[codyw912]

Change and mathematical scope

Adds research/schur-invariant-transitions/: a self-contained binary split-pivot recurrence, image/kernel block stabilizers, explicit ordered residual intertwiners, and a sufficient rank-one invariant with proof. It answers the invariant-transition handoff from #4689 without generating GL4 or GL5.

The existing seven-state table and 93,564 / 9,999,360 = 2599/277760 full-guard count remain credited to Mihir Srivastava. The contribution is their state-aware algebraic derivation, not new numerical counts, a new κ, a physical supplier, or selected-frontier promotion.

The recurrence applies to noncommuting residual binary idempotents. The rank-one invariant includes ambient dimension and the ordered data ([u=v], [phi=psi], phi(v), psi(u)). A written extension argument proves sufficiency, rather than inferring it from finite samples. The rank-two classification certificate remains restricted to the specified initial profile (1,1,1,2).

Task and overlap

Started from canonical main b58ad9ef4d4e861b0eaa5dba41742bdc072dca51, priority revision 7; selected κ remains 60403/20000000 (#4470). formal-interface-lemma is explicitly changed to this algebraic side result. Only the five new package files change; no shared implementation, frozen source, environment setup or selected-frontier file changes.

Nearest predecessors: #4689 (head 3a72fbad148294aa9ca7f900a48dbb10e64232be), #4678/#4671 and #4597. Pending sources are attributed immutable citations, not runtime or Git-history dependencies. Scope and overlap coverage are in TASK.json. Claim posting was attempted but unsuccessful; no public reservation is claimed.

Validation

Executed by the orchestrator on Linux x86_64, CPython 3.12.13:

python3.12 -B scripts/research_package.py --package research/schur-invariant-transitions
python3.12 -B -m unittest discover -s tests -p 'test_research_*.py' -v

• Ordinary and optimized registered commands both passed, using explicit exceptions under -O.
• All 496 initial splits, 172 admitted splits, seven states and their multiplicities, seven 120-split next domains, and 784 ordered rank-one pairs were checked.
• Nine mathematical mutation controls reject lost completion factors, insufficient/incorrect state quotients, inconsistent terminal constraints, omitted Schur correction and first-guard/full-guard confusion.
• Shared onboarding suite: 331 tests, 11 Windows-only skips; no failures.
• Separate Astra mathematical review found no blocking issue. This is AI review, not independent human or formal-kernel approval. Hosted CI is not claimed.
• Publication preflight passed for commit 4afbea22e79e19d1d65702eb4090d454ee39d967: exact package bytes and source-history ancestry. Its five-file committed export passed integrity checking and both direct normal/optimized replays in a fresh Git-free directory.

No selected scientific cold replay, Lean, Docker or physical odd-q caller replay was run. Physical selection, emission, reads, restoration, stock and the full consumer bill remain unpriced, not zero.

Credit

Mihir Srivastava (#4689), ssafwann (#4678/#4671), Nick Petalas (#4597 and earlier Schur/header lineage), Ryanson Jonathan (full-guard convention and separate-family census), lee-phys, Cinna Davis and the retained normalizer contributors. Prior assistance disclosures and exact heads remain in the package. Prepared for codyw912 with substantial OpenAI Codex (GPT-6 Astra) assistance and separate AI review. Apache-2.0.

Open next questions

README/TASK retain actionable next questions: a sufficient higher-rank invariant beyond this finite profile, and an actual caller-specific paid header cover. Neither follows merely from the exact binary guard probability.