{
  "schemaVersion": "1.0",
  "kind": "selected-constructive-pilot",
  "startedAt": "2026-10-15T13:41:43.249Z",
  "finishedAt": "2026-10-15T13:41:43.272Z",
  "elapsedMs": 21.524667,
  "intakeId": "atlas-transversal-pilot-20261015",
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  "intakeSha256": "114cca23afe415897e034ff7e7b7e8bcfb6f36850c2d5af6ad1e04a33f80eaec",
  "node": "v25.2.1",
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    "fixed_matching_cross_edges": [
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    "claimed_invariants": {
      "minimum_maximal_matching": 15,
      "independent_domination": 16
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    "status": "unrefereed candidate; bundled proof tree verified"
  },
  "result": {
    "before": {
      "routes": [
        {
          "id": "enumerate-transversals",
          "title": "Find and independently check an explicit transversal",
          "requires": [
            {
              "field": "transversalEncoding",
              "op": "eq",
              "value": true
            },
            {
              "field": "matchingSize",
              "op": "lte",
              "value": 20
            }
          ],
          "produces": [
            "boundedSearchAvailable"
          ],
          "action": "matching-transversal",
          "methods": [
            "serial-search",
            "bookkeeping",
            "prove-too-much"
          ],
          "source": {
            "url": "https://arxiv.org/html/2608.22498v1",
            "location": "Proposition 6; finite assignment enumeration supplied by this pilot"
          },
          "basis": "Enumerate all 2^|M| assignments, reconstruct each first satisfying selection, then check domination from the original edges without using the clauses.",
          "limit": "At most 20 matching edges and 1,048,576 assignments. An exhausted user budget is inconclusive. No-transversal excludes only this restricted route.",
          "ready": true,
          "missing": [],
          "score": 0
        },
        {
          "id": "external-order48",
          "title": "Prior-art stop: the order-48 inequality is already covered",
          "requires": [
            {
              "field": "degree",
              "op": "eq",
              "value": 3
            },
            {
              "field": "order",
              "op": "lte",
              "value": 48
            }
          ],
          "produces": [
            "knownGeneralResultApplicable"
          ],
          "methods": [
            "named-obstructions",
            "lemma-recombination"
          ],
          "source": {
            "url": "https://arxiv.org/html/2608.22498v1",
            "location": "Proposition 7, importing Zhang–Peitl–Szeider 2024"
          },
          "basis": "The cited preprint states gamma <= gamma_e for cubic graphs of order at most 48, using a minimum unsatisfiable bounded-occurrence CNF classification.",
          "limit": "Imported preprint and classification, not reverified here. Use this as a duplicate-research warning; the direct finite witness removes the imported classification from the pilot's conclusion.",
          "ready": true,
          "missing": [],
          "score": 0
        },
        {
          "id": "graph-to-sat",
          "title": "Reformulate domination as endpoint-transversal SAT",
          "requires": [
            {
              "field": "finiteSimple",
              "op": "eq",
              "value": true
            },
            {
              "field": "maximalMatching",
              "op": "eq",
              "value": true
            }
          ],
          "produces": [
            "transversalEncoding"
          ],
          "methods": [
            "reformulation",
            "distant-imports"
          ],
          "source": {
            "url": "https://arxiv.org/html/2608.22498v1",
            "location": "Definition 2 and Proposition 6"
          },
          "basis": "One Boolean variable selects an endpoint of each matching edge. A clause for every unmatched vertex requires a neighbour to be selected. Vertices adjacent to both endpoints of one edge give tautologies. Matched vertices are automatically dominated.",
          "limit": "SAT is equivalent to a dominating transversal of this matching, not to an arbitrary dominating set. The implemented equivalence also handles non-cubic finite simple graphs.",
          "ready": true,
          "missing": [],
          "score": 0
        },
        {
          "id": "matching-count-bound",
          "title": "Certify the supplied matching is minimum",
          "requires": [
            {
              "field": "maximalMatching",
              "op": "eq",
              "value": true
            },
            {
              "field": "matchingSize",
              "op": "eq",
              "otherField": "matchingLowerBound"
            }
          ],
          "produces": [
            "minimumMatchingByCounting"
          ],
          "methods": [
            "bookkeeping",
            "chaining"
          ],
          "source": {
            "url": "https://evidencepress.org/releases/txgraffiti-order48-successor/",
            "location": "Exact result for specialists"
          },
          "basis": "A maximal matching edge-dominates all edges. Each edge covers at most 2 Delta - 1 edges; ceiling(m/(2 Delta - 1)) is a lower bound. Equality with a checked witness establishes minimality.",
          "limit": "Works when this elementary lower bound is attained. Otherwise minimality remains unknown, not false.",
          "ready": true,
          "missing": [],
          "score": 0
        },
        {
          "id": "checked-domination-bound",
          "title": "Chain the checked witness into an ordinary-domination upper bound",
          "requires": [
            {
              "field": "witnessVerified",
              "op": "eq",
              "value": true
            }
          ],
          "produces": [
            "ordinaryUpperBoundCertified"
          ],
          "methods": [
            "chaining"
          ],
          "source": {
            "url": "https://arxiv.org/html/2609.10783v1",
            "location": "Section 1: definition of domination; Section 4: witness upper bound"
          },
          "basis": "Every vertex is either in the selected set or adjacent to it, so gamma(G) is at most the witness size.",
          "limit": "An upper bound is not an exact minimum. No lower bound is inferred from failure of a restricted search.",
          "ready": false,
          "missing": [
            {
              "field": "witnessVerified",
              "op": "eq",
              "value": true
            }
          ],
          "score": 0
        },
        {
          "id": "independent-upper",
          "title": "Independent domination transfers only an upper bound",
          "requires": [
            {
              "field": "independentWitnessVerified",
              "op": "eq",
              "value": true
            }
          ],
          "produces": [
            "independentWitnessGivesOrdinaryUpper"
          ],
          "methods": [
            "prove-too-much",
            "named-obstructions"
          ],
          "source": {
            "url": "https://arxiv.org/html/2609.10783v1",
            "location": "Section 8: ordinary versus independent domination"
          },
          "basis": "Every independent dominating set is dominating: gamma <= i. A lower bound on i is NOT a lower bound on gamma.",
          "limit": "Knowing i=16 does not prove gamma>=16. This invalid transfer is deliberately absent from the rule library.",
          "ready": false,
          "missing": [
            {
              "field": "independentWitnessVerified",
              "op": "eq",
              "value": true
            }
          ],
          "score": 0
        },
        {
          "id": "compare-matching",
          "title": "Decide that this graph is not an ordinary-domination counterexample",
          "requires": [
            {
              "field": "ordinaryUpperBoundCertified",
              "op": "eq",
              "value": true
            },
            {
              "field": "minimumMatchingByCounting",
              "op": "eq",
              "value": true
            },
            {
              "field": "dominationUpper",
              "op": "lte",
              "otherField": "matchingSize"
            }
          ],
          "produces": [
            "ordinaryCounterexampleRuledOut"
          ],
          "methods": [
            "lemma-recombination",
            "chaining"
          ],
          "source": {
            "url": "https://arxiv.org/html/2608.22498v1",
            "location": "Proposition 1 (dominating transversal reduction)"
          },
          "basis": "Combine a graph-checked dominating set of size at most |M| with a checked minimum maximal matching M. This proves gamma(G) <= gamma_e(G) for this exact graph.",
          "limit": "No universal statement about regular graphs; the two graph objects and parameter meanings must be identical.",
          "ready": false,
          "missing": [
            {
              "field": "ordinaryUpperBoundCertified",
              "op": "eq",
              "value": true
            },
            {
              "field": "dominationUpper",
              "op": "lte",
              "otherField": "matchingSize"
            }
          ],
          "score": 0
        }
      ],
      "chains": [
        {
          "id": "matching-count-bound",
          "requires": [
            {
              "field": "maximalMatching",
              "op": "eq",
              "value": true
            },
            {
              "field": "matchingSize",
              "op": "eq",
              "otherField": "matchingLowerBound"
            }
          ],
          "produces": [
            "minimumMatchingByCounting"
          ],
          "source": {
            "url": "https://evidencepress.org/releases/txgraffiti-order48-successor/",
            "location": "Exact result for specialists"
          },
          "limit": "Works when this elementary lower bound is attained. Otherwise minimality remains unknown, not false."
        },
        {
          "id": "graph-to-sat",
          "requires": [
            {
              "field": "finiteSimple",
              "op": "eq",
              "value": true
            },
            {
              "field": "maximalMatching",
              "op": "eq",
              "value": true
            }
          ],
          "produces": [
            "transversalEncoding"
          ],
          "source": {
            "url": "https://arxiv.org/html/2608.22498v1",
            "location": "Definition 2 and Proposition 6"
          },
          "limit": "SAT is equivalent to a dominating transversal of this matching, not to an arbitrary dominating set. The implemented equivalence also handles non-cubic finite simple graphs."
        },
        {
          "id": "enumerate-transversals",
          "requires": [
            {
              "field": "transversalEncoding",
              "op": "eq",
              "value": true
            },
            {
              "field": "matchingSize",
              "op": "lte",
              "value": 20
            }
          ],
          "produces": [
            "boundedSearchAvailable"
          ],
          "source": {
            "url": "https://arxiv.org/html/2608.22498v1",
            "location": "Proposition 6; finite assignment enumeration supplied by this pilot"
          },
          "limit": "At most 20 matching edges and 1,048,576 assignments. An exhausted user budget is inconclusive. No-transversal excludes only this restricted route."
        },
        {
          "id": "external-order48",
          "requires": [
            {
              "field": "degree",
              "op": "eq",
              "value": 3
            },
            {
              "field": "order",
              "op": "lte",
              "value": 48
            }
          ],
          "produces": [
            "knownGeneralResultApplicable"
          ],
          "source": {
            "url": "https://arxiv.org/html/2608.22498v1",
            "location": "Proposition 7, importing Zhang–Peitl–Szeider 2024"
          },
          "limit": "Imported preprint and classification, not reverified here. Use this as a duplicate-research warning; the direct finite witness removes the imported classification from the pilot's conclusion."
        }
      ],
      "observed": {
        "finiteSimple": true,
        "order": 48,
        "degree": 3,
        "matchingSize": 15,
        "maximalMatching": true,
        "matchingLowerBound": 15,
        "minimumMatching": true,
        "unmatched": 18
      },
      "conditionalFacts": {
        "finiteSimple": true,
        "order": 48,
        "degree": 3,
        "matchingSize": 15,
        "maximalMatching": true,
        "matchingLowerBound": 15,
        "minimumMatching": true,
        "unmatched": 18,
        "minimumMatchingByCounting": true,
        "transversalEncoding": true,
        "boundedSearchAvailable": true,
        "knownGeneralResultApplicable": true
      },
      "ceiling": "Matched conditions generate conditional research routes, not proof or novelty. Only executed witness checks verify finite witnesses."
    },
    "run": {
      "version": "1.0.0",
      "operator": "matching-transversal",
      "status": "witness-found",
      "variables": 15,
      "clauseCount": 18,
      "safeVertices": [],
      "examined": 32768,
      "totalAssignments": 32768,
      "complete": true,
      "solutions": 317,
      "witness": [
        0,
        2,
        4,
        7,
        8,
        10,
        13,
        14,
        16,
        18,
        21,
        22,
        24,
        26,
        28
      ],
      "witnessCheck": {
        "valid": true,
        "size": 15,
        "uncovered": [],
        "independent": false
      },
      "scope": "One supplied maximal matching only. No-transversal is NOT a lower bound for ordinary domination. No optimal domination number or novelty is inferred.",
      "facts": {
        "finiteSimple": true,
        "order": 48,
        "degree": 3,
        "matchingSize": 15,
        "maximalMatching": true,
        "matchingLowerBound": 15,
        "minimumMatching": true,
        "unmatched": 18,
        "transversalFound": true,
        "witnessVerified": true,
        "dominationUpper": 15
      }
    },
    "after": {
      "routes": [
        {
          "id": "checked-domination-bound",
          "title": "Chain the checked witness into an ordinary-domination upper bound",
          "requires": [
            {
              "field": "witnessVerified",
              "op": "eq",
              "value": true
            }
          ],
          "produces": [
            "ordinaryUpperBoundCertified"
          ],
          "methods": [
            "chaining"
          ],
          "source": {
            "url": "https://arxiv.org/html/2609.10783v1",
            "location": "Section 1: definition of domination; Section 4: witness upper bound"
          },
          "basis": "Every vertex is either in the selected set or adjacent to it, so gamma(G) is at most the witness size.",
          "limit": "An upper bound is not an exact minimum. No lower bound is inferred from failure of a restricted search.",
          "ready": true,
          "missing": [],
          "score": 0
        },
        {
          "id": "compare-matching",
          "title": "Decide that this graph is not an ordinary-domination counterexample",
          "requires": [
            {
              "field": "ordinaryUpperBoundCertified",
              "op": "eq",
              "value": true
            },
            {
              "field": "minimumMatchingByCounting",
              "op": "eq",
              "value": true
            },
            {
              "field": "dominationUpper",
              "op": "lte",
              "otherField": "matchingSize"
            }
          ],
          "produces": [
            "ordinaryCounterexampleRuledOut"
          ],
          "methods": [
            "lemma-recombination",
            "chaining"
          ],
          "source": {
            "url": "https://arxiv.org/html/2608.22498v1",
            "location": "Proposition 1 (dominating transversal reduction)"
          },
          "basis": "Combine a graph-checked dominating set of size at most |M| with a checked minimum maximal matching M. This proves gamma(G) <= gamma_e(G) for this exact graph.",
          "limit": "No universal statement about regular graphs; the two graph objects and parameter meanings must be identical.",
          "ready": true,
          "missing": [],
          "score": 0
        },
        {
          "id": "enumerate-transversals",
          "title": "Find and independently check an explicit transversal",
          "requires": [
            {
              "field": "transversalEncoding",
              "op": "eq",
              "value": true
            },
            {
              "field": "matchingSize",
              "op": "lte",
              "value": 20
            }
          ],
          "produces": [
            "boundedSearchAvailable"
          ],
          "action": "matching-transversal",
          "methods": [
            "serial-search",
            "bookkeeping",
            "prove-too-much"
          ],
          "source": {
            "url": "https://arxiv.org/html/2608.22498v1",
            "location": "Proposition 6; finite assignment enumeration supplied by this pilot"
          },
          "basis": "Enumerate all 2^|M| assignments, reconstruct each first satisfying selection, then check domination from the original edges without using the clauses.",
          "limit": "At most 20 matching edges and 1,048,576 assignments. An exhausted user budget is inconclusive. No-transversal excludes only this restricted route.",
          "ready": true,
          "missing": [],
          "score": 0
        },
        {
          "id": "external-order48",
          "title": "Prior-art stop: the order-48 inequality is already covered",
          "requires": [
            {
              "field": "degree",
              "op": "eq",
              "value": 3
            },
            {
              "field": "order",
              "op": "lte",
              "value": 48
            }
          ],
          "produces": [
            "knownGeneralResultApplicable"
          ],
          "methods": [
            "named-obstructions",
            "lemma-recombination"
          ],
          "source": {
            "url": "https://arxiv.org/html/2608.22498v1",
            "location": "Proposition 7, importing Zhang–Peitl–Szeider 2024"
          },
          "basis": "The cited preprint states gamma <= gamma_e for cubic graphs of order at most 48, using a minimum unsatisfiable bounded-occurrence CNF classification.",
          "limit": "Imported preprint and classification, not reverified here. Use this as a duplicate-research warning; the direct finite witness removes the imported classification from the pilot's conclusion.",
          "ready": true,
          "missing": [],
          "score": 0
        },
        {
          "id": "graph-to-sat",
          "title": "Reformulate domination as endpoint-transversal SAT",
          "requires": [
            {
              "field": "finiteSimple",
              "op": "eq",
              "value": true
            },
            {
              "field": "maximalMatching",
              "op": "eq",
              "value": true
            }
          ],
          "produces": [
            "transversalEncoding"
          ],
          "methods": [
            "reformulation",
            "distant-imports"
          ],
          "source": {
            "url": "https://arxiv.org/html/2608.22498v1",
            "location": "Definition 2 and Proposition 6"
          },
          "basis": "One Boolean variable selects an endpoint of each matching edge. A clause for every unmatched vertex requires a neighbour to be selected. Vertices adjacent to both endpoints of one edge give tautologies. Matched vertices are automatically dominated.",
          "limit": "SAT is equivalent to a dominating transversal of this matching, not to an arbitrary dominating set. The implemented equivalence also handles non-cubic finite simple graphs.",
          "ready": true,
          "missing": [],
          "score": 0
        },
        {
          "id": "matching-count-bound",
          "title": "Certify the supplied matching is minimum",
          "requires": [
            {
              "field": "maximalMatching",
              "op": "eq",
              "value": true
            },
            {
              "field": "matchingSize",
              "op": "eq",
              "otherField": "matchingLowerBound"
            }
          ],
          "produces": [
            "minimumMatchingByCounting"
          ],
          "methods": [
            "bookkeeping",
            "chaining"
          ],
          "source": {
            "url": "https://evidencepress.org/releases/txgraffiti-order48-successor/",
            "location": "Exact result for specialists"
          },
          "basis": "A maximal matching edge-dominates all edges. Each edge covers at most 2 Delta - 1 edges; ceiling(m/(2 Delta - 1)) is a lower bound. Equality with a checked witness establishes minimality.",
          "limit": "Works when this elementary lower bound is attained. Otherwise minimality remains unknown, not false.",
          "ready": true,
          "missing": [],
          "score": 0
        },
        {
          "id": "independent-upper",
          "title": "Independent domination transfers only an upper bound",
          "requires": [
            {
              "field": "independentWitnessVerified",
              "op": "eq",
              "value": true
            }
          ],
          "produces": [
            "independentWitnessGivesOrdinaryUpper"
          ],
          "methods": [
            "prove-too-much",
            "named-obstructions"
          ],
          "source": {
            "url": "https://arxiv.org/html/2609.10783v1",
            "location": "Section 8: ordinary versus independent domination"
          },
          "basis": "Every independent dominating set is dominating: gamma <= i. A lower bound on i is NOT a lower bound on gamma.",
          "limit": "Knowing i=16 does not prove gamma>=16. This invalid transfer is deliberately absent from the rule library.",
          "ready": false,
          "missing": [
            {
              "field": "independentWitnessVerified",
              "op": "eq",
              "value": true
            }
          ],
          "score": 0
        }
      ],
      "chains": [
        {
          "id": "matching-count-bound",
          "requires": [
            {
              "field": "maximalMatching",
              "op": "eq",
              "value": true
            },
            {
              "field": "matchingSize",
              "op": "eq",
              "otherField": "matchingLowerBound"
            }
          ],
          "produces": [
            "minimumMatchingByCounting"
          ],
          "source": {
            "url": "https://evidencepress.org/releases/txgraffiti-order48-successor/",
            "location": "Exact result for specialists"
          },
          "limit": "Works when this elementary lower bound is attained. Otherwise minimality remains unknown, not false."
        },
        {
          "id": "graph-to-sat",
          "requires": [
            {
              "field": "finiteSimple",
              "op": "eq",
              "value": true
            },
            {
              "field": "maximalMatching",
              "op": "eq",
              "value": true
            }
          ],
          "produces": [
            "transversalEncoding"
          ],
          "source": {
            "url": "https://arxiv.org/html/2608.22498v1",
            "location": "Definition 2 and Proposition 6"
          },
          "limit": "SAT is equivalent to a dominating transversal of this matching, not to an arbitrary dominating set. The implemented equivalence also handles non-cubic finite simple graphs."
        },
        {
          "id": "enumerate-transversals",
          "requires": [
            {
              "field": "transversalEncoding",
              "op": "eq",
              "value": true
            },
            {
              "field": "matchingSize",
              "op": "lte",
              "value": 20
            }
          ],
          "produces": [
            "boundedSearchAvailable"
          ],
          "source": {
            "url": "https://arxiv.org/html/2608.22498v1",
            "location": "Proposition 6; finite assignment enumeration supplied by this pilot"
          },
          "limit": "At most 20 matching edges and 1,048,576 assignments. An exhausted user budget is inconclusive. No-transversal excludes only this restricted route."
        },
        {
          "id": "checked-domination-bound",
          "requires": [
            {
              "field": "witnessVerified",
              "op": "eq",
              "value": true
            }
          ],
          "produces": [
            "ordinaryUpperBoundCertified"
          ],
          "source": {
            "url": "https://arxiv.org/html/2609.10783v1",
            "location": "Section 1: definition of domination; Section 4: witness upper bound"
          },
          "limit": "An upper bound is not an exact minimum. No lower bound is inferred from failure of a restricted search."
        },
        {
          "id": "compare-matching",
          "requires": [
            {
              "field": "ordinaryUpperBoundCertified",
              "op": "eq",
              "value": true
            },
            {
              "field": "minimumMatchingByCounting",
              "op": "eq",
              "value": true
            },
            {
              "field": "dominationUpper",
              "op": "lte",
              "otherField": "matchingSize"
            }
          ],
          "produces": [
            "ordinaryCounterexampleRuledOut"
          ],
          "source": {
            "url": "https://arxiv.org/html/2608.22498v1",
            "location": "Proposition 1 (dominating transversal reduction)"
          },
          "limit": "No universal statement about regular graphs; the two graph objects and parameter meanings must be identical."
        },
        {
          "id": "external-order48",
          "requires": [
            {
              "field": "degree",
              "op": "eq",
              "value": 3
            },
            {
              "field": "order",
              "op": "lte",
              "value": 48
            }
          ],
          "produces": [
            "knownGeneralResultApplicable"
          ],
          "source": {
            "url": "https://arxiv.org/html/2608.22498v1",
            "location": "Proposition 7, importing Zhang–Peitl–Szeider 2024"
          },
          "limit": "Imported preprint and classification, not reverified here. Use this as a duplicate-research warning; the direct finite witness removes the imported classification from the pilot's conclusion."
        }
      ],
      "observed": {
        "finiteSimple": true,
        "order": 48,
        "degree": 3,
        "matchingSize": 15,
        "maximalMatching": true,
        "matchingLowerBound": 15,
        "minimumMatching": true,
        "unmatched": 18,
        "transversalFound": true,
        "witnessVerified": true,
        "dominationUpper": 15
      },
      "conditionalFacts": {
        "finiteSimple": true,
        "order": 48,
        "degree": 3,
        "matchingSize": 15,
        "maximalMatching": true,
        "matchingLowerBound": 15,
        "minimumMatching": true,
        "unmatched": 18,
        "transversalFound": true,
        "witnessVerified": true,
        "dominationUpper": 15,
        "minimumMatchingByCounting": true,
        "transversalEncoding": true,
        "boundedSearchAvailable": true,
        "ordinaryUpperBoundCertified": true,
        "ordinaryCounterexampleRuledOut": true,
        "knownGeneralResultApplicable": true
      },
      "ceiling": "Matched conditions generate conditional research routes, not proof or novelty. Only executed witness checks verify finite witnesses."
    }
  },
  "negativeControl": {
    "input": {
      "title": "Public 50-vertex ordinary-domination obstruction (negative control)",
      "source": {
        "url": "https://arxiv.org/html/2609.10783v1",
        "location": "Section 2, twenty defining clauses and numeric labelling",
        "attribution": "Koyar Afrasyab (2026), using the previously public TxGraffiti graph. Encoding transcribed for a scoped transversal control; not a replay of the ordinary-domination lower-bound proof."
      },
      "order": 50,
      "edges": [
        [
          0,
          1
        ],
        [
          2,
          3
        ],
        [
          4,
          5
        ],
        [
          6,
          7
        ],
        [
          8,
          9
        ],
        [
          10,
          11
        ],
        [
          12,
          13
        ],
        [
          14,
          15
        ],
        [
          16,
          17
        ],
        [
          18,
          19
        ],
        [
          20,
          21
        ],
        [
          22,
          23
        ],
        [
          24,
          25
        ],
        [
          26,
          27
        ],
        [
          28,
          29
        ],
        [
          1,
          30
        ],
        [
          17,
          30
        ],
        [
          20,
          30
        ],
        [
          6,
          31
        ],
        [
          18,
          31
        ],
        [
          24,
          31
        ],
        [
          1,
          32
        ],
        [
          16,
          32
        ],
        [
          26,
          32
        ],
        [
          3,
          33
        ],
        [
          11,
          33
        ],
        [
          27,
          33
        ],
        [
          8,
          34
        ],
        [
          10,
          34
        ],
        [
          29,
          34
        ],
        [
          3,
          35
        ],
        [
          4,
          35
        ],
        [
          29,
          35
        ],
        [
          5,
          36
        ],
        [
          9,
          36
        ],
        [
          16,
          36
        ],
        [
          10,
          37
        ],
        [
          22,
          37
        ],
        [
          28,
          37
        ],
        [
          0,
          38
        ],
        [
          9,
          38
        ],
        [
          20,
          38
        ],
        [
          7,
          39
        ],
        [
          12,
          39
        ],
        [
          28,
          39
        ],
        [
          2,
          40
        ],
        [
          15,
          40
        ],
        [
          22,
          40
        ],
        [
          5,
          41
        ],
        [
          17,
          41
        ],
        [
          21,
          41
        ],
        [
          6,
          42
        ],
        [
          14,
          42
        ],
        [
          19,
          42
        ],
        [
          13,
          43
        ],
        [
          14,
          43
        ],
        [
          25,
          43
        ],
        [
          7,
          44
        ],
        [
          13,
          44
        ],
        [
          24,
          44
        ],
        [
          2,
          45
        ],
        [
          12,
          45
        ],
        [
          27,
          45
        ],
        [
          4,
          46
        ],
        [
          21,
          46
        ],
        [
          26,
          46
        ],
        [
          15,
          47
        ],
        [
          19,
          47
        ],
        [
          23,
          47
        ],
        [
          18,
          48
        ],
        [
          23,
          48
        ],
        [
          25,
          48
        ],
        [
          0,
          49
        ],
        [
          8,
          49
        ],
        [
          11,
          49
        ]
      ],
      "maximal_matching": [
        [
          0,
          1
        ],
        [
          2,
          3
        ],
        [
          4,
          5
        ],
        [
          6,
          7
        ],
        [
          8,
          9
        ],
        [
          10,
          11
        ],
        [
          12,
          13
        ],
        [
          14,
          15
        ],
        [
          16,
          17
        ],
        [
          18,
          19
        ],
        [
          20,
          21
        ],
        [
          22,
          23
        ],
        [
          24,
          25
        ],
        [
          26,
          27
        ],
        [
          28,
          29
        ]
      ]
    },
    "inputSha256": "d529eec43fb57b2277e171e0765becf2506deccfd9c27053125570026bca8b53",
    "result": {
      "before": {
        "routes": [
          {
            "id": "enumerate-transversals",
            "title": "Find and independently check an explicit transversal",
            "requires": [
              {
                "field": "transversalEncoding",
                "op": "eq",
                "value": true
              },
              {
                "field": "matchingSize",
                "op": "lte",
                "value": 20
              }
            ],
            "produces": [
              "boundedSearchAvailable"
            ],
            "action": "matching-transversal",
            "methods": [
              "serial-search",
              "bookkeeping",
              "prove-too-much"
            ],
            "source": {
              "url": "https://arxiv.org/html/2608.22498v1",
              "location": "Proposition 6; finite assignment enumeration supplied by this pilot"
            },
            "basis": "Enumerate all 2^|M| assignments, reconstruct each first satisfying selection, then check domination from the original edges without using the clauses.",
            "limit": "At most 20 matching edges and 1,048,576 assignments. An exhausted user budget is inconclusive. No-transversal excludes only this restricted route.",
            "ready": true,
            "missing": [],
            "score": 0
          },
          {
            "id": "graph-to-sat",
            "title": "Reformulate domination as endpoint-transversal SAT",
            "requires": [
              {
                "field": "finiteSimple",
                "op": "eq",
                "value": true
              },
              {
                "field": "maximalMatching",
                "op": "eq",
                "value": true
              }
            ],
            "produces": [
              "transversalEncoding"
            ],
            "methods": [
              "reformulation",
              "distant-imports"
            ],
            "source": {
              "url": "https://arxiv.org/html/2608.22498v1",
              "location": "Definition 2 and Proposition 6"
            },
            "basis": "One Boolean variable selects an endpoint of each matching edge. A clause for every unmatched vertex requires a neighbour to be selected. Vertices adjacent to both endpoints of one edge give tautologies. Matched vertices are automatically dominated.",
            "limit": "SAT is equivalent to a dominating transversal of this matching, not to an arbitrary dominating set. The implemented equivalence also handles non-cubic finite simple graphs.",
            "ready": true,
            "missing": [],
            "score": 0
          },
          {
            "id": "matching-count-bound",
            "title": "Certify the supplied matching is minimum",
            "requires": [
              {
                "field": "maximalMatching",
                "op": "eq",
                "value": true
              },
              {
                "field": "matchingSize",
                "op": "eq",
                "otherField": "matchingLowerBound"
              }
            ],
            "produces": [
              "minimumMatchingByCounting"
            ],
            "methods": [
              "bookkeeping",
              "chaining"
            ],
            "source": {
              "url": "https://evidencepress.org/releases/txgraffiti-order48-successor/",
              "location": "Exact result for specialists"
            },
            "basis": "A maximal matching edge-dominates all edges. Each edge covers at most 2 Delta - 1 edges; ceiling(m/(2 Delta - 1)) is a lower bound. Equality with a checked witness establishes minimality.",
            "limit": "Works when this elementary lower bound is attained. Otherwise minimality remains unknown, not false.",
            "ready": true,
            "missing": [],
            "score": 0
          },
          {
            "id": "checked-domination-bound",
            "title": "Chain the checked witness into an ordinary-domination upper bound",
            "requires": [
              {
                "field": "witnessVerified",
                "op": "eq",
                "value": true
              }
            ],
            "produces": [
              "ordinaryUpperBoundCertified"
            ],
            "methods": [
              "chaining"
            ],
            "source": {
              "url": "https://arxiv.org/html/2609.10783v1",
              "location": "Section 1: definition of domination; Section 4: witness upper bound"
            },
            "basis": "Every vertex is either in the selected set or adjacent to it, so gamma(G) is at most the witness size.",
            "limit": "An upper bound is not an exact minimum. No lower bound is inferred from failure of a restricted search.",
            "ready": false,
            "missing": [
              {
                "field": "witnessVerified",
                "op": "eq",
                "value": true
              }
            ],
            "score": 0
          },
          {
            "id": "external-order48",
            "title": "Prior-art stop: the order-48 inequality is already covered",
            "requires": [
              {
                "field": "degree",
                "op": "eq",
                "value": 3
              },
              {
                "field": "order",
                "op": "lte",
                "value": 48
              }
            ],
            "produces": [
              "knownGeneralResultApplicable"
            ],
            "methods": [
              "named-obstructions",
              "lemma-recombination"
            ],
            "source": {
              "url": "https://arxiv.org/html/2608.22498v1",
              "location": "Proposition 7, importing Zhang–Peitl–Szeider 2024"
            },
            "basis": "The cited preprint states gamma <= gamma_e for cubic graphs of order at most 48, using a minimum unsatisfiable bounded-occurrence CNF classification.",
            "limit": "Imported preprint and classification, not reverified here. Use this as a duplicate-research warning; the direct finite witness removes the imported classification from the pilot's conclusion.",
            "ready": false,
            "missing": [
              {
                "field": "order",
                "op": "lte",
                "value": 48
              }
            ],
            "score": 0
          },
          {
            "id": "independent-upper",
            "title": "Independent domination transfers only an upper bound",
            "requires": [
              {
                "field": "independentWitnessVerified",
                "op": "eq",
                "value": true
              }
            ],
            "produces": [
              "independentWitnessGivesOrdinaryUpper"
            ],
            "methods": [
              "prove-too-much",
              "named-obstructions"
            ],
            "source": {
              "url": "https://arxiv.org/html/2609.10783v1",
              "location": "Section 8: ordinary versus independent domination"
            },
            "basis": "Every independent dominating set is dominating: gamma <= i. A lower bound on i is NOT a lower bound on gamma.",
            "limit": "Knowing i=16 does not prove gamma>=16. This invalid transfer is deliberately absent from the rule library.",
            "ready": false,
            "missing": [
              {
                "field": "independentWitnessVerified",
                "op": "eq",
                "value": true
              }
            ],
            "score": 0
          },
          {
            "id": "compare-matching",
            "title": "Decide that this graph is not an ordinary-domination counterexample",
            "requires": [
              {
                "field": "ordinaryUpperBoundCertified",
                "op": "eq",
                "value": true
              },
              {
                "field": "minimumMatchingByCounting",
                "op": "eq",
                "value": true
              },
              {
                "field": "dominationUpper",
                "op": "lte",
                "otherField": "matchingSize"
              }
            ],
            "produces": [
              "ordinaryCounterexampleRuledOut"
            ],
            "methods": [
              "lemma-recombination",
              "chaining"
            ],
            "source": {
              "url": "https://arxiv.org/html/2608.22498v1",
              "location": "Proposition 1 (dominating transversal reduction)"
            },
            "basis": "Combine a graph-checked dominating set of size at most |M| with a checked minimum maximal matching M. This proves gamma(G) <= gamma_e(G) for this exact graph.",
            "limit": "No universal statement about regular graphs; the two graph objects and parameter meanings must be identical.",
            "ready": false,
            "missing": [
              {
                "field": "ordinaryUpperBoundCertified",
                "op": "eq",
                "value": true
              },
              {
                "field": "dominationUpper",
                "op": "lte",
                "otherField": "matchingSize"
              }
            ],
            "score": 0
          }
        ],
        "chains": [
          {
            "id": "matching-count-bound",
            "requires": [
              {
                "field": "maximalMatching",
                "op": "eq",
                "value": true
              },
              {
                "field": "matchingSize",
                "op": "eq",
                "otherField": "matchingLowerBound"
              }
            ],
            "produces": [
              "minimumMatchingByCounting"
            ],
            "source": {
              "url": "https://evidencepress.org/releases/txgraffiti-order48-successor/",
              "location": "Exact result for specialists"
            },
            "limit": "Works when this elementary lower bound is attained. Otherwise minimality remains unknown, not false."
          },
          {
            "id": "graph-to-sat",
            "requires": [
              {
                "field": "finiteSimple",
                "op": "eq",
                "value": true
              },
              {
                "field": "maximalMatching",
                "op": "eq",
                "value": true
              }
            ],
            "produces": [
              "transversalEncoding"
            ],
            "source": {
              "url": "https://arxiv.org/html/2608.22498v1",
              "location": "Definition 2 and Proposition 6"
            },
            "limit": "SAT is equivalent to a dominating transversal of this matching, not to an arbitrary dominating set. The implemented equivalence also handles non-cubic finite simple graphs."
          },
          {
            "id": "enumerate-transversals",
            "requires": [
              {
                "field": "transversalEncoding",
                "op": "eq",
                "value": true
              },
              {
                "field": "matchingSize",
                "op": "lte",
                "value": 20
              }
            ],
            "produces": [
              "boundedSearchAvailable"
            ],
            "source": {
              "url": "https://arxiv.org/html/2608.22498v1",
              "location": "Proposition 6; finite assignment enumeration supplied by this pilot"
            },
            "limit": "At most 20 matching edges and 1,048,576 assignments. An exhausted user budget is inconclusive. No-transversal excludes only this restricted route."
          }
        ],
        "observed": {
          "finiteSimple": true,
          "order": 50,
          "degree": 3,
          "matchingSize": 15,
          "maximalMatching": true,
          "matchingLowerBound": 15,
          "minimumMatching": true,
          "unmatched": 20
        },
        "conditionalFacts": {
          "finiteSimple": true,
          "order": 50,
          "degree": 3,
          "matchingSize": 15,
          "maximalMatching": true,
          "matchingLowerBound": 15,
          "minimumMatching": true,
          "unmatched": 20,
          "minimumMatchingByCounting": true,
          "transversalEncoding": true,
          "boundedSearchAvailable": true
        },
        "ceiling": "Matched conditions generate conditional research routes, not proof or novelty. Only executed witness checks verify finite witnesses."
      },
      "run": {
        "version": "1.0.0",
        "operator": "matching-transversal",
        "status": "no-transversal",
        "variables": 15,
        "clauseCount": 20,
        "safeVertices": [],
        "examined": 32768,
        "totalAssignments": 32768,
        "complete": true,
        "solutions": 0,
        "witness": null,
        "witnessCheck": null,
        "scope": "One supplied maximal matching only. No-transversal is NOT a lower bound for ordinary domination. No optimal domination number or novelty is inferred.",
        "facts": {
          "finiteSimple": true,
          "order": 50,
          "degree": 3,
          "matchingSize": 15,
          "maximalMatching": true,
          "matchingLowerBound": 15,
          "minimumMatching": true,
          "unmatched": 20,
          "transversalFound": false,
          "witnessVerified": false,
          "dominationUpper": null
        }
      },
      "after": {
        "routes": [
          {
            "id": "enumerate-transversals",
            "title": "Find and independently check an explicit transversal",
            "requires": [
              {
                "field": "transversalEncoding",
                "op": "eq",
                "value": true
              },
              {
                "field": "matchingSize",
                "op": "lte",
                "value": 20
              }
            ],
            "produces": [
              "boundedSearchAvailable"
            ],
            "action": "matching-transversal",
            "methods": [
              "serial-search",
              "bookkeeping",
              "prove-too-much"
            ],
            "source": {
              "url": "https://arxiv.org/html/2608.22498v1",
              "location": "Proposition 6; finite assignment enumeration supplied by this pilot"
            },
            "basis": "Enumerate all 2^|M| assignments, reconstruct each first satisfying selection, then check domination from the original edges without using the clauses.",
            "limit": "At most 20 matching edges and 1,048,576 assignments. An exhausted user budget is inconclusive. No-transversal excludes only this restricted route.",
            "ready": true,
            "missing": [],
            "score": 0
          },
          {
            "id": "graph-to-sat",
            "title": "Reformulate domination as endpoint-transversal SAT",
            "requires": [
              {
                "field": "finiteSimple",
                "op": "eq",
                "value": true
              },
              {
                "field": "maximalMatching",
                "op": "eq",
                "value": true
              }
            ],
            "produces": [
              "transversalEncoding"
            ],
            "methods": [
              "reformulation",
              "distant-imports"
            ],
            "source": {
              "url": "https://arxiv.org/html/2608.22498v1",
              "location": "Definition 2 and Proposition 6"
            },
            "basis": "One Boolean variable selects an endpoint of each matching edge. A clause for every unmatched vertex requires a neighbour to be selected. Vertices adjacent to both endpoints of one edge give tautologies. Matched vertices are automatically dominated.",
            "limit": "SAT is equivalent to a dominating transversal of this matching, not to an arbitrary dominating set. The implemented equivalence also handles non-cubic finite simple graphs.",
            "ready": true,
            "missing": [],
            "score": 0
          },
          {
            "id": "matching-count-bound",
            "title": "Certify the supplied matching is minimum",
            "requires": [
              {
                "field": "maximalMatching",
                "op": "eq",
                "value": true
              },
              {
                "field": "matchingSize",
                "op": "eq",
                "otherField": "matchingLowerBound"
              }
            ],
            "produces": [
              "minimumMatchingByCounting"
            ],
            "methods": [
              "bookkeeping",
              "chaining"
            ],
            "source": {
              "url": "https://evidencepress.org/releases/txgraffiti-order48-successor/",
              "location": "Exact result for specialists"
            },
            "basis": "A maximal matching edge-dominates all edges. Each edge covers at most 2 Delta - 1 edges; ceiling(m/(2 Delta - 1)) is a lower bound. Equality with a checked witness establishes minimality.",
            "limit": "Works when this elementary lower bound is attained. Otherwise minimality remains unknown, not false.",
            "ready": true,
            "missing": [],
            "score": 0
          },
          {
            "id": "checked-domination-bound",
            "title": "Chain the checked witness into an ordinary-domination upper bound",
            "requires": [
              {
                "field": "witnessVerified",
                "op": "eq",
                "value": true
              }
            ],
            "produces": [
              "ordinaryUpperBoundCertified"
            ],
            "methods": [
              "chaining"
            ],
            "source": {
              "url": "https://arxiv.org/html/2609.10783v1",
              "location": "Section 1: definition of domination; Section 4: witness upper bound"
            },
            "basis": "Every vertex is either in the selected set or adjacent to it, so gamma(G) is at most the witness size.",
            "limit": "An upper bound is not an exact minimum. No lower bound is inferred from failure of a restricted search.",
            "ready": false,
            "missing": [
              {
                "field": "witnessVerified",
                "op": "eq",
                "value": true
              }
            ],
            "score": 0
          },
          {
            "id": "external-order48",
            "title": "Prior-art stop: the order-48 inequality is already covered",
            "requires": [
              {
                "field": "degree",
                "op": "eq",
                "value": 3
              },
              {
                "field": "order",
                "op": "lte",
                "value": 48
              }
            ],
            "produces": [
              "knownGeneralResultApplicable"
            ],
            "methods": [
              "named-obstructions",
              "lemma-recombination"
            ],
            "source": {
              "url": "https://arxiv.org/html/2608.22498v1",
              "location": "Proposition 7, importing Zhang–Peitl–Szeider 2024"
            },
            "basis": "The cited preprint states gamma <= gamma_e for cubic graphs of order at most 48, using a minimum unsatisfiable bounded-occurrence CNF classification.",
            "limit": "Imported preprint and classification, not reverified here. Use this as a duplicate-research warning; the direct finite witness removes the imported classification from the pilot's conclusion.",
            "ready": false,
            "missing": [
              {
                "field": "order",
                "op": "lte",
                "value": 48
              }
            ],
            "score": 0
          },
          {
            "id": "independent-upper",
            "title": "Independent domination transfers only an upper bound",
            "requires": [
              {
                "field": "independentWitnessVerified",
                "op": "eq",
                "value": true
              }
            ],
            "produces": [
              "independentWitnessGivesOrdinaryUpper"
            ],
            "methods": [
              "prove-too-much",
              "named-obstructions"
            ],
            "source": {
              "url": "https://arxiv.org/html/2609.10783v1",
              "location": "Section 8: ordinary versus independent domination"
            },
            "basis": "Every independent dominating set is dominating: gamma <= i. A lower bound on i is NOT a lower bound on gamma.",
            "limit": "Knowing i=16 does not prove gamma>=16. This invalid transfer is deliberately absent from the rule library.",
            "ready": false,
            "missing": [
              {
                "field": "independentWitnessVerified",
                "op": "eq",
                "value": true
              }
            ],
            "score": 0
          },
          {
            "id": "compare-matching",
            "title": "Decide that this graph is not an ordinary-domination counterexample",
            "requires": [
              {
                "field": "ordinaryUpperBoundCertified",
                "op": "eq",
                "value": true
              },
              {
                "field": "minimumMatchingByCounting",
                "op": "eq",
                "value": true
              },
              {
                "field": "dominationUpper",
                "op": "lte",
                "otherField": "matchingSize"
              }
            ],
            "produces": [
              "ordinaryCounterexampleRuledOut"
            ],
            "methods": [
              "lemma-recombination",
              "chaining"
            ],
            "source": {
              "url": "https://arxiv.org/html/2608.22498v1",
              "location": "Proposition 1 (dominating transversal reduction)"
            },
            "basis": "Combine a graph-checked dominating set of size at most |M| with a checked minimum maximal matching M. This proves gamma(G) <= gamma_e(G) for this exact graph.",
            "limit": "No universal statement about regular graphs; the two graph objects and parameter meanings must be identical.",
            "ready": false,
            "missing": [
              {
                "field": "ordinaryUpperBoundCertified",
                "op": "eq",
                "value": true
              },
              {
                "field": "dominationUpper",
                "op": "lte",
                "otherField": "matchingSize"
              }
            ],
            "score": 0
          }
        ],
        "chains": [
          {
            "id": "matching-count-bound",
            "requires": [
              {
                "field": "maximalMatching",
                "op": "eq",
                "value": true
              },
              {
                "field": "matchingSize",
                "op": "eq",
                "otherField": "matchingLowerBound"
              }
            ],
            "produces": [
              "minimumMatchingByCounting"
            ],
            "source": {
              "url": "https://evidencepress.org/releases/txgraffiti-order48-successor/",
              "location": "Exact result for specialists"
            },
            "limit": "Works when this elementary lower bound is attained. Otherwise minimality remains unknown, not false."
          },
          {
            "id": "graph-to-sat",
            "requires": [
              {
                "field": "finiteSimple",
                "op": "eq",
                "value": true
              },
              {
                "field": "maximalMatching",
                "op": "eq",
                "value": true
              }
            ],
            "produces": [
              "transversalEncoding"
            ],
            "source": {
              "url": "https://arxiv.org/html/2608.22498v1",
              "location": "Definition 2 and Proposition 6"
            },
            "limit": "SAT is equivalent to a dominating transversal of this matching, not to an arbitrary dominating set. The implemented equivalence also handles non-cubic finite simple graphs."
          },
          {
            "id": "enumerate-transversals",
            "requires": [
              {
                "field": "transversalEncoding",
                "op": "eq",
                "value": true
              },
              {
                "field": "matchingSize",
                "op": "lte",
                "value": 20
              }
            ],
            "produces": [
              "boundedSearchAvailable"
            ],
            "source": {
              "url": "https://arxiv.org/html/2608.22498v1",
              "location": "Proposition 6; finite assignment enumeration supplied by this pilot"
            },
            "limit": "At most 20 matching edges and 1,048,576 assignments. An exhausted user budget is inconclusive. No-transversal excludes only this restricted route."
          }
        ],
        "observed": {
          "finiteSimple": true,
          "order": 50,
          "degree": 3,
          "matchingSize": 15,
          "maximalMatching": true,
          "matchingLowerBound": 15,
          "minimumMatching": true,
          "unmatched": 20,
          "transversalFound": false,
          "witnessVerified": false,
          "dominationUpper": null
        },
        "conditionalFacts": {
          "finiteSimple": true,
          "order": 50,
          "degree": 3,
          "matchingSize": 15,
          "maximalMatching": true,
          "matchingLowerBound": 15,
          "minimumMatching": true,
          "unmatched": 20,
          "transversalFound": false,
          "witnessVerified": false,
          "dominationUpper": null,
          "minimumMatchingByCounting": true,
          "transversalEncoding": true,
          "boundedSearchAvailable": true
        },
        "ceiling": "Matched conditions generate conditional research routes, not proof or novelty. Only executed witness checks verify finite witnesses."
      }
    }
  },
  "ablation": {
    "removed": "graph-to-sat",
    "result": {
      "status": "route-unavailable",
      "scope": "No executable hypothesis-matched bridge in the supplied library."
    }
  },
  "searchSpace": {
    "unrestricted15Subsets": "1093260079344",
    "restrictedAssignments": 32768,
    "meaning": "A mathematical candidate-count comparison only, not a runtime benchmark against an optimized solver or a human."
  },
  "outcome": "Concrete ordinary dominating witness found for the exact 48-vertex graph; finite counterexample route stopped. External order-48 theorem was already known. No new general theorem.",
  "limits": [
    "One selected internal pilot, not a blinded comparison or causal acceleration result.",
    "50-vertex no-transversal control does not replay its published ordinary-domination lower bound.",
    "Independent domination 16 is inherited from the release, not reverified here.",
    "No graph relationship or scholarly assurance automatically promoted."
  ]
}