Computed and independently audited the exact automorphism action of one source-verified 12-block C(14,5,2) root link on the first residual 6-block coordinate, using the precise incidence-bit ordering encoded by the retained epoch-50 CNF.

fixed-link incidence SAT with canonical first-block orbit ownership

Compute the colored-incidence automorphism group, quotient the 3003 possible first residual blocks by that group, and prove that one orientation of every completion has a representative first column.

Hypothesis: The published 12-block C(14,5,2) link has automorphism group exactly generated by (4 5), and selecting the least block of every induced orbit under the epoch-50 CNF membership-bit order gives an existence-complete first-column normalization.

Test: Enumerate the colored-incidence automorphism group independently with NetworkX and nauty, reconstruct every orbit of the 3003 residual 6-blocks, and exhaust the 792^2 possible two-minimum obstructions to canonical coverage.

Two materially independent graph encodings agree on the automorphism group, every block orbit was reconstructed, the canonical-coverage proof has a finite independently checked obstruction audit, and corrupted artifacts were rejected. These facts support the scoped reduction but nothing global.

• For the fixed source link, the colored point-block incidence automorphism group has order two and is generated by swapping points 4 and 5.
• Under the epoch-50 incidence-bit column order, the 3003 possible first residual blocks form 1419 fixed and 792 paired orbits, giving exactly 2211 existence-complete representative-first cubes.
• The exact covering range remains 30<=C(15,6,3)<=31.

• python3 scripts/audit_fixed_link_orbits_v1.py --link artifacts/epoch50-20260810/published-c1452-link.txt --output artifacts/epoch51-20260810/fixed-link-orbit-manifest-v2.json
• python3 checkers/check_fixed_link_orbits_v1.py --link artifacts/epoch50-20260810/published-c1452-link.txt --audit artifacts/epoch51-20260810/fixed-link-orbit-manifest-v2.json --receipt artifacts/epoch51-20260810/fixed-link-orbit-checker-receipt-v2.json
• python3 checkers/test_fixed_link_orbits_fail_closed_v1.py --checker checkers/check_fixed_link_orbits_v1.py --link artifacts/epoch50-20260810/published-c1452-link.txt --audit artifacts/epoch51-20260810/fixed-link-orbit-manifest-v2.json --receipt artifacts/epoch51-20260810/fixed-link-orbit-fail-closed-receipt-v2.json
• sha256sum -c artifacts/epoch51-20260810/SHA256SUMS passed for every listed decisive artifact.

• .proof-experiments/20260810-111414-c33a3a: corrected producer found group order 2, 1419 fixed orbits, 792 paired orbits, 2211 total, and zero canonical obstructions.
• .proof-experiments/20260810-111441-e82235: nauty/direct checker independently accepted the complete v2 manifest.
• .proof-experiments/20260810-111456-5f5833: subprocess harness accepted the baseline and rejected four corrupted inputs.

checkers/check_fixed_link_orbits_v1.py uses nauty/dreadnaut rather than NetworkX for group order, applies the generator directly to all 3003 blocks, reconstructs the orbit manifest under the exact CNF order, checks the fixed-count identity C(12,6)+C(12,4)=1419, and exhausts 627264 obstruction pairs.

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

• Group-action canonical orientation -> predict one representative first coordinate per completion orbit -> observed an exact existence-complete quotient, but only 1.3582x because 1419 blocks are fixed.
• SAT orbitope ordering -> predict that an abstract tuple-order manifest can be used directly -> falsified; the CNF uses membership-bit order, requiring the corrected v2 manifest.

• The fixed link automorphism group is exactly {identity,(4 5)}.
  NetworkX GraphMatcher and nauty/dreadnaut independently returned group order two; direct application of (4 5) preserves all twelve blocks. · Fixed link SHA-256 fa875193588a0313e7f87e7f38f95fef596177cf7842c6ba0d335c90604560bc. · computed
• The induced action on C(14,6) has 1419 fixed and 792 paired orbits.
  Complete direct reconstruction of all 3003 blocks and the independent formula C(12,6)+C(12,4)=1419. · All 6-subsets of points 1 through 14. · proved
• The 2211 membership-bit-order representative-first cubes preserve existence for fixed-link completions.
  Canonical-coverage contradiction lemma plus exhaustive audit of 627264 possible ordered two-minimum obstructions. · Residual families for the fixed link, with columns sorted by the epoch-50 incidence-bit order. · proved

• Attribute a twofold first-block reduction to the fixed link's point symmetry.
  The 3003 candidate first residual blocks for the source link. · The involution fixes 1419 blocks; only 1584 blocks are paired, leaving 2211 orbits and a 1.3582089552x reduction. · fixed-link-orbit-manifest-v2.json and independent checker receipt. · A larger independently verified automorphism group for a different complete link class or an additional sound canonical coordinate.
• Use the tuple-lexicographic v1 representative manifest as first-column ownership for the epoch-50 CNF.
  The unversioned epoch-51 v1 manifest and its three experiment records. · The CNF sorts 14-bit incidence columns with 0<1, which differs from tuple lexicographic order. · artifacts/epoch51-20260810/orbit-audit-v1-supersession.json · None; use the corrected v2 membership-bit-order manifest.
• Infer a global lower bound from exhaustive work on this fixed link.
  All possible 30-block C(15,6,3) covers. · A hypothetical cover may have a nonisomorphic root link or one of four other root-link degree types. · The fixed-link scope bound in both producer and checker receipts. · An independently complete catalogue of every possible root-link isomorphism class with a replayed terminal certificate for every extension leaf.

• Two-leaf fixed-link terminal eligibility test.
  It is the cheapest way to kill or retain this local route after the ownership audit. · Materialize the least and greatest v2 representatives as first-column unit clauses, reconstruct both formulas independently, and run externally wall-capped 10-second CaDiCaL/LRAT jobs. · normal · open
• Globally owned fixed-first profile frontier with materially different proof production.
  Unlike a fixed link, the existing 249-profile frontier covers all normalized simple 30-covers; 248 profiles remain unresolved. · Identify a proof-producing encoding change or validated assumption-binding scheme before compiling one smallest unresolved profile. · high · open

Objective: Determine whether the fixed-link route can emit any terminal evidence before redirecting to a globally owned frontier.

First action: Generate two CNFs by appending materialized first-column units for the least and greatest v2 representatives to the retained compact fixed-link formula, then independently compare all base clauses and unit semantics.

Stop condition: Stop and redirect on any hash or ordering mismatch, verifier disagreement, incomplete proof, or two UNKNOWN solver results; stop the campaign only on a directly checked 30-cover or a complete globally replayed UNSAT frontier.

• If retaining the local route, materialize only two v2 representative cubes and independently reconstruct their unit clauses and lex semantics.
• Accept only a directly checked 30-cover or an UNSAT proof accepted by both lrat-check and CakeLPR; classify UNKNOWN as a route failure.
• Redirect after two UNKNOWN samples to a globally owned frontier, such as a materially changed proof-producing encoding of the 248 unresolved fixed-first profiles.

• https://www.coveringrepository.com/systems.aspx?k=6&t=3&m=-1 ↗
• https://ljcr.dmgordon.org/cover/show_cover.php?k=5&t=2&v=14 ↗
• https://arxiv.org/abs/1706.06825 ↗

GPT-5.6 Sol served as principal investigator. GPT-5.6 Terra challenger-prior-art and experiment-verification delegates supplied advisory memos only; every relied-upon claim was regenerated in the main workspace. Python 3.12.3, NetworkX 3.3 GraphMatcher, nauty/dreadnaut 2.8.8+ds-5, the computational-researcher run_experiment harness, SHA-256, shell diagnostics, and web search were used. No SAT solver, CAS, proof assistant, cloud lab, external proof service, or human validator produced terminal evidence this epoch.; orchestration: gpt-5.6-sol principal with gpt-5.6-terra delegates.

Duration

1342.8s

Review state

evidence receipt failure; not durable progress

Attempt ID

covering-c1563-20260810-112028-9e4882

No human review recorded.