Computed and independently verified a complete rooted link-block orbit quotient for the pair-excess skeleton [3,2^6], replacing 1001 labelled first-block choices by 14 exhaustive representative branches.

pair-excess-skeleton link-orbit case split

Fix a doubled-edge endpoint as root, compute the rooted skeleton stabilizer by generator closure, and quotient all four-point remainders of blocks through the root into exact group orbits.

Hypothesis: For the canonical [3,2^6] pair-excess skeleton, after fixing point 0 as an endpoint of a doubled edge, the rooted stabilizer has exactly 14 orbits on the 1001 possible four-point remainders of a selected block through point 0.

Test: Compare a generator-closure orbit census with an independent direct-product construction of S3 x (S2 wreath S5), requiring exact agreement on all 1001 members, orbit representatives, features, sizes, group hash, and mutation rejection.

The result is exact because both implementations enumerate the full finite universe, agree on the group hash and all orbit members, reproduce the orbit sizes by closed formulas, and fail closed under mutations. Its scope is local to one skeleton type, so only progress rather than candidate status is justified.

• The rooted [3,2^6] skeleton stabilizer has order 23040.
• Its action on the 1001 four-point remainders of blocks through a doubled-edge endpoint has exactly 14 orbits.
• The 14 representative-unit branches collectively cover every global extension in that rooted skeleton fibre.
• No representative extension branch was excluded or constructed in this epoch.

• Producer command recorded in experiment 20260810-121634-3d56a1 returned orbit_count=14, stabilizer_order=23040, and universe_size=1001.
• Result SHA-256 680419c565b8642a3b442fb0d6323879eaf21760415480681bdb0f5d1be9f024.
• Independent experiment 20260810-121647-22bbf3 reported valid=true for twelve checks and five mutation controls.
• Independent-check SHA-256 74d793891a2b52e818ace21a8120f14cd64de961dc0673f83481d3905ffe23c3.
• sha256sum -c artifacts/skeleton-326-root-link-orbits-20260810/manifest.sha256 accepted every listed artifact.

• .proof-experiments/20260810-121634-3d56a1: generator closure computed a 23040-element stabilizer and 14 orbits covering 1001 masks in 2.532 seconds.
• .proof-experiments/20260810-121647-22bbf3: independent direct-product and feature-fibre checker accepted all checks and mutations in 0.925 seconds.

checkers/check_skeleton_326_root_link_orbits_v1.py constructs S3 x (S2 wreath S5) directly rather than by generator closure, derives orbits from feature fibres and closed size formulas rather than action traversal, checks exact disjoint coverage, and rejects five mutations.

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

• C(12,6,4) certificate case splitting -> predict that a small symmetry-complete local frontier can precede proof solving -> observed an exact 14-branch frontier for one C(15,5,3) skeleton fibre.
• Wreath-product symmetry of paired components -> predict classification by complete/singleton pair counts -> observed feature fibres exactly equal all generated orbits.
• Certificate-first SAT practice -> reject a longer monolithic 25000-conflict rerun without structural change -> selected an exact quotient before solver scale-up.

• The automorphism group of the [3,2^6] skeleton fixing a doubled-edge endpoint is S3 x (S2 wreath S5) and has order 23040.
  Generator-closure and direct-product constructions have identical permutation-set hashes. · Canonical labels with doubled edge 01, other doubled edges 23,45,67,89,10-11, and triangle 12-13-14. · proved
• There are exactly 14 rooted-stabilizer orbits on the 1001 possible four-point block remainders.
  Exact orbit traversal and independent feature-fibre reconstruction agree member-for-member and total 1001. · Blocks containing rooted point 0 in the [3,2^6] skeleton fibre. · computed
• UNSAT proofs for all 14 representative-unit branches would exclude the entire [3,2^6] skeleton fibre.
  Every extension contains a block through point 0, and a rooted skeleton automorphism sends its remainder to one representative while preserving all constraints. · Conditional on correct proof-producing encodings and replayed proofs for every representative branch. · proved

• Treat a larger 25000-conflict monolithic root-link-cap run as this epoch's contribution.
  The previously UNKNOWN globally complete root-link-cap encoding without a new split or decisive certificate. · It is only a larger arbitrary cutoff and cannot produce evidence unless it terminates with a checked model or replayed proof. · Prior 5000-conflict runs remained LIMIT; the Terra recommendation supplied no new proof artifact. · A structural decomposition, measured encoding improvement, directly checked SAT model, or complete replayable UNSAT proof.
• Infer any branch exclusion or numerical covering bound from the 14-orbit census.
  All 14 representatives and the global value C(15,5,3). · The census establishes only symmetry coverage; no extension formula was solved. · The result and independent checker explicitly limit their scope. · A directly checked 54-cover or replayed UNSAT proofs for a complete relevant branch union.

• Proof-producing [3,2^6] representative extension leaves.
  The exact 14-branch quotient is now small enough for a one-leaf propagation pilot and potentially a complete certificate set. · Generate the pair-upper-bound base CNF, independently reconstruct it, proof-smoke it, and run the unique size-1 orbit representative. · high · open
• Canonical pair-normalized global SAT branches.
  They cover the entire problem through four common-family types, but all remain UNKNOWN and the type-2 cap transfer regressed. · Only revisit after a materially new symmetry or encoding improvement, not a larger cutoff. · normal · open
• Constructive coverage-aware degree-preserving search.
  A defect below the validated plateau of 10 would be immediately informative and a 54-cover terminal. · Design a new move family coupled to exact skeleton pair targets before another bounded basin run. · low · open

Objective: Determine whether the 14-branch [3,2^6] quotient yields tractable proof-producing global extension leaves.

First action: Implement triple-coverage plus pair-upper-target CNF generation for [3,2^6], derive the size-1 representative unit branch [0,1,12,13,14], and independently reconstruct every clause before solving.

Stop condition: Stop on any semantic mismatch; validate SAT immediately; replay UNSAT completely; redirect on UNKNOWN without a predeclared propagation improvement.

• Build a forward-only pair-upper-bound CNF enforcing triple coverage and the exact [3,2^6] pair targets.
• Independently reconstruct the base formula and all 14 representative variable mappings.
• Run DRAT-to-LRAT smoke and wrong-formula controls before any live leaf.
• Pilot the size-1 representative branch under a predeclared small conflict cap.
• Validate SAT directly as a 54-cover; accept UNSAT only after full LRAT replay.
• Redirect if the first leaf remains UNKNOWN without a measured propagation improvement.

• 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 ↗
• https://arxiv.org/abs/math/9502238 ↗

GPT-5.6 Sol acted as principal investigator. GPT-5.6 Terra delegates supplied advisory reconnaissance only; their model assertions were not treated as evidence, and the selected 14-orbit assertion was independently audited. Python 3.12.3 performed exact group and subset enumeration, independent checking, and mutation tests. The computational-researcher harness recorded commands, limits, logs, versions, and hashes. SHA-256 bound the artifacts. Web search checked the maintained status and narrow prior-art trail. No SAT solver, CAS, proof assistant, or checkpointed lab compute was used in this epoch.; orchestration: gpt-5.6-sol principal with gpt-5.6-terra delegates.

Duration

917.8s

Review state

evidence receipt failure; not durable progress

Attempt ID

covering-c1553-20260810-122112-49bd4e

No human review recorded.