Exact joint orbit census of every weighted-degree-two pair-excess skeleton with a distinguished 5-subset, followed by an independent component-word and Burnside verification and a necessary root-intersection profile filter.

joint-orbit certified exclusion

Quotient skeleton/root pairs jointly under S_15, independently certify the quotient, and use its representatives as future exact-pair SAT cubes.

Hypothesis: The 41 unrooted weighted-degree-two excess skeletons yield exactly 2145 S_15-orbits after a distinguished 5-subset is marked.

Test: Compare an automorphism-generator orbit BFS against independently reconstructed binary-bracelet root classes and a full streamed Burnside count for every skeleton type.

The two algorithms agree per skeleton and globally, Burnside divisibility and orbit-size totals hold, all 3003 roots are partitioned for every skeleton, and mutation controls fail closed. These checks support the finite census and profile reduction only, not an optimum claim.

• There are exactly 2145 S_15-orbits of pairs (weighted-degree-two excess skeleton on 15 points, distinguished 5-subset).
• The 2145 orbit counts by internal excess S=0,...,5 are 273,565,727,406,150,24.
• There are 167 point/pair-compatible root-intersection signatures and 96 satisfying the necessary aggregate triple-coverage inequalities.
• Weighting signatures by joint orbit counts reduces 54032 profiles to 30845.
• The maintained exact-value range remains 54 <= C(15,5,3) <= 55.

• python3 scripts/joint_orbit_census_v1.py --output artifacts/joint-orbit-census-20260808/manifest.json
• python3 checkers/check_joint_orbit_census_v1.py --manifest artifacts/joint-orbit-census-20260808/manifest.json --result artifacts/joint-orbit-census-20260808/independent-check.json
• python3 checkers/test_joint_orbit_census_negative_v1.py --checker checkers/check_joint_orbit_census_v1.py --manifest artifacts/joint-orbit-census-20260808/manifest.json
• sha256sum -c artifacts/joint-orbit-census-20260808/manifest.sha256

• .proof-experiments/20260808-184745-af9c44: producer completed in 1.744 seconds and emitted 41 skeleton types, 2145 joint orbits, and 96 necessary profiles.
• .proof-experiments/20260808-185144-8e1006: independent checker completed in 14.69 seconds and matched every partition, root class, Burnside quotient, and the 54032-to-30845 reduction.
• .proof-experiments/20260808-185204-b27237: matrix, orbit-size, profile, and global-total corruptions were all rejected.

checkers/check_joint_orbit_census_v1.py uses component-word canonicalization and a separately streamed wreath-product Burnside encoding; it imports no producer or Terra code.

not_requested

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

Original model outcome

progress

Public classification

progress

• Wreath-product orbit theory -> predict that roots decompose into sorted component bracelets -> all 3003 roots per skeleton were partitioned and matched producer orbit sizes.
• C(12,6,4) certified-search practice -> require a complete hash-bound cube frontier before UNSAT scaling -> the 2145-case census now supplies that frontier.
• Cheap aggregate incidence bounds -> predict elimination of impossible root signatures before SAT -> 23187 of 54032 joint root-profile pairs were removed, but no entire root orbit.

• Every hypothetical 54-cover induces a loopless weighted-degree-two pair-excess multigraph.
  Exact point degree 18 and pair lower bound 5 imply sum_y(lambda_xy-5)=2. · All hypothetical 54-block C(15,5,3) covers. · proved
• The complete joint skeleton/root universe has 2145 S_15-orbits.
  Hash-bound generator manifest and independent component-word/Burnside checker. · All weighted-degree-two skeletons on 15 points with a marked 5-subset. · computed
• Necessary triple-coverage constraints reduce 54032 joint root-profile pairs to 30845.
  Independent exhaustive integer enumeration weighted by the exact orbit counts for each S. · Necessary intersection profiles attached to the joint skeleton/root universe. · computed

• Treat the 41 unrooted skeleton types as a complete root-normalized case split.
  Any exhaustive search that fixes a distinguished block. · Marking the block refines the quotient to 2145 joint orbits. · artifacts/joint-orbit-census-20260808/manifest.json and independent checker. · None under the same group action; use the certified joint frontier.
• Claim that the root-intersection inequalities eliminate complete joint root orbits.
  All 2145 joint orbits. · Every internal excess S=0,...,5 retains at least ten necessary profiles. · Profile counts 10,12,15,17,20,22 by S. · A stronger sound invariant that makes the profile set empty for a specific orbit.
• Use agreement with the Terra scratch count as independent validation.
  The 2145 census. · Model agreement is not evidence. · Validation instead comes from the separately written deterministic checker and Burnside computation. · Not applicable.

• Proof-producing calibration of the first certified joint skeleton/root orbit.
  It directly measures whether the complete frontier supports a feasible negative-certificate campaign and could also expose a checked 54-cover. · Generate exact pair counters for partition [15], root 01234, then run one fixed proof-producing solver budget and replay the certificate. · high · open

Objective: Determine whether one exact-pair joint representative is proof-tractable.

First action: Implement the versioned pair-target generator for manifest partition [15], root mask 31, using residual targets 4+s_xy inside the root and 5+s_xy otherwise.

Stop condition: Stop or redirect on UNKNOWN, unreplayed UNSAT, semantic-control failure, or prohibitive proof-byte forecast; promote only a model accepted by both cover checkers or a replayed proof for the explicitly local leaf.

• Create a versioned exact-pair extension of the root-block CNF generator.
• Use manifest partition [15], matrix SHA-256 3b082585065ae705d6e65567f78076e2d3f4d3ba572b6c5907bbace8516a70f9, root mask 31.
• Encode residual pair targets 4+s_xy for root-internal pairs and 5+s_xy otherwise.
• Run one fixed proof-producing calibration and independently replay any UNSAT proof.
• Do not scale the 2145-case frontier until proof size, replay cost, and cube semantics pass controls.

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

GPT-5.6 Sol principal designed, implemented, and audited the durable work after reviewing advisory leads from two GPT-5.6 Terra delegates. Deterministic Python 3.12.3 produced the orbit census, independent checker, Burnside enumeration, arithmetic profiles, and mutation controls. The computational-researcher experiment harness recorded commands, limits, hashes, and resource use. Web search checked the maintained and historical status. No CAS, proof assistant, SAT solver, or cloud lab was used in this epoch.; orchestration: gpt-5.6-sol principal with gpt-5.6-terra delegates.

Duration

1351.2s

Review state

not a result claim

Attempt ID

covering-c1553-20260808-185948-86dc42

No human review recorded.