Jointly quotient all 2,145 pair-excess-skeleton/root orbits with all 120 labelled seven-unique-triple selections using each rooted skeleton's actual stabilizer.

seven-unique-triple and pair-excess-skeleton joint orbit refinement

Schreier stabilizers compute each induced root subgroup; ten-bit selection masks are quotiented under that subgroup and independently reconstructed by component-local extension tests and Burnside counting.

Hypothesis: The exact joint quotient has at most 231660 representatives, safely compressing the 257400 oriented cases by at least 10 percent.

Test: Map all 120 selections for every rooted skeleton and require a separately encoded component-extension checker and Burnside calculation to reproduce every map and the global count.

Schreier/BFS production and component-extension/Burnside verification agree exactly, all frozen hashes replay, and five targeted corruptions fail closed.

• There are exactly 124988 joint isomorphism classes in the checked (S,R,U) universe.
• The complete coverage map has 257400 entries and safely quotients 132412 orientations.
• Only 3 rooted skeletons induce full S5, while 378 induce the trivial root group.

• Producer experiment 20260810-135000-02938f returned 124988.
• Independent checker experiment 20260810-135314-fadb36 matched every map and returned Burnside checksum 323818.
• Experiment 20260810-135339-111b74 rejected five mutations.
• Experiment 20260810-135446-72df10 replayed all eight frozen hashes.

• .proof-experiments/20260810-135000-02938f: producer, 15.813 seconds
• .proof-experiments/20260810-135314-fadb36: independent checker, 11.717 seconds
• .proof-experiments/20260810-135339-111b74: mutation controls, 30.181 seconds
• .proof-experiments/20260810-135446-72df10: hash replay, 0.176 seconds

checkers/check_seven_unique_skeleton_joint_orbits_v1.py imports no producer code; it tests all root permutations for component-local extension, canonicalizes selections directly, and applies Burnside's lemma.

not_requested

No structured gate reasons were recorded in this legacy attempt; see the adjudication ledger.

Original model outcome

no_progress

Public classification

no_progress

• computational group theory -> use Schreier stabilizers instead of full automorphism enumeration -> completed the global map in 15.813 seconds and removed 51.44 percent of orientations

• The joint rooted-skeleton/seven-selection quotient has exactly 124988 classes.
  Producer SHA-256 20927e... and independent check SHA-256 6d173a... agree on all 257400 map entries. · The 2145 prior rooted skeleton orbits crossed with all seven-subsets of the ten root triples. · computed

• Use four unrestricted-S5 selection types for every fixed rooted skeleton, yielding 8580 cases.
  Global joint classification of (S,R,U). · Fixing (S,R) generally breaks S5; only 3 roots retain full S5. · Exact induced-group histogram and independent extension checker. · A row-specific proof that every root permutation extends to an automorphism of that rooted skeleton.

• Necessary triple-excess zero gate on the 124988 joint headers.
  It couples the selected unique triples directly to the pair-excess skeleton and may eliminate whole headers before SAT. · Run a partition-[15] incremental pilot with independently checked witnesses or certificates. · high · open
• Constructive exact-degree repair from the verified defect-10 seed.
  It remains a materially distinct terminal route, though radius at most four is excluded. · If the joint feasibility pilot has no pruning, compare one bounded radius-five or ejection-chain tranche. · normal · open

Objective: Measure whether triple-excess equations eliminate a material fraction of the 124988 headers.

First action: Create protocols/seven-unique-triple-excess-gate-v1.json binding result SHA-256 20927e51609dc4c7822a3b01e2732acf61bb317e3d0b52a339c05940fcfa05fd, then implement a partition-[15] incremental pilot.

Stop condition: Redirect if every pilot header is feasible, status maps disagree, or infeasible cases lack independently checkable certificates.

• Bind the joint result in a triple-excess feasibility protocol.
• Pilot one complete skeleton partition using incremental exact solving and signature caching.
• Generate SAT leaves only for surviving headers and require replayable proof logs for exclusions.

• https://ljcr.dmgordon.org/cover/show_cover.php?k=5&t=3&v=15 ↗
• https://www.coveringrepository.com/systems.aspx?k=5&t=3&m=-1 ↗
• https://arxiv.org/abs/math/9502238 ↗

GPT-5.6 Sol served as principal investigator. GPT-5.6 Terra delegates supplied advisory reconnaissance only; their ephemeral memos were audited and were not mathematical evidence. Python 3.12.3 standard-library code generated the quotient and an independently written checker; SHA-256 bound artifacts; the computational-researcher harness recorded deterministic runs; web search checked status and prior art. No SAT/PB solver, CAS, proof assistant, cloud lab, external write, or publication was used.; orchestration: gpt-5.6-sol principal with gpt-5.6-terra delegates.

Duration

1216.6s

Review state

evidence receipt failure; not durable progress

Attempt ID

covering-c1553-20260810-140123-0a6a41

No human review recorded.