step is extending the six-map criterion from scalar pivots to
+arbitrary invertible pivots, with the matrix order retained throughout.
+This resolves Open next question 1 of both
+[#4774](https://github.com/CrocSwap/integer-mult-bounds/pull/4774) and
+[#4785](https://github.com/CrocSwap/integer-mult-bounds/pull/4785), and goes
+beyond their unequal-scalar-block question. The ordered Schur factorization
+itself is the existing result of
+[#4597](https://github.com/CrocSwap/integer-mult-bounds/pull/4597).
+This package is a new theorem and construction, not a replication of those
+predecessors. [Claim and coordination: #5121](https://github.com/CrocSwap/integer-mult-bounds/discussions/5121).
+
+## Scope and evidence
+
+The proof is universal; the computations are finite corroboration. The
+standard-library exact constructor [verify.py](verify.py) checks:
+
+* All 576 ordered triples `A,B in GL_2(F_2), H in M_2(F_2)`.
+* 648 ternary triples: eight explicitly enumerated noncommuting pivot pairs,
+  each with all 81 choices of H.
+* 162 further structured cases over F_2, F_3, F_5, F_7 and Q, in dimensions
+  d=2,3,4. These include noncommuting pivots and singular compressed products.
+* The idempotence and commutation of the full matrices, all three prescribed
+  blocks, literal Schur complements, the ordered factor identity, sector
+  dimensions, input and residual ranks, and the exact minimum formula.
+* Six controls with one active X or Y map deficient, two singular-pivot
+  rejections, and factor-order mutations. Swapping the H and B-inverse
+  factors in either ordered factor is detected on 1,017 cases.
+
+This is 1,386 constructed cases, plus two explicit noncommuting rational and
+ternary examples and the deficient-map controls. The rational example uses
+
+```
+A=[[2,1],[0,2]], B=[[2,0],[1,2]], H=[[3,1],[1,3]],
+```
+
+and attains n=6, input ranks 4 and Schur-commutator rank 4. All three pairwise
+commutators of A,B,H are nonzero. The same example works modulo three.
+
+[binary_check.py](binary_check.py) independently implements raw matrix
+arithmetic with bitsets. It imports no primary-verifier code and reads the
+576 complete matrix fixtures in [FIXTURES.json](FIXTURES.json). It verifies
+exact coverage, the defining equations and all claimed ranks, including
+288 inputs with noncommuting A,B and 180 with pairwise noncommuting A,B,H.
+Its seven rejection controls cover matrix corruption, a false dimension,
+altered H, singular A, and missing or duplicate coverage. [RESULT.json](RESULT.json)
+records the primary counts, full rational/ternary examples and fixture digest.
+
+No independent human proof acceptance or formal proof-kernel check is claimed.
+The finite domains are not an exhaustive
+search for all full matrices or all dimensions; minimality follows from
+the sector argument in PROOF.md. No physical header, dirty-owner restoration,
+precision, stock, fixed-tape, recursive-compilation or outer-transfer claim
+is made. The selected frontier and kappa are unchanged.
+
+## Reproduce
+
+Python 3.12, standard library only. Each command is read-only; `-O` replay
+also checks that verification does not rely on Python assertions.
+
+```sh
+python3.12 -I -W error -B research/arbitrary-pivot-schur-minimum/verify.py
+python3.12 -I -O -W error -B research/arbitrary-pivot-schur-minimum/verify.py
+python3.12 -I -W error -B research/arbitrary-pivot-schur-minimum/binary_check.py
+python3.12 -I -O -W error -B research/arbitrary-pivot-schur-minimum/binary_check.py
+python3.12 -B scripts/check_research_commit.py --package research/arbitrary-pivot-schur-minimum --check-source-history
+```
+
+The first four commands can also be run by absolute path from a read-only,
+Git-free export with an empty PATH. `VERIFY.json` registers them. Explicit
+`verify.py --write` regenerates only FIXTURES.json and RESULT.json; normal
+verification regenerates in memory and requires byte-for-byte agreement.
+All external source files in SOURCES.json are provenance, not replay inputs.
+
+## Open next questions
+
+1. For the same prescribed A,B,H, which smaller Schur-commutator ranks are
+   feasible at each ambient dimension? The six-map equivalence here treats
+   only the maximal rank 2d. A bounded first step is to classify rank-two
+   residual commutators for d=2 over F_2 with fixed noncommuting A,B,H,
+   before attempting a general theorem. No such classification is claimed.
+2. Can the explicit n=6 ternary witness be realized by an admitted complete
+   header with fully priced producer and consumer? First identify an actual
+   caller and its interface requirements. Algebraic attainment alone gives
+   neither a physical construction nor a multiplication improvement.
+
+## Credit and provenance
+
+Nick Petalas's #4774 and #4785 state the open unequal-block-pivot questions;
+#4771 and #4746 supply the scalar and all-field identity-pivot predecessors.
+ssafwann's #4657/#4671 supply the sector and split-factor method. Nick
+Petalas's #4597/#4278 supply the ordered block and scalar factorizations.
+Ryanson Jonathan's #4609/#4617 and lee-phys's #4634 supply earlier sharpness
+constructions. Nick Petalas's #4259/#4228 retain the laminar-header and
+single-request context. Cinna Davis supplies the BIT request setting;
+source527, commdata2338 and Jose Manuel retain the normalizer lineage credit.
+No historical priority is claimed for the underlying linear algebra.
+
+The checkout starts at canonical main
+`81e9bd69e4a5b00ba9c713a1620e36d576d5522e`. TASK.json preserves the original
+priority-revision-7 card and explicitly changes its scope to this algebraic
+side result. SOURCES.json pins the predecessor documents at that base.
+The predecessor scientific heads are #4597
+`dfeb8ca729a04aea4aa290438f2ed50a37a9c7cb`, #4774
+`a012548529a25a103345e8e8a8afa707a2b3a3e6`, and #4785
+`1c9996b87d497a1690630302a0ac65748affbf16`. Their proofs supply the starting
+point and attribution; their earlier computational receipts are not re-audited.