Joined the exact multiplicity signature of the q=6 fixed-point (14,4,2) link bin to every one of the 41 pair-excess skeleton types through the triple-excess budget identity.

fixed-point link / pair-excess budget join

A fixed-point link pins e_pxy=mu_xy-1; comparing it with sum_{T containing xy} e_T=2+3s_xy turns the multiplicity-six pair in a q=6 link into a forced triangle in the weighted-degree-two pair-excess skeleton.

Hypothesis: The exact triple-excess budget pinned by a q=6 fixed-point link makes at least one of the 41 pair-excess skeleton types incompatible with that rooted link branch.

Test: Reconstruct 121 checked q=6 links, compute their pinned pair multiplicities, and compare the resulting lower bounds on s_xy with every rooted placement in all 41 independently certified skeleton matrices.

The multiplicity-six link pair contributes five units to one exact triple-excess row, while a zero-excess skeleton pair has budget only two. It therefore forces a positive skeleton edge between the root's two unit-excess neighbors; weighted degree two then forces an isolated triangle. The complete partition census makes the 21/20 classification exact.

• In any hypothetical 54-block C(15,5,3) cover, a point whose fixed link is in the q=6, degree-6,6,5^12 bin lies in a unit-weight triangle component of the pair-excess skeleton.
• Exactly 21 of the 41 unrooted pair-excess skeleton types are compatible with the q=6 fixed-point necessary-budget join; the 20 types without a triangle are incompatible.
• The 121 checked q=6 link representatives yield 2,541 compatible and 2,420 incompatible cells against the 41 skeleton types.

• python3 scripts/fixed_point_q6_skeleton_budget_v1.py ... returned checked_cells=4961, compatible_types=21, incompatible_types=20.
• python3 checkers/check_fixed_point_q6_skeleton_budget_v1.py ... returned valid=true and independently reconstructed 121 links and 41 matrices.
• sha256sum -c artifacts/fixed-point-q6-skeleton-budget-20260812/manifest.sha256 returned OK for every entry.
• Fresh temporary regeneration produced byte-identical result, independent-check, and mutation-control files.

• .proof-experiments/20260812-034045-62d613: producer completed in 0.136 seconds and found 21 compatible versus 20 incompatible types per q=6 link.
• .proof-experiments/20260812-034053-a6da50: independent checker completed in 0.130 seconds, agreed on 4,961 cells, and rejected three mutations.

checkers/check_fixed_point_q6_skeleton_budget_v1.py reconstructs q=6 links from residual indices and skeleton matrices from component partitions, rather than trusting producer expansions.

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

• Certified-link decomposition in C(12,6,4) -> predict fixed-point link multiplicities expose complete branch constraints in C(15,5,3) -> the q=6 multiplicity forces a triangle and removes 20 of 41 skeleton types in that branch.

• A q=6 fixed-point link forces its root and the two degree-six link points to form an isolated unit-weight triangle in the pair-excess skeleton.
  artifacts/fixed-point-q6-skeleton-budget-20260812/derivation.md and independent-check.json · Any hypothetical 54-block C(15,5,3) cover with that point-link bin. · proved
• Twenty-one of the complete 41 skeleton partitions contain a 3-cycle and twenty do not.
  Complete result type_rows independently rebuilt from all 41 component partitions. · Unrooted loopless weighted-degree-two multigraphs on 15 vertices. · computed
• All 121 checked q=6 representatives have the required multiplicity-six pair signature and the same 21/20 skeleton compatibility split.
  result.json and independent-check.json; 4,961 cells checked. · The hash-bound 121-representative q=6 control corpus. · computed

• A q=6 fixed-point link can be joined to a pair-excess skeleton type without a triangle component.
  The exact necessary triple-excess budget in any hypothetical 54-cover. · The multiplicity-six link pair forces a positive edge between the root's two unit-excess neighbors, hence an isolated triangle. · derivation.md; complete 41-type result and independent checker · Only an error in the exact counting identity, the 41-type component census, or the fixed q=6 definition; all are independently checked.
• Enumerate the full q=6 point-link catalogue before extracting any global consequence.
  Resource allocation for the current q=6 budget join. · The compatibility criterion depends only on the common multiplicity-six signature, not link isomorphism class. · All 121 diverse representatives share the same high-pair signature and complete compatibility map. · A new global constraint that depends on internal q=6 link structure beyond its pair multiplicity signature.

• Compile triangle-forcing as at-most-three neighbor-triple constraints in a complete no-triangle skeleton CNF.
  This transfers the verified 20-of-41 structural reduction into the certificate-producing proof route and satisfies Terra's requirement for a nonzero semantic delta before solver time. · Generate the 15-cycle skeleton branch with and without the 15 derived at-most-three constraints, independently reconstruct the primary models on small boundary controls, then run three matched seeds only if clauses are genuinely added. · high · open
• Generalize the static budget catalogue across all mutual multiplicities q of the two degree-six link points.
  The same inequality predicts that q>=4 forces a triangle, while q<=3 supplies no edge force; an exact bin interface could partition every no-triangle skeleton branch without enumerating link isomorphism classes. · Write a symbolic q=1..6 budget table and independently exhaust integer link-degree excess allocations before changing any global encoding. · normal · open

Objective: Measure whether the triangle/q-bin lemma improves a complete no-triangle proof branch.

First action: Run `python3 scripts/pair_cube_cnf_v1.py --help`, freeze the existing 15-cycle branch inputs, and predeclare a static clause-delta plus three-seed A/B protocol for adding multiplicity-at-most-three constraints on its 15 neighbor triples.

Stop condition: Redirect if the semantic delta is zero, the independent model map disagrees, the three-seed improvement is below 20 percent, or any proof output fails DRAT/LRAT replay; promote immediately on a directly checked 54-cover or complete replayed UNSAT certificate.

• Audit a complete no-triangle skeleton CNF for a nonzero semantic delta from the derived at-most-three neighbor-triple constraints.
• If the delta is nonzero, run the fixed three-seed matched A/B pilot and retain the encoding only for at least 20 percent improvement or a checked SAT/replayed LRAT result.
• Do not resume the full q=6 link catalogue or the fixed-pair link rerun; neither is needed for this branch reduction.

• 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/2607.23766 ↗

GPT-5.6 Sol acted as principal investigator. Two pre-epoch GPT-5.6 Terra delegates supplied advisory route-audit memos; Sol rejected their stale fixed-pair rerun and did not treat model agreement as validation. Python 3.12.3 performed deterministic enumeration and independent checking; GNU sha256sum/cmp verified hashes and byte-identical regeneration; the web reader checked the maintained LJCR status and searched for newer exact work. No SAT solver, CAS, proof assistant, cloud lab, package installation, external publication, or remote write was used in this epoch.; orchestration: gpt-5.6-sol principal with gpt-5.6-terra delegates.

Duration

1141.0s

Review state

evidence receipt failure; not durable progress

Attempt ID

covering-c1553-20260812-035029-c33e7d

No human review recorded.