Derive invariant Schur guard transitions without GL4/GL5 enumeration (no new κ)

research/schur-invariant-transitions/README.md

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+# Split-pivot recurrence and sufficient residual invariants
+
+Prepared by **codyw912**, with substantial **OpenAI Codex (GPT-6 Astra)**
+assistance. Apache-2.0. **No new κ and no selected-frontier change.**
+
+This package supplies an intrinsic recurrence, a block-intertwiner certificate
+for the seven ordered rank-two residuals, and a sufficient rank-one invariant.
+The seven-state numerical table and the total **93,564 / 9,999,360** are credited
+to **Mihir Srivastava, PR4689**, not claimed as new numbers. The new contribution
+is their derivation through small state transitions without generating GL4 or
+GL5, together with a proof that the terminal invariant is sufficient.
+
+**Evidence status:** parent replay on CPython 3.12.13, Linux x86_64, completed
+with exit status zero using `python3.12 -B scripts/research_package.py --package
+research/schur-invariant-transitions`. Both registered ordinary
+`-I -W error -B` and optimized `-I -W error -O -B` commands emitted
+`PASS_SCHUR_INVARIANT_TRANSITIONS`; the runner emitted
+`PASS_REGISTERED_RESEARCH_PACKAGE`. The replay reproduced 496 initial splits,
+172 admitted splits, 784 rank-one pairs, seven exact states, nine rejected
+mutations and 93,564/9,999,360 = 2599/277760, including all embedded state and
+depth expectations. The expectations remain credited predecessor results;
+this is observed regeneration, not a claim to new numerical discoveries.
+A separate AI mathematical review found no blocking issue in the recurrence,
+stabilizers or invariant-sufficiency proof. Neither that review nor the local
+replay constitutes independent human, proof-kernel or hosted approval. The
+implementation worker ran no research code, gates, builds or formatters.
+Final separate AI static review also reported PASS with no blockers. Parent
+reported the shared 331-test suite passing with 11 Windows-specific skips;
+this tooling evidence is separate from the mathematical package replay.
+No public Discussion claim was posted. Upstream PR creation requires operator
+approval; publication status is reported in the handoff.
+
+## Domain and complete-header recurrence
+
+All vector spaces in this package are over F2. Requests remain **ordered**. For
+idempotents A,B on V of dimension n, let H(n,k;A,B) count complete ordered bases
+M such that both M^-1 A M and M^-1 B M have nonzero leading principal minors of
+orders 1 through k. Require 0 <= k <= n; GL0 has order one. Neither commutation
+nor an equal-rank hypothesis is needed for the recurrence. An impossible guard
+has count zero, in particular when k exceeds either rank.
+
+For a vector a and functional beta satisfying
+
+```
+beta(a) = beta(Aa) = beta(Ba) = 1,
+W = ker(beta),
+A'(x) = Ax + Aa beta(Ax),  B'(x) = Bx + Ba beta(Bx),  x in W,
+```
+
+we have the exact, basis-independent recurrence
+
+```
+H(n,0;A,B) = |GL_n(F2)| = product_{j=0}^{n-1}(2^n - 2^j),
+H(n,k;A,B) = sum_{admitted (a,beta)} H(n-1,k-1;A',B').
+```
+
+The minus sign of the ordinary Schur formula equals plus in F2. The maps land
+in W because beta(Aa)=beta(Ba)=1. For A, put c=Aa and t=beta(Ax).
+Then A(Ax+ct)=Ax+ct since A²=A and Ac=c. Its beta-value is zero, so applying
+the same Schur operator again changes nothing: A'²=A'. The argument for B is
+identical. In a basis (a,w1,...,w[n-1]) with the w's in W, A has block form
+`[[1,R],[C,D]]`, and A' is `D+CR`. Invertible block elimination gives
+`rank(A')=rank(A)-1`. This also proves agreement with the intrinsic map.
+
+**Completion and guard proof.** A complete header M determines exactly the
+split consisting of its first column a and the first row beta of M^-1.
+Conversely, for a fixed split, appending any ordered basis of ker(beta) gives
+exactly one header with that first column and inverse first row. Thus its
+completion fibre has size |GL[n-1]|, not one and not `2^(n-1)|GL[n-1]|`.
+At a unit pivot the determinant of each leading (j+1)-block equals the
+corresponding leading j-block determinant of the Schur complement. Partitioning
+headers by split and applying this identity proves the recurrence. A basis
+change on W bijects its complete headers, so the intrinsic notation is valid.
+The recurrence applies at every subsequent, possibly **noncommuting**, residual.
+
+The concrete input is
+
+```
+P0 = diag(0,1,0,1,1), Q0 = diag(0,0,1,1,1).
+```
+
+Only guard orders 1,2,3 are required. The 31 nonzero first columns have 16
+splitting functionals each, giving 496 splits and `496*20160=9999360` complete
+headers. After an admitted first split the two residuals have dimension four
+and rank two; after the next, dimension three and rank one.
+
+## Rank-two blocks, stabilizers, and certified classification
+
+Matrices use integer **rows**, bit j representing column j. All representatives
+have `S=(0,0,4,8)=diag(0_2,I_2)`. In `ker S ⊕ im S`, write
+`T=[[E,F],[G,H]]`. An invertible map commuting with S preserves both summands,
+so it is exactly `K=diag(U,V)` with U,V in GL2. It stabilizes T exactly when
+
+```
+UE=EU, UF=FV, VG=GU, VH=HV.
+```
+
+This proves that only 6*6=36 candidates, not GL4, need examination. It also
+proves the normalization equivalence criterion: after normalizing S, two T's
+are equivalent precisely by this block action.
+
+Here are analytic stabilizer derivations. Let `D=diag(0,1)` and let
+`J=e2 e1^t` on F2². In F2 a one-dimensional invertible scalar is one.
+
+* State 1: E=0, H=I, F=G=0. Every U,V works: 36.
+* State 2: E=0, H=I, F=0, G=J. The one nontrivial equation is
+  `V e2 e1^t = e2 e1^t U`. Equality of nonzero rank-one tensors forces
+  `V e2=e2` and `e1^t U=e1^t`; each has two choices: 4.
+* State 3: E=H=D and F=G=0. Centralizing D preserves its two distinct
+  eigenlines, hence U=V=I: 1.
+* State 4: E=H=D, F=0, G=e1 e2^t. The preceding diagonal equations already
+  force U=V=I, which satisfies the off-diagonal equation: 1.
+* State 5: E=0, H=I, G=0, F=J. Now `U e2=e2` and `e1^t V=e1^t`:
+  again 2*2=4.
+* State 6: E=0, H=I, F=G=J. Both U and V must fix e2 and its annihilating
+  functional e1^t, hence each is `[[1,0],[c,1]]`, c in F2: 4.
+* State 7: E=H=D, G=0, F=J. Again U=V=I: 1.
+
+The implementation corroborates these constraints on the 36 GL2² pairs per
+representative and prints every stabilizer. It builds only their small **block
+orbits**, checking these are pairwise disjoint. For each of the admitted initial
+splits, it computes a deterministic basis C of `ker S' ⊕ im S'`, finds a block K
+with `C^-1 T' C = K T_i K^-1`, and checks the explicit witness L=CK:
+
+```
+S' L = L S,   T' L = L T_i,   rank(L)=4.
+```
+
+All 172 splits, their state assignments, and these invertible witnesses are
+printed. Unmatched residuals or overlaps fail; no rank-signature heuristic is
+used to force membership. This is completeness for the initial split domain,
+**not** a classification of every pair of rank-two projectors in dimension four.
+
+## Sufficient rank-one invariant: proof, not a guess
+
+Over F2, each nonzero rank-one idempotent has a unique factorization
+`R=u phi`, with u and phi nonzero and phi(u)=1. For an ordered pair
+`(R,T)=(u phi,v psi)` on an n-dimensional space use
+
+```
+I(R,T) = ([u=v], [phi=psi], phi(v), psi(u)), together with n.
+```
+
+These data are necessary under simultaneous conjugation. They are also
+**sufficient**. Define `X:F2² -> V` with columns u,v and `Y:V -> F2²` with rows
+phi,psi. The first flag determines ker X (zero or span(1,1)). The second flag
+determines im Y (F2² or span(1,1)). The two cross-values determine
+`YX=[[1,phi(v)],[psi(u),1]]`.
+
+Here is the full extension argument, including the potentially missing datum.
+Consider two systems X,Y and X',Y' with these three data equal and ambient
+dimensions equal. Let L=im X, K=ker Y, and similarly L',K'. Equality of ker X
+makes `f0(Xz)=X'z` a well-defined isomorphism L->L'. Equality of YX gives
+`Y'f0=Y|L`, so f0 sends L∩K isomorphically onto L'∩K'. Equality of im Y and
+ambient dimensions gives dim K=dim K'. Extend this restriction to any
+isomorphism fK:K->K'. The maps f0,fK agree on the intersection and therefore
+combine to an isomorphism f1:L+K -> L'+K'. It satisfies `Y'f1=Y`.
+Finally choose a basis of a complement of Y(L) in the **common** im Y, and
+choose a lift in each ambient space for every vector of that basis. Those lifts
+complement L+K and L'+K', respectively: the induced maps from these quotients
+to im Y/Y(L) are isomorphisms. Mapping corresponding lifts extends f1 to an
+invertible f on V with `fX=X'` and `Y'f=Y`. Consequently
+`f(u phi)=(u' phi')f` and `f(v psi)=(v' psi')f`.
+
+Thus no extra datum is missing: the intersection information is already fixed
+by ker X, im Y and YX. This is an all-dimension binary proof, not a consequence
+of the finite terminal census. Dimension must be retained. For example, when
+both flags are zero and both cross-values are one, rank(YX)=1; in dimension two
+X and Y would both be invertible, an impossibility. Dimension three admits it.
+When either equality flag is one both cross-values must be one. Hence in
+**dimension three** there are exactly seven possibilities, all realized below.
+
+## Terminal counts from consistent linear constraints
+
+For rank-one requests an admitted split requires
+`phi(a)=psi(a)=1` and `beta(a)=beta(u)=beta(v)=1`. For fixed a the last three
+constraints are consistent iff every relation among a,u,v has even coefficient
+sum. Equivalently the coefficient and augmented matrices have equal ranks.
+Thus the exact number of terminal splits is
+
+```
+g(R,T) = sum_{a: phi(a)=psi(a)=1}
+         [consistent a,u,v -> 1] * 2^(n-rank(a,u,v)),
+H(3,1;R,T) = 6*g(R,T).
+```
+
+The zero vector never satisfies the first conditions. In particular a=u+v
+with u,v distinct gives the odd relation a+u+v=0 and contributes **zero**.
+Dropping consistency is not a valid use of the rank formula.
+
+The following representatives give the terminal class table. Vector and
+functional integers have the same least-significant-bit coordinate convention.
+The last four rows list the distinct cross-pairings in order.
+
+| Invariant | (u,phi,v,psi) | g | Derivation in dimension three |
+|---|---|---:|---|
+| 1111 | (1,1,1,1) | 10 | a=u contributes 4; three other a contribute 2 each |
+| 1011 | (1,1,1,3) | 6 | a=u contributes 4; the other allowed a contributes 2 |
+| 0111 | (1,1,3,1) | 6 | a=u,v contribute 2 each; two other a contribute 1 |
+| 0000 | (1,1,2,2) | 1 | a=3 is inconsistent; a=7 contributes 1 |
+| 0001 | (1,1,2,3) | 3 | a=1 contributes 2; a=5 contributes 1 |
+| 0010 | (1,3,2,2) | 3 | a=2 contributes 2; a=6 contributes 1 |
+| 0011 | (1,3,2,7) | 4 | a=1,2 each contribute 2 |
+
+The verifier constructs this table and uses invariant lookup in the recurrence.
+As an independent finite corroboration, all 28²=784 ordered rank-one pairs are
+checked against literal `(a,beta)` split constraints, with no basis enumeration.
+Every next-pivot transition is counted with multiplicity; deterministic stdout
+contains all seven terminal-type entries, including zeros, for each state.
+The ordered-pair populations in the displayed class order are respectively
+28,84,84,168,168,168,84, also embedded as exact targets. The seven H(4,2)
+values are additionally compared against direct recursive split evaluation
+with the GL-order zero-guard base case; memoization is in-memory only.
+
+
+## Expected state and guard-depth tables
+
+The T rows, b,z,o,f, header weights and full guards below are Mihir's PR4689
+results. Here m counts the next admitted splits, G sums their terminal g-values,
+and `H(4,2)=6G`. These m,G and depth targets are the predecessor's depth data
+and arithmetic consequences of its table, reproduced by the parent replay.
+
+| State | T rows | b | z | m | G | o=20160/z | f=6G/z | Header weight | Full guards |
+|---|---|---:|---:|---:|---:|---:|---:|---:|---:|
+| 1 | (0,0,4,8) | 1 | 36 | 60 | 600 | 560 | 100 | 20160 | 3600 |
+| 2 | (0,0,4,9) | 3 | 4 | 44 | 312 | 5040 | 468 | 60480 | 5616 |
+| 3 | (0,2,0,8) | 126 | 1 | 30 | 61 | 20160 | 366 | 2540160 | 46116 |
+| 4 | (0,2,2,8) | 18 | 1 | 34 | 136 | 20160 | 816 | 362880 | 14688 |
+| 5 | (0,4,4,8) | 3 | 4 | 44 | 312 | 5040 | 468 | 60480 | 5616 |
+| 6 | (0,4,4,9) | 3 | 4 | 36 | 180 | 5040 | 270 | 60480 | 3240 |
+| 7 | (0,6,0,8) | 18 | 1 | 34 | 136 | 20160 | 816 | 362880 | 14688 |
+
+After the initial admitted guard, let depth d=0,1,2 be the number of further
+common guards passed. A next split has |GL3|=168 unrestricted completions,
+whereas a successful two-step continuation has |GL2|=6 completions. Conjugation
+by GL4 has uniform fibre z onto its orbit (orbit-stabilizer theorem). Therefore
+
+```
+d0=(20160-168m)/z, d1=(168m-6G)/z, d2=6G/z,
+headers per orbit element = b*z,
+full guards in initial state = b*6G.
+```
+
+| State | d0 | d1 | d2 |
+|---|---:|---:|---:|
+| 1 | 280 | 180 | 100 |
+| 2 | 3192 | 1380 | 468 |
+| 3 | 15120 | 4674 | 366 |
+| 4 | 14448 | 4896 | 816 |
+| 5 | 3192 | 1380 | 468 |
+| 6 | 3528 | 1242 | 270 |
+| 7 | 14448 | 4896 | 816 |
+
+Summing gives 172 first-admitted splits and 3,467,520 first-guard headers, but
+only 93,564 full-guard headers. The full fraction is 2599/277760. State 6 retains
+the credited one-block sharp-header count 60,480 and full continuation 3,240.
+No uniform weighting of states or orbit elements across distinct initial
+fibres is assumed.
+
+## Reproduction, controls, and boundaries
+
+Python 3.12 standard library, five files, no third-party imports, external reads,
+pending-head objects, network calls, subprocesses, persistent caches or write mode in the
+scientific verifier. Direct exported-package execution needs no Git:
+
+```sh
+python3.12 -I -W error -B research/schur-invariant-transitions/verify.py
+python3.12 -I -W error -O -B research/schur-invariant-transitions/verify.py
+```
+
+Each command prints deterministic JSON followed by
+`PASS_SCHUR_INVARIANT_TRANSITIONS` only after every explicit check succeeds.
+The package runner also checks historical source metadata:
+
+```sh
+python3.12 -B scripts/research_package.py --package research/schur-invariant-transitions
+```
+
+VERIFY.json proposes a 60-second cap per run, not a measured runtime. The
+scientific domain is 496 initial splits, seven 120-split next domains, 36 block
+changes per representative, and 784 terminal pairs, plus direct recursive
+split evaluation of the seven H(4,2) representatives. There is no GL4/GL5
+generator, no 4x4 raw-matrix census, no native compiler, Lean or selected cold
+replay. All failure checks use exceptions rather than disabled-by-`-O` asserts.
+Permanent mutation controls reject missing GL2/GL4 completion factors,
+commutator-only and product/sum/commutator-rank identification, transposition,
+unordered terminal requests, ignored odd relations, omitted Schur correction,
+and replacing full guards by first-guard counts. States 1 and 3 have equal
+commutator rank but different continuations; states 2 and 5 share the listed
+ranks yet are distinct ordered orbits. Transposition exchanges 2/5 and 4/7.
+
+The recurrence is proved generally for binary idempotents; the seven-state
+certificate is only for the displayed sector profile (1,1,1,2). No larger-field
+invariant classification is claimed. The binary event is not admission into
+the retained odd-q executable normalizer. Physical header selection/emission,
+reads, restoration, storage, stock, dispatch and the complete consumer bill
+remain unpriced, not zero. No formal proof-kernel or independent human theorem
+acceptance is claimed.
+
+## Provenance, overlap, and credit
+
+Canonical base is `b58ad9ef4d4e861b0eaa5dba41742bdc072dca51`, priority revision 7,
+selected κ=60403/20000000=0.00302015 from #4470. TASK.json explicitly changes the
+`formal-interface-lemma` card to this algebraic side result; it neither discharges
+the selected all-size interface obligations nor repeats stale endpoint arithmetic.
+SOURCES.json pins only historical canonical-main provenance; VERIFY.json has no
+external live inputs. Pending works below are immutable **citations**, not Git
+ancestry or executable dependencies. Embedded numeric expectations are openly
+transcribed fixtures, not runtime-authenticated remote files.
+
+* **Mihir Srivastava**, substantial OpenAI Codex assistance, [PR4689](https://github.com/CrocSwap/integer-mult-bounds/pull/4689),
+  head `3a72fbad148294aa9ca7f900a48dbb10e64232be`:
+  [seven-state table and proof](https://github.com/CrocSwap/integer-mult-bounds/blob/3a72fbad148294aa9ca7f900a48dbb10e64232be/research/schur-residual-orbits/README.md),
+  [depth receipt](https://github.com/CrocSwap/integer-mult-bounds/blob/3a72fbad148294aa9ca7f900a48dbb10e64232be/research/schur-residual-orbits/RESULT.json).
+  All seven numerical states and the total retain this attribution.
+* **ssafwann**, substantial OpenAI Codex assistance, [PR4678](https://github.com/CrocSwap/integer-mult-bounds/pull/4678),
+  head `af9cf805c501e4e9e947c98c0b97b2281b6b6e97`: one-block counts, equal
+  completion fibres, profile and [released handoff](https://github.com/CrocSwap/integer-mult-bounds/discussions/4653#discussioncomment-19126925).
+  [PR4671](https://github.com/CrocSwap/integer-mult-bounds/pull/4671), head
+  `8d2e3bc76aa7d43e5c629daaf1e72d4dd5a81ed9`, supplies the sector criterion.
+* **Nick Petalas**, [PR4597](https://github.com/CrocSwap/integer-mult-bounds/pull/4597),
+  head `dfeb8ca729a04aea4aa290438f2ed50a37a9c7cb`: block Schur bound and orbit
+  question; retained [scalar Schur proof](../schur-commutator-criterion/README.md)
+  supplies residual idempotency/rank and the earlier header lineage.
+* **Ryanson Jonathan**, substantial Claude Code (Claude Opus 5.5) assistance:
+  [full-guard convention and separate-family census](../full-guard-header-census/README.md).
+  Retain **lee-phys**'s binary sharpness/dimension contributions, **Cinna Davis**'s
+  PR4100 BIT request context and its normalizer lineage (source527, commdata2338,
+  Jose Manuel). No predecessor code is imported or copied into this verifier.
+
+At the final parent checkpoint, canonical origin/main remained
+`b58ad9ef4d4e861b0eaa5dba41742bdc072dca51`. The refreshed overlap snapshot
+covered 469 open PRs at 2026-10-22T16:48:00.710568+00:00 and 3,624 discussion
+records at 2026-10-22T16:48:34.859558+00:00, with complete pagination. All 5,860
+tracked local records were covered; the local snapshot had 5,864 records,
+including four from this contribution work. Direct predecessor heads #4689,
+#4678, #4671 and #4597 remained unchanged from the cited pins. Coverage does
+not include arbitrary remote diffs, closed remote PR bodies or unpublished work;
+no novelty or reservation over those domains is asserted. No public Discussion
+claim was posted. Upstream PR creation requires operator approval; publication
+status is reported in the handoff.
+
+## Open next questions
+
+1. **Can a sufficient higher-rank state invariant replace block-orbit matching?**
+   This is relevant to scalable recursion beyond one binary profile. First derive
+   the representation data of two image/kernel decompositions and test proposed
+   invariants against the explicit witnesses here. Block intertwiners certify
+   this finite domain but are not a general classification; that is the blocker.
+2. **Can the state recurrence support an admitted, paid header cover?** The finite
+   full-guard probability alone supplies no executable cost bound. First state
+   the actual caller's field and header-selection contract and price emission,
+   physical reads and restoration. Odd-q admission and the complete consumer bill
+   remain separate blockers. No expansion or follow-up claim is held here.