Research: certify a class-C V2 stabilizer quotient

[1337isnot1337]

Summary

This contributes a bounded route-B result to Discussion #2954. It certifies a stabilizer-subgroup orbit quotient for the (5,6,4,5) ordering of config 6554 = (6,5,5,4) and one first-flat orbit.

The package checks the 12 pinned label permutations on all of F_2^9, their induced permutations of the 18,100 typed rows in data88.txt, and 12 exact invertible rational 8 x 8 witnesses carrying every listed line or plane direction to its image. For the selected first-flat orbit, it exhaustively partitions the 776,806 distinct six-flat root masks into 47,850 subgroup orbits, a factor 776806/47850 = 16.234190177638x reduction in the V2 candidate list. The largest orbit contains 32 masks.

The proof is equivariance: a subgroup fixing V1 maps V2, its label cover, the finite Off-direction set, and the corresponding real direction spans bijectively. Therefore every V2 in a subgroup orbit has the same checker outcome and one representative suffices. The least-mask convention is used for this witness; integration with all existing union-find V1 representatives remains open.

Verification

• python3 -B research/k2-class-c-stabilizer-orbit-quotient/verify.py — PASS under CPython 3.14, 178.459 seconds; exact source checks, rational matrix checks, exhaustive RREF/mask enumeration, orbit partition and rejecting controls.
• python3 scripts/research_package.py --package research/k2-class-c-stabilizer-orbit-quotient --check-only — PASS.
• python3 scripts/check_research_commit.py --package research/k2-class-c-stabilizer-orbit-quotient — PASS for commit ba3cfcea4bdc485e663a4a48f91b66da4dbc01ed.

The registered replay specifies Python 3.12. This container has only Python 3.14, so the repository wrapper declined the full registered run with “run replay with Python 3.12”; the verifier itself completed directly under 3.14. It uses only the standard library.

Scope and remaining work

The result assumes the pinned source's statement that data88.txt is complete for |Off(h)| <= 88. The count applies to one first-flat orbit, not all V1 orbits or all configs. No s4z pair closure was run, no row is killed, and no RRC premise is checked. No supplier, general lower bound, or kappa change is claimed; selected kappa remains 0.001672660197076918.

Merged PRs #2975 and #3000 improve other stages. Merged #3315 audits the config census; open #3334 establishes the same-threshold dominance reduction to 6554@T7 for the hard rows but no geometric span pass. This PR covers the separate V2 stabilizer quotient. The open Issue #913 is editorial and unrelated.

The changed-scope task metadata, source pins, exact matrix witnesses, attribution and open questions are in research/k2-class-c-stabilizer-orbit-quotient/. The 5 MB data88.txt input is copied unchanged and credited in NOTICE.md. Prepared by OpenAI Codex with substantial AI assistance; no independent human proof review is claimed.