Complete symmetry classification and point-capacity filtering of the missing triangle/coincident-mark two-link interface.

missing-interface capacity classification

Expand one distinguished coincident mark over the validated 94 unmarked six-triple families, quotient under S_13 with colored incidence graphs, then reject types violating q_z <= 3*(5+[z=mark]-h_z).

Hypothesis: The complete coincident-mark interface quotient has at most 2000 types and at least one type violates the necessary point-capacity inequality.

Test: Canonicalize all 94*13 marked representatives, reconstruct every marked-point orbit independently with VF2 automorphisms, and recompute the capacity decision for every canonical row.

The exact orbit count, per-type decisions, regeneration, and mutation controls agree independently. The odd-partition argument proves A/B case coverage. However, a local interface rejection is not an outer-profile exclusion, and every profile retains untested survivors, so the result is genuine structural progress rather than a candidate exact optimum.

• There are exactly 575 S_13-isomorphism types of six distinct triples with union 13 and one distinguished coincident mark.
• Exactly 222 of those 575 types violate q_z <= 3*(5+[z=mark]-h_z); exactly 353 survive.
• The prior A corpus and the new B corpus together contain a selectable multiplicity-six interface for all 41 pair-excess skeleton profiles.
• The combined capacity-surviving A/B frontier has 2,171 + 353 = 2,524 canonical types.
• The exact covering number remains unresolved in the range 54 <= C(15,5,3) <= 55.

• Producer experiment 20260812-151936-c1ed0e completed in 1.028 seconds with counts 575/222/353.
• Independent experiment 20260812-152108-f2a4f7 mapped 575 canonical rows, checked 7,475 point cells, and reproduced 222/353.
• Mutation experiment 20260812-152232-130bb2 rejected truncation, cleared-violation, and capacity mutations after hashes were rebound.
• Regeneration experiment 20260812-152232-ebfdb8 reproduced five decisive files byte-identically.
• sha256sum -c artifacts/two-link-coincident-mark-capacity-20260812/manifest.sha256 passed for every bound artifact.

• .proof-experiments/20260812-151936-c1ed0e: producer PASS, 575 canonical types, 222 rejected, 353 surviving.
• .proof-experiments/20260812-152108-f2a4f7: independent checker PASS, 575 orbit mappings and 7,475 checked cells.
• .proof-experiments/20260812-152232-130bb2: three semantic or coverage mutations rejected.
• .proof-experiments/20260812-152232-ebfdb8: five producer files regenerated byte-identically.

checkers/check_two_link_coincident_mark_capacity_v1.py does not import the producer. It reconstructs the 94 automorphism groups with NetworkX VF2, derives marked-point orbits, maps every canonical graph independently, and recomputes all degree, shadow, q_z, d_z, and violation data.

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

• Forced-link covering certificates -> classify literal local interfaces before SAT -> the coincident-mark quotient was only 575 types and the capacity lemma removed 222 without solving.
• Odd-component parity in 2-regular multigraphs -> predict every A-omitted 15-vertex profile contains a triangle -> all three omitted profiles are covered by the B interface.

• The coincident-mark interface has exactly 575 canonical types, with 222 capacity rejects and 353 survivors.
  Receipt SHA-256 cd0705a25d950041fe02af9243b3c46a3cd434429c5563b2aa457ba185bf6c05 and independent-check status PASS. · All six-common-triple union-13 interfaces with one coincident mark. · computed
• A/B interfaces cover all 41 pair-excess skeleton profiles.
  The prior scope audit identifies the A omissions as [3^5], [3^3,2^3], and [3,2^6]; each contains a triangle, as formalized in notes/two-link-interface-union-scope-lemma-20260812.md. · All unlabelled weighted-degree-two pair-excess multigraphs on 15 points. · proved
• For each B interface point z, q_z <= 3*(5+[z=mark]-h_z) is necessary.
  Each of the d_z residual 4-subsets containing z covers at most three required pairs incident with z. · Either residual side of a triangle/coincident-mark multiplicity-six pair. · proved

• The 222 rejected B types can occur as the selected triangle interface in a 54-cover.
  The exact 222 canonical rows in rejected-types.jsonl. · At least one point has more unshadowed incident-pair demand than its residual incidences can cover. · Complete reject ledger and independent direct recomputation. · A counterexample to the capacity lemma or an independently demonstrated producer/checker semantic error.
• A seven-common-triple doubled-edge corpus is required solely to obtain coverage of all 41 outer skeleton profiles.
  Case coverage only, not potential pruning usefulness. · Any A-omitted partition of odd total 15 into only 2s and 3s contains a triangle and therefore supplies a B edge. · notes/two-link-interface-union-scope-lemma-20260812.md and branch-scope-audit.json. · An explicit pair-excess skeleton type lacking both a cycle of length at least four and a triangle.

• Triangle-skeleton compatibility classification.
  It is the cheapest complete-family opportunity to quotient or eliminate the 353 B survivors before exact solving. · Canonical compatibility ledger for [3^5], [3^3,2^3], and [3,2^6], rooted at the marked triangle edge. · high · open
• Constructive repair beyond exhausted 2-for-2 neighborhoods.
  A checked 54-cover is terminal and remains materially distinct from the exclusion route. · Define a degree-preserving move family not contained in prior 2-for-2 and fixed 4-for-3 scans, then pre-count it before optimization. · normal · open
• Alternative exact residual encoding.
  Four prior representative A kernels remained UNKNOWN, so a matched encoding discriminator is required before further proof-scale solving. · Compare one fixed kernel under a materially different PB, meet-in-the-middle, or cubed encoding at the same 5,000-conflict budget. · low · open

Objective: Bulk-prune the complete 353-type coincident-mark frontier through exact outer-skeleton compatibility.

First action: Generate a hash-bound compatibility ledger for [3^5], [3^3,2^3], and [3,2^6], rooted at the marked triangle edge, and independently reconstruct every accepted cell.

Stop condition: Stop on producer/checker coverage disagreement or zero measurable quotient/elimination; switch immediately to direct witness validation or complete proof replay on a decisive result.

• Generate an exact compatibility ledger between the 353 B survivors and the three triangle-only skeleton profiles.
• Independently reconstruct the ledger and measure whole-class quotienting or rejection before producing SAT kernels.
• Retain doubled-edge seven-common-triple interfaces only as an alternative decomposition, not as a prerequisite for 41-profile coverage.
• On any 54-block witness, stop and check all 455 triples directly; on any complete UNSAT leaf family, require replayed DRAT/LRAT evidence and exhaustive split coverage.

• 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. The two injected GPT-5.6 Terra delegates supplied advisory reconnaissance only; their agreement was not treated as validation, and the suggested fixed-link rerun and repeated PB calibration were rejected after artifact audit. Deterministic work used Python 3.12.3, NetworkX 3.3, nauty-shortg 2.8.8, SHA-256, the computational-researcher experiment harness, an independent VF2 checker, mutation controls, and byte-identical regeneration. The web reader and curl were used for source checking, but sandbox DNS prevented a fresh live-page retrieval; the supplied official status and pre-acquired same-day sources were retained. No SAT solver, CAS, proof assistant, cloud lab, package installation, system change, external write, or subagent was used in the decisive experiment.; orchestration: gpt-5.6-sol principal with gpt-5.6-terra delegates.

Duration

1144.6s

Review state

evidence receipt failure; not durable progress

Attempt ID

covering-c1553-20260812-152958-ccebf5

No human review recorded.