Classify seven residual Schur-pair states and full-guard counts (no new κ)

research/schur-residual-orbits/README.md

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+# Seven residual Schur-pair states and their full-guard counts
+
+**Exact finite classification and counting, with a written multiplicity
+argument. No new kappa.** Prepared by Mihir Srivastava with substantial
+OpenAI Codex assistance, by one agent. Apache-2.0.
+
+This answers one bounded instance of the residual-state handoff in
+[ssafwann's PR4678](https://github.com/CrocSwap/integer-mult-bounds/pull/4678),
+continuing the orbit question in Nick Petalas's PR4597. For the binary
+commuting projector pair with joint-sector dimensions **(1,1,1,2)**, exactly
+**seven ordered simultaneous-conjugacy classes** occur after a common scalar
+pivot. Their complete-header weights are not uniform. The census yields
+**93,564 full-guard headers out of 9,999,360**, a fraction **2599/277760**.
+
+The sharp first-step event from PR4678 occupies just one residual class:
+its **60,480** headers produce **5,040** distinct coordinate pairs, each
+exactly **12** times. Only **3,240** of those headers pass the full guard
+chain, a conditional fraction **3/56**. The 60,480 one-block count is credited
+corroboration; its orbit classification and continuation count are the new
+quantities here. Commutator rank alone cannot determine the continuation law.
+
+## Fixed sample space and guards
+
+Work over F2 and fix, before choosing the header,
+
+```text
+P0 = diag(0,1,0,1,1),  Q0 = diag(0,0,1,1,1).
+```
+
+The axes have joint sectors 00,10,01,11,11. Both projectors have rank three.
+Choose a complete ordered header M uniformly from GL5(F2) and set
+`P=M^-1 P0 M`, `Q=M^-1 Q0 M`. A full guard means that **both** leading
+principal minors of each order 1,2,3 are nonzero. Orders 4 and 5 are not
+required, since the requests have rank three. This is equivalent to three
+successive nonzero scalar pivots without row exchanges. Using `G=M^-1`
+gives the earlier `G P0 G^-1` convention and the same uniform sample space.
+
+Conditional on P[0,0]=Q[0,0]=1, write the Schur complements S,T, of size four.
+They are rank-two idempotents. States are orbits of the **ordered** pair
+`(S,T)` under `(S,T) -> (K^-1 S K, K^-1 T K)` for K in GL4(F2).
+Swapping the requests or transposing them is not part of this equivalence.
+
+The same result holds for any fixed commuting binary idempotent pair with
+these four sector dimensions: simultaneous diagonalization sends it to this
+pair, and the fixed basis change permutes the complete-header sample space.
+No claim is made for a different sector profile, field or restricted cover.
+
+## State table
+
+Every representative below has `S=(0,0,4,8)`. Matrices are encoded as integer
+**rows**, with bit j representing column j, least significant bit first.
+For example `(0,4,4,9)` has rows `0000`, `0010`, `0010`, `1001` when
+columns are printed in increasing index order. The representatives minimize
+the ordered tuple of all S rows followed by all T rows within each orbit.
+
+Here b counts admitted split pairs reaching the state, o is its orbit size,
+z its stabilizer size, and f counts orbit elements whose two remaining
+guards both pass. Thus `o*z=20160`, header weight is `b*20160`, and the
+full-guard contribution is `b*z*f`.
+
+| State | T rows | rank [S,T] | b | o | z | f | Header weight | Full guards |
+|---|---|---:|---:|---:|---:|---:|---:|---:|
+| 1 | (0,0,4,8) | 0 | 1 | 560 | 36 | 100 | 20,160 | 3,600 |
+| 2 | (0,0,4,9) | 1 | 3 | 5,040 | 4 | 468 | 60,480 | 5,616 |
+| 3 | (0,2,0,8) | 0 | 126 | 20,160 | 1 | 366 | 2,540,160 | 46,116 |
+| 4 | (0,2,2,8) | 1 | 18 | 20,160 | 1 | 816 | 362,880 | 14,688 |
+| 5 | (0,4,4,8) | 1 | 3 | 5,040 | 4 | 468 | 60,480 | 5,616 |
+| 6 | (0,4,4,9) | 2 | 3 | 5,040 | 4 | 270 | 60,480 | 3,240 |
+| 7 | (0,6,0,8) | 1 | 18 | 20,160 | 1 | 816 | 362,880 | 14,688 |
+| Total | | | 172 | | | | 3,467,520 | 93,564 |
+
+[RESULT.json](RESULT.json) includes every one of the 172 admitted split pairs,
+their state assignments, and the exact guard-depth distributions within
+each orbit. The total header denominator is `496*20160=9999360`.
+
+The sharp state 6 has split pairs `(15,23)`, `(23,15)`, `(31,31)` in the
+fixed five-axis basis. Its fraction among all first-guard headers is 3/172,
+as in PR4678; it is 3/496 among all headers. Its later full-guard fraction
+is 270/5040=3/56, giving 3*4*270=3240 headers.
+
+Two tiny complete-header witnesses use the same integer-row convention in
+dimension five. `M=(17,9,11,5,18)` has sharp first Schur commutator and
+passes all three leading minors for both requests. `M=(23,3,5,9,16)` also
+has sharp first Schur commutator, but fails the next common guard.
+RESULT.json includes a passing and a first-guard-only header for every state.
+
+Two explicit losses of information are useful for future recursion design:
+
+- States 1 and 3 both have commutator rank zero, but their conditional
+  full-guard fractions are respectively **5/28** and **61/3360**.
+  States with commutator rank one also have unequal continuation fractions.
+- States 2 and 5 have the same ranks of S, T, ST, TS, S+T and [S,T].
+  They are still different orbits: `dim(ker S intersect ker T)` is one
+  for state 2 and two for state 5. Their continuation fractions happen to
+  agree. This witness refutes identification of those states by the listed
+  ranks; it does not claim that their guard counts differ.
+
+## Why the finite census counts complete headers exactly
+
+**Split pair and completion.** Let a be the first column of M and b the first
+row of M^-1. Then ba=1. Conversely every pair with ba=1 extends to exactly
+`|GL4(F2)|` headers: append an arbitrary ordered basis of ker b to a.
+There are 31 choices of nonzero a and 16 choices of b with ba=1, hence 496
+split pairs. This is the credited equal-completion argument in PR4678 and
+the preceding Petalas header work. No extra factor of 16 is applied after
+the split pair has already been chosen.
+
+The verifier constructs one deterministic kernel basis for each split pair,
+inverts the resulting header, and checks the two scalar pivots literally.
+Exactly 172 split pairs pass. For any fixed one of them, all other completions
+are uniquely `M diag(1,K)` with K in GL4. Block substitution gives
+
+```text
+Schur(diag(1,K)^-1 P diag(1,K)) = K^-1 Schur(P) K.
+```
+
+Therefore a split pair contributes exactly one entire simultaneous orbit,
+with each orbit element reached z times. Split pairs whose seed lies in
+the same orbit have identical completion distributions. Summing their b
+fibres gives the table's `b*z` multiplicity per orbit element.
+
+**Residual ranks.** In characteristic two, write
+`P=[[1,R],[C,D]]`, so `S=D+CR`. Idempotency gives RC=0, RD=0, DC=0 and
+D^2+CR=D. Hence S^2=S; block elimination gives `rank(P)=1+rank(S)`.
+The same applies to Q. Thus both residual ranks are two, and exactly two
+further scalar guards are required. The determinant identity
+`det(P[:j+1,:j+1])=det(S[:j,:j])` at the first unit pivot proves equivalence
+with the original orders 2 and 3. Repeating it justifies the guard-depth test.
+
+**Completeness of classification.** The independent-row generator visits
+each ordered basis of F2^4 exactly once: at each level it extends by every
+row outside the current span. Its size is `(16-1)(16-2)(16-4)(16-8)=20160`.
+For each uncovered seed, the checker explicitly generates its entire orbit,
+checks uniform fibre size and orbit-stabilizer equality, and verifies that
+it is disjoint from all previously constructed orbits. Every admitted split
+pair is assigned. This exhausts the stated finite domain, yielding seven
+classes; it is not a random search or an extrapolated general classification.
+
+For an additional direct check, choose one split pair in each of the seven
+classes and form **all 20,160 complete five-dimensional headers** in its
+completion fibre. Invert and conjugate the original P0,Q0 and evaluate their
+leading minors using the Leibniz determinant formula, without Schur
+elimination. All **141,120** direct header cases reproduce the per-class
+continuation counts. Applying those counts to the other split pairs uses
+the proved orbit-equivariance argument. This is not a raw enumeration of
+all 9,999,360 headers.
+
+## Reproduce and evidence boundaries
+
+Python 3.12 standard library; no pending executable or external input needed:
+
+```sh
+python3.12 -B scripts/research_package.py --package research/schur-residual-orbits
+python3.12 -B scripts/check_research_commit.py --package research/schur-residual-orbits --check-source-history
+```
+
+The registered normal and optimized (`-O`) modes use explicit exceptions and
+compare every scientific field with RESULT.json. Direct Git-free replay is
+`python3.12 -I -W error -O -B verify.py` from an exported package. The only
+writing mode is explicit `verify.py --write`, which regenerates RESULT.json.
+
+Verification covers all 496 split pairs, full GL4 orbits for all seven
+classes, and the 141,120 literal complete headers above. A separately
+enumerated raw domain of all **65,536** four-by-four binary matrices uses
+the Leibniz determinant to check that the generated GL4 set is exact.
+Seven rejection controls distinguish a singular header, a nonsplit pair,
+a zero pivot, omission of the completion multiplier, replacing the full
+guard with the first guard, collapsing states by commutator rank, and
+collapsing the distinct same-rank states 2 and 5.
+
+This is author-checked exact finite evidence plus the written reduction;
+independent human or proof-kernel acceptance is not claimed. No Lean, Docker,
+selected scientific cold replay, large-d classification or physical caller
+replay is part of this package. Hosted CI is reported separately in the PR.
+
+The counted binary shared-header event is an algebraic model. As explained
+in the credited projector-header notes, the retained normalizer's executable
+domain uses odd q; this F2 census is not an admission check for that caller.
+Header selection/emission, physical reads, dirty restoration, owned storage,
+stock, dispatch and the full consumer bill remain unpriced. No cost is
+implicitly zero and no selected-frontier change follows from these counts.
+
+## Scope, overlap and credit
+
+This starts from canonical main `b58ad9ef4d4e861b0eaa5dba41742bdc072dca51`,
+priority revision 7, selected conditional kappa 0.00302015. It takes the fresh
+PR4678 handoff in [Discussion4653](https://github.com/CrocSwap/integer-mult-bounds/discussions/4653#discussioncomment-19126925).
+The `formal-interface-lemma` card is marked **changed** to this algebraic
+side result. The first working object is the handoff's explicit n=5 profile;
+the stale balanced-transfer baseline is not used. The family decision is
+**narrow** to one state distribution and its continuation law, with no
+automatic construction or adjacent-profile census.
+
+TASK.json records overlap coverage and exact citation pins. At entry, 449
+open PR bodies and 3,608 Discussion records were refreshed. Before submission,
+coverage was expanded to all 5,860 tracked in-scope local notes, 453 open PR
+bodies (15:59 UTC) and 3,614 Discussion records (16:00 UTC), with no coverage
+warnings. Main was refreshed unchanged. Remote closed PR bodies, arbitrary
+code diffs, nested source notes and unpublished work remain coverage limits.
+PR4678 counts one block for general profiles; this contribution classifies
+the subsequent ordered states and counts the remaining guards for one
+profile. The earlier full-guard census covers other named families, not
+this seven-state distribution. No priority over unknown work is asserted.
+
+Predecessor credit is retained explicitly:
+
+- **ssafwann, PR4678**, head `af9cf805c501e4e9e947c98c0b97b2281b6b6e97`,
+  `research/schur-header-count/README.md`, blob
+  `5e644cf316312c0fdd489c8d33911583f1b88af2`: the exact one-block count,
+  this profile, completion-fibre argument and residual-state handoff;
+  substantial OpenAI Codex assistance. PR4671 supplies its sector criterion.
+- **Nick Petalas, PR4597**, head `dfeb8ca729a04aea4aa290438f2ed50a37a9c7cb`:
+  block-Schur commutator factorization and orbit question. His retained
+  PR4278 scalar Schur proof and earlier PR4259/4246/4228 header work supply
+  the commutator and equal-completion lineage. The local scalar note is
+  provenance-pinned in SOURCES.json; none of its code is imported.
+- **Ryanson Jonathan**, the retained `full-guard-header-census`, with
+  substantial Claude Code (Claude Opus 5.5) assistance: literal leading-minor
+  guard convention and separate-family census. PR4609/4617 and **lee-phys's
+  PR4634** retain their binary sharpness and dimension contributions.
+- **Cinna Davis**, PR4100's BIT request context, and the earlier normalizer
+  lineage credited there (source527, commdata2338 and Jose Manuel).
+
+The source-bound tooling retains its existing contributors' attribution.
+The pending PRs are citations, not runtime or provenance dependencies that
+would require unpublished Git ancestry. All new files are Apache-2.0.
+
+## Open next questions
+
+Can an invariant-based transition rule recover these seven states and their
+continuation counts without enumerating GL4? That would turn this exact
+finite table into a useful test fixture for a scalable state-aware header
+recursion. Start by deriving the stabilizers of the seven displayed pairs
+and the next-pivot transitions using their image/kernel decompositions.
+The blocker is a proved sufficient state invariant and its transition
+multiplicities; commutator rank and the listed product/sum ranks are
+insufficient for orbit identity. Larger fields/profiles and an actual
+production header cover require a separate scope and cost allowance.
+No follow-up claim is held by this package.