Exact pair-count compatibility test coupling all 353 surviving coincident-mark interfaces to the three omitted triangle/doubled-edge pair-excess skeleton profiles.

skeleton-conditioned pair-count feasibility

Represent each residual skeleton by triangle and doubled-edge exact-cover selectors, then couple it to exact pair multiplicities, point degrees, and coverage lower bounds for the two root-only and neither-root block categories.

Hypothesis: At least one of the 1059 coincident-mark interface/profile cells is infeasible in the exact pair-count relaxation.

Test: Solve all 353 by 3 exact integer feasibility cells and independently verify a complete integral witness for every feasible cell; require separate certification before accepting any infeasible result.

Every possible rejection predicted by this relaxation is falsified by an explicit integral assignment. Because the relaxation omits decomposition into actual blocks, these assignments are only positive relaxation witnesses and cannot support a covering claim.

• Every one of the 1059 scoped interface/profile cells has an integral solution to the declared necessary pair-count relaxation.
• The independent checker verified 82,602 pair balances and 41,301 category point-degree equations.
• No coincident-mark interface or pair-excess skeleton profile is excluded by this result.
• The exact covering number remains unresolved in the maintained range 54 to 55.

• Final captured experiment 20260812-160649-6c5af0: return code 0, duration 54.936 seconds, peak child memory 80,364 KiB.
• sha256sum -c artifacts/triangle-skeleton-count-compatibility-20260812/manifest.sha256: every bound file passed.
• Independent check: 1059 cells, 82,602 pair balances, 41,301 point-degree equations.
• Mutation controls rejected a broken pair balance, a deleted ledger cell, and an incorrect profile component list.
• Result, witness ledger, and telemetry regenerated byte-identically.

• .proof-experiments/20260812-160649-6c5af0: all 1059 cells feasible; independent checks, mutations, regeneration, and manifest validation passed.
• .proof-experiments/20260812-160348-d08fff: failed control exposed volatile timing in regeneration comparison; no mathematical claim was taken from it.

checkers/check_triangle_skeleton_count_compatibility_v1.py uses only the Python standard library and independently verifies the complete witness ledger without SciPy or HiGHS.

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 forced-link analysis for C(12,6,4) -> profile-wide link information should prune more strongly than isolated margins -> pair-count profile coupling rejected zero of 1059 cells, so any useful transfer must retain literal block structure.

• The declared pair-count equations are necessary for every genuine cover in the scoped triangle-interface branches.
  notes/triangle-skeleton-count-relaxation-lemma-20260812.md and independent arithmetic reconstruction. · The 353 surviving coincident-mark interfaces under profiles [3^5], [3^3,2^3], and [3,2^6]. · proved
• All 1059 scoped cells satisfy those necessary equations integrally.
  Complete witness ledger SHA-256 3297d2f314bdfd1b2c66227bb7c7460146ac82b0f7fd7f10351c4e5aa5599b4c and independent check. · Pair-count relaxation only. · computed

• Use exact profile-conditioned pair and point counts alone to eliminate a capacity-surviving coincident-mark interface.
  All 353 interfaces crossed with all three omitted profiles, exactly 1059 cells. · Every cell has an independently checked integral witness. · artifacts/triangle-skeleton-count-compatibility-20260812/independent-check.json · Provide a proved constraint depending on actual block decomposition or higher-order incidence, not another consequence of the same pair and point marginals.

• Canonical complete 18-block root-link pilot.
  Fixing a full point link retains literal block information and follows the successful certificate architecture for C(12,6,4). · After scope approval, enumerate a bounded tranche in each forced degree profile and compare matched completion propagation with the pair-normalized baseline. · high · open
• Constructive degree-preserving move family beyond prior neighborhoods.
  A defect reduction below nine or a 54-cover is directly valuable and materially distinct from local exclusion. · Define one move family not expressible as the exhausted 2-for-2 or tested 4-for-3 moves and scan a fixed small tranche around the defect-nine seed. · normal · open

Objective: Determine whether complete canonical point-link fixing gives a measurable propagation advantage over pair normalization.

First action: Obtain human approval for the exact bounded pilot scope, then write a protocol normalizing the two forced link-degree profiles and fixing matched solver limits.

Stop condition: Stop on canonical-coverage disagreement, less than a twofold matched propagation improvement, all pilot leaves remaining UNKNOWN, or any proof-replay failure.

• Obtain human approval for the exact scope of a bounded canonical full-root-link pilot before dispatching it.
• Normalize the excess neighbor or neighbor pair in degree profiles (7,5^13) and (6,6,5^12), then enumerate a small hash-bound tranche of canonical 18-block C(14,4,2) links.
• Measure matched completion propagation against the multiplicity-five-pair baseline; continue only on at least a twofold reduction or a directly checked SAT/UNSAT leaf.
• Keep a constructive route active using a degree-preserving move family outside the exhausted 2-for-2 and tested 4-for-3 neighborhoods.

• 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 served as principal investigator, designed and audited the experiment, and interpreted the result. Two injected GPT-5.6 Terra delegates supplied advisory reconnaissance; their agreement was not treated as validation. Python 3.12.3, SciPy 1.11.4 with bundled HiGHS, a separate Python-standard-library checker, SHA-256, and the computational-researcher experiment harness were used. No subagent was spawned. No SAT solver, CAS, proof assistant, cloud lab, installation, system change, or external write was used.; orchestration: gpt-5.6-sol principal with gpt-5.6-terra delegates.

Duration

1198.8s

Review state

not a result claim

Attempt ID

covering-c1553-20260812-161148-9c7ef6

No human review recorded.