{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261015-advance-20261115-r06-minimum-vertex-cover-0015",
  "pilot_number": "R0051-26040",
  "number": 26040,
  "family_id": "qf-mod46-mincover-exhaustive-minimizers",
  "title": {
    "zh-Hant": "第 26040 題：最小頂點覆蓋",
    "en": "Question 26040: Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261015-advance-20261115-r06-minimum-vertex-cover-0015",
    "batch_id": "batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-minimum-vertex-cover"
    ],
    "title_zh": "最小頂點覆蓋",
    "stem_zh": "bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-exhaustive-minimizers；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v0\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([v0, v1, v2, v3, v4, v5, v6, v7, v8], [[v0, v1], [v0, v8], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8]], [[v0, v1, v3, v5, v7], [v0, v2, v3, v5, v7], [v0, v1, v2, v3, v5, v7], [v0, v2, v4, v5, v7], [v0, v1, v2, v4, v5, v7], [v0, v1, v3, v4, v5, v7], [v0, v2, v3, v4, v5, v7], [v0, v1, v2, v3, v4, v5, v7], [v0, v2, v4, v6, v7], [v0, v1, v2, v4, v6, v7], [v0, v1, v3, v4, v6, v7], [v0, v2, v3, v4, v6, v7], [v0, v1, v2, v3, v4, v6, v7], [v0, v1, v3, v5, v6, v7], [v0, v2, v3, v5, v6, v7], [v0, v1, v2, v3, v5, v6, v7], [v0, v2, v4, v5, v6, v7], [v0, v1, v2, v4, v5, v6, v7], [v0, v1, v3, v4, v5, v6, v7], [v0, v2, v3, v4, v5, v6, v7], [v0, v1, v2, v3, v4, v5, v6, v7], [v0, v2, v4, v6, v8], [v1, v2, v4, v6, v8], [v0, v1, v2, v4, v6, v8], [v1, v3, v4, v6, v8], [v0, v1, v3, v4, v6, v8], [v0, v2, v3, v4, v6, v8], [v1, v2, v3, v4, v6, v8], [v0, v1, v2, v3, v4, v6, v8], [v1, v3, v5, v6, v8], [v0, v1, v3, v5, v6, v8], [v0, v2, v3, v5, v6, v8], [v1, v2, v3, v5, v6, v8], [v0, v1, v2, v3, v5, v6, v8], [v0, v2, v4, v5, v6, v8], [v1, v2, v4, v5, v6, v8], [v0, v1, v2, v4, v5, v6, v8], [v1, v3, v4, v5, v6, v8], [v0, v1, v3, v4, v5, v6, v8], [v0, v2, v3, v4, v5, v6, v8], [v1, v2, v3, v4, v5, v6, v8], [v0, v1, v2, v3, v4, v5, v6, v8], [v1, v3, v5, v7, v8], [v0, v1, v3, v5, v7, v8], [v0, v2, v3, v5, v7, v8], [v1, v2, v3, v5, v7, v8], [v0, v1, v2, v3, v5, v7, v8], [v0, v2, v4, v5, v7, v8], [v1, v2, v4, v5, v7, v8], [v0, v1, v2, v4, v5, v7, v8], [v1, v3, v4, v5, v7, v8], [v0, v1, v3, v4, v5, v7, v8], [v0, v2, v3, v4, v5, v7, v8], [v1, v2, v3, v4, v5, v7, v8], [v0, v1, v2, v3, v4, v5, v7, v8], [v0, v2, v4, v6, v7, v8], [v1, v2, v4, v6, v7, v8], [v0, v1, v2, v4, v6, v7, v8], [v1, v3, v4, v6, v7, v8], [v0, v1, v3, v4, v6, v7, v8], [v0, v2, v3, v4, v6, v7, v8], [v1, v2, v3, v4, v6, v7, v8], [v0, v1, v2, v3, v4, v6, v7, v8], [v1, v3, v5, v6, v7, v8], [v0, v1, v3, v5, v6, v7, v8], [v0, v2, v3, v5, v6, v7, v8], [v1, v2, v3, v5, v6, v7, v8], [v0, v1, v2, v3, v5, v6, v7, v8], [v0, v2, v4, v5, v6, v7, v8], [v1, v2, v4, v5, v6, v7, v8], [v0, v1, v2, v4, v5, v6, v7, v8], [v1, v3, v4, v5, v6, v7, v8], [v0, v1, v3, v4, v5, v6, v7, v8], [v0, v2, v3, v4, v5, v6, v7, v8], [v1, v2, v3, v4, v5, v6, v7, v8], [v0, v1, v2, v3, v4, v5, v6, v7, v8]], 5, [[v0, v1, v3, v5, v7], [v0, v2, v3, v5, v7], [v0, v2, v4, v5, v7], [v0, v2, v4, v6, v7], [v0, v2, v4, v6, v8], [v1, v2, v4, v6, v8], [v1, v3, v4, v6, v8], [v1, v3, v5, v6, v8], [v1, v3, v5, v7, v8]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)",
      "explanation_zh": "Wave6第九層核對=([v0, v1, v2, v3, v4, v5, v6, v7, v8], [[v0, v1], [v0, v8], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8]], [[v0, v1, v3, v5, v7], [v0, v2, v3, v5, v7], [v0, v1, v2, v3, v5, v7], [v0, v2, v4, v5, v7], [v0, v1, v2, v4, v5, v7], [v0, v1, v3, v4, v5, v7], [v0, v2, v3, v4, v5, v7], [v0, v1, v2, v3, v4, v5, v7], [v0, v2, v4, v6, v7], [v0, v1, v2, v4, v6, v7], [v0, v1, v3, v4, v6, v7], [v0, v2, v3, v4, v6, v7], [v0, v1, v2, v3, v4, v6, v7], [v0, v1, v3, v5, v6, v7], [v0, v2, v3, v5, v6, v7], [v0, v1, v2, v3, v5, v6, v7], [v0, v2, v4, v5, v6, v7], [v0, v1, v2, v4, v5, v6, v7], [v0, v1, v3, v4, v5, v6, v7], [v0, v2, v3, v4, v5, v6, v7], [v0, v1, v2, v3, v4, v5, v6, v7], [v0, v2, v4, v6, v8], [v1, v2, v4, v6, v8], [v0, v1, v2, v4, v6, v8], [v1, v3, v4, v6, v8], [v0, v1, v3, v4, v6, v8], [v0, v2, v3, v4, v6, v8], [v1, v2, v3, v4, v6, v8], [v0, v1, v2, v3, v4, v6, v8], [v1, v3, v5, v6, v8], [v0, v1, v3, v5, v6, v8], [v0, v2, v3, v5, v6, v8], [v1, v2, v3, v5, v6, v8], [v0, v1, v2, v3, v5, v6, v8], [v0, v2, v4, v5, v6, v8], [v1, v2, v4, v5, v6, v8], [v0, v1, v2, v4, v5, v6, v8], [v1, v3, v4, v5, v6, v8], [v0, v1, v3, v4, v5, v6, v8], [v0, v2, v3, v4, v5, v6, v8], [v1, v2, v3, v4, v5, v6, v8], [v0, v1, v2, v3, v4, v5, v6, v8], [v1, v3, v5, v7, v8], [v0, v1, v3, v5, v7, v8], [v0, v2, v3, v5, v7, v8], [v1, v2, v3, v5, v7, v8], [v0, v1, v2, v3, v5, v7, v8], [v0, v2, v4, v5, v7, v8], [v1, v2, v4, v5, v7, v8], [v0, v1, v2, v4, v5, v7, v8], [v1, v3, v4, v5, v7, v8], [v0, v1, v3, v4, v5, v7, v8], [v0, v2, v3, v4, v5, v7, v8], [v1, v2, v3, v4, v5, v7, v8], [v0, v1, v2, v3, v4, v5, v7, v8], [v0, v2, v4, v6, v7, v8], [v1, v2, v4, v6, v7, v8], [v0, v1, v2, v4, v6, v7, v8], [v1, v3, v4, v6, v7, v8], [v0, v1, v3, v4, v6, v7, v8], [v0, v2, v3, v4, v6, v7, v8], [v1, v2, v3, v4, v6, v7, v8], [v0, v1, v2, v3, v4, v6, v7, v8], [v1, v3, v5, v6, v7, v8], [v0, v1, v3, v5, v6, v7, v8], [v0, v2, v3, v5, v6, v7, v8], [v1, v2, v3, v5, v6, v7, v8], [v0, v1, v2, v3, v5, v6, v7, v8], [v0, v2, v4, v5, v6, v7, v8], [v1, v2, v4, v5, v6, v7, v8], [v0, v1, v2, v4, v5, v6, v7, v8], [v1, v3, v4, v5, v6, v7, v8], [v0, v1, v3, v4, v5, v6, v7, v8], [v0, v2, v3, v4, v5, v6, v7, v8], [v1, v2, v3, v4, v5, v6, v7, v8], [v0, v1, v2, v3, v4, v5, v6, v7, v8]], 5, [[v0, v1, v3, v5, v7], [v0, v2, v3, v5, v7], [v0, v2, v4, v5, v7], [v0, v2, v4, v6, v7], [v0, v2, v4, v6, v8], [v1, v2, v4, v6, v8], [v1, v3, v4, v6, v8], [v1, v3, v5, v6, v8], [v1, v3, v5, v7, v8]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "最小頂點覆蓋"
      ]
    },
    "variant": {
      "family_id": "qf-mod46-mincover-exhaustive-minimizers",
      "seed": 202611156015,
      "parameters": {
        "vertices": [
          "v0",
          "v1",
          "v2",
          "v3",
          "v4",
          "v5",
          "v6",
          "v7",
          "v8"
        ],
        "edges": [
          [
            "v0",
            "v1"
          ],
          [
            "v1",
            "v2"
          ],
          [
            "v2",
            "v3"
          ],
          [
            "v3",
            "v4"
          ],
          [
            "v4",
            "v5"
          ],
          [
            "v5",
            "v6"
          ],
          [
            "v6",
            "v7"
          ],
          [
            "v7",
            "v8"
          ],
          [
            "v8",
            "v0"
          ]
        ],
        "domain": "Complete_finite_minimum_vertex_cover",
        "operation": "mincover-exhaustive-minimizers",
        "semantic_template": "qf-mod46-mincover-exhaustive-minimizers/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v8\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_or_symbolic_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-15",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261015-advance-20261115-r06-minimum-vertex-cover-0015",
    "batch_id": "batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "753bcbc17c630fbe5968b00ce8d6db3c67b2238e13fdeaf6e2cdd2a4cd4a4984",
      "family_id": "qf-mod46-mincover-exhaustive-minimizers",
      "seed": 202611156015,
      "semantic_template": "qf-mod46-mincover-exhaustive-minimizers/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v8\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"]}"
    },
    "fields": {
      "title": "Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits",
      "stem": "Use complete unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8] and loop-free duplicate-free undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v0]]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (vertices, edges, all_covers, minimum_size, minimum_covers, certificate).",
      "answer": {
        "kind": "expression",
        "value": "([v0, v1, v2, v3, v4, v5, v6, v7, v8], [[v0, v1], [v0, v8], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8]], [[v0, v1, v3, v5, v7], [v0, v2, v3, v5, v7], [v0, v1, v2, v3, v5, v7], [v0, v2, v4, v5, v7], [v0, v1, v2, v4, v5, v7], [v0, v1, v3, v4, v5, v7], [v0, v2, v3, v4, v5, v7], [v0, v1, v2, v3, v4, v5, v7], [v0, v2, v4, v6, v7], [v0, v1, v2, v4, v6, v7], [v0, v1, v3, v4, v6, v7], [v0, v2, v3, v4, v6, v7], [v0, v1, v2, v3, v4, v6, v7], [v0, v1, v3, v5, v6, v7], [v0, v2, v3, v5, v6, v7], [v0, v1, v2, v3, v5, v6, v7], [v0, v2, v4, v5, v6, v7], [v0, v1, v2, v4, v5, v6, v7], [v0, v1, v3, v4, v5, v6, v7], [v0, v2, v3, v4, v5, v6, v7], [v0, v1, v2, v3, v4, v5, v6, v7], [v0, v2, v4, v6, v8], [v1, v2, v4, v6, v8], [v0, v1, v2, v4, v6, v8], [v1, v3, v4, v6, v8], [v0, v1, v3, v4, v6, v8], [v0, v2, v3, v4, v6, v8], [v1, v2, v3, v4, v6, v8], [v0, v1, v2, v3, v4, v6, v8], [v1, v3, v5, v6, v8], [v0, v1, v3, v5, v6, v8], [v0, v2, v3, v5, v6, v8], [v1, v2, v3, v5, v6, v8], [v0, v1, v2, v3, v5, v6, v8], [v0, v2, v4, v5, v6, v8], [v1, v2, v4, v5, v6, v8], [v0, v1, v2, v4, v5, v6, v8], [v1, v3, v4, v5, v6, v8], [v0, v1, v3, v4, v5, v6, v8], [v0, v2, v3, v4, v5, v6, v8], [v1, v2, v3, v4, v5, v6, v8], [v0, v1, v2, v3, v4, v5, v6, v8], [v1, v3, v5, v7, v8], [v0, v1, v3, v5, v7, v8], [v0, v2, v3, v5, v7, v8], [v1, v2, v3, v5, v7, v8], [v0, v1, v2, v3, v5, v7, v8], [v0, v2, v4, v5, v7, v8], [v1, v2, v4, v5, v7, v8], [v0, v1, v2, v4, v5, v7, v8], [v1, v3, v4, v5, v7, v8], [v0, v1, v3, v4, v5, v7, v8], [v0, v2, v3, v4, v5, v7, v8], [v1, v2, v3, v4, v5, v7, v8], [v0, v1, v2, v3, v4, v5, v7, v8], [v0, v2, v4, v6, v7, v8], [v1, v2, v4, v6, v7, v8], [v0, v1, v2, v4, v6, v7, v8], [v1, v3, v4, v6, v7, v8], [v0, v1, v3, v4, v6, v7, v8], [v0, v2, v3, v4, v6, v7, v8], [v1, v2, v3, v4, v6, v7, v8], [v0, v1, v2, v3, v4, v6, v7, v8], [v1, v3, v5, v6, v7, v8], [v0, v1, v3, v5, v6, v7, v8], [v0, v2, v3, v5, v6, v7, v8], [v1, v2, v3, v5, v6, v7, v8], [v0, v1, v2, v3, v5, v6, v7, v8], [v0, v2, v4, v5, v6, v7, v8], [v1, v2, v4, v5, v6, v7, v8], [v0, v1, v2, v4, v5, v6, v7, v8], [v1, v3, v4, v5, v6, v7, v8], [v0, v1, v3, v4, v5, v6, v7, v8], [v0, v2, v3, v4, v5, v6, v7, v8], [v1, v2, v3, v4, v5, v6, v7, v8], [v0, v1, v2, v3, v4, v5, v6, v7, v8]], 5, [[v0, v1, v3, v5, v7], [v0, v2, v3, v5, v7], [v0, v2, v4, v5, v7], [v0, v2, v4, v6, v7], [v0, v2, v4, v6, v8], [v1, v2, v4, v6, v8], [v1, v3, v4, v6, v8], [v1, v3, v5, v6, v8], [v1, v3, v5, v7, v8]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)",
        "explanation": "Retain all covers in source subset order, then minimum cardinality and all covers attaining it. This complete bounded search is not a polynomial-time algorithm claim, approximation guarantee or use of Konig theorem. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5, v6, v7, v8], [[v0, v1], [v0, v8], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8]], [[v0, v1, v3, v5, v7], [v0, v2, v3, v5, v7], [v0, v1, v2, v3, v5, v7], [v0, v2, v4, v5, v7], [v0, v1, v2, v4, v5, v7], [v0, v1, v3, v4, v5, v7], [v0, v2, v3, v4, v5, v7], [v0, v1, v2, v3, v4, v5, v7], [v0, v2, v4, v6, v7], [v0, v1, v2, v4, v6, v7], [v0, v1, v3, v4, v6, v7], [v0, v2, v3, v4, v6, v7], [v0, v1, v2, v3, v4, v6, v7], [v0, v1, v3, v5, v6, v7], [v0, v2, v3, v5, v6, v7], [v0, v1, v2, v3, v5, v6, v7], [v0, v2, v4, v5, v6, v7], [v0, v1, v2, v4, v5, v6, v7], [v0, v1, v3, v4, v5, v6, v7], [v0, v2, v3, v4, v5, v6, v7], [v0, v1, v2, v3, v4, v5, v6, v7], [v0, v2, v4, v6, v8], [v1, v2, v4, v6, v8], [v0, v1, v2, v4, v6, v8], [v1, v3, v4, v6, v8], [v0, v1, v3, v4, v6, v8], [v0, v2, v3, v4, v6, v8], [v1, v2, v3, v4, v6, v8], [v0, v1, v2, v3, v4, v6, v8], [v1, v3, v5, v6, v8], [v0, v1, v3, v5, v6, v8], [v0, v2, v3, v5, v6, v8], [v1, v2, v3, v5, v6, v8], [v0, v1, v2, v3, v5, v6, v8], [v0, v2, v4, v5, v6, v8], [v1, v2, v4, v5, v6, v8], [v0, v1, v2, v4, v5, v6, v8], [v1, v3, v4, v5, v6, v8], [v0, v1, v3, v4, v5, v6, v8], [v0, v2, v3, v4, v5, v6, v8], [v1, v2, v3, v4, v5, v6, v8], [v0, v1, v2, v3, v4, v5, v6, v8], [v1, v3, v5, v7, v8], [v0, v1, v3, v5, v7, v8], [v0, v2, v3, v5, v7, v8], [v1, v2, v3, v5, v7, v8], [v0, v1, v2, v3, v5, v7, v8], [v0, v2, v4, v5, v7, v8], [v1, v2, v4, v5, v7, v8], [v0, v1, v2, v4, v5, v7, v8], [v1, v3, v4, v5, v7, v8], [v0, v1, v3, v4, v5, v7, v8], [v0, v2, v3, v4, v5, v7, v8], [v1, v2, v3, v4, v5, v7, v8], [v0, v1, v2, v3, v4, v5, v7, v8], [v0, v2, v4, v6, v7, v8], [v1, v2, v4, v6, v7, v8], [v0, v1, v2, v4, v6, v7, v8], [v1, v3, v4, v6, v7, v8], [v0, v1, v3, v4, v6, v7, v8], [v0, v2, v3, v4, v6, v7, v8], [v1, v2, v3, v4, v6, v7, v8], [v0, v1, v2, v3, v4, v6, v7, v8], [v1, v3, v5, v6, v7, v8], [v0, v1, v3, v5, v6, v7, v8], [v0, v2, v3, v5, v6, v7, v8], [v1, v2, v3, v5, v6, v7, v8], [v0, v1, v2, v3, v5, v6, v7, v8], [v0, v2, v4, v5, v6, v7, v8], [v1, v2, v4, v5, v6, v7, v8], [v0, v1, v2, v4, v5, v6, v7, v8], [v1, v3, v4, v5, v6, v7, v8], [v0, v1, v3, v4, v5, v6, v7, v8], [v0, v2, v3, v4, v5, v6, v7, v8], [v1, v2, v3, v4, v5, v6, v7, v8], [v0, v1, v2, v3, v4, v5, v6, v7, v8]], 5, [[v0, v1, v3, v5, v7], [v0, v2, v3, v5, v7], [v0, v2, v4, v5, v7], [v0, v2, v4, v6, v7], [v0, v2, v4, v6, v8], [v1, v2, v4, v6, v8], [v1, v3, v4, v6, v8], [v1, v3, v5, v6, v8], [v1, v3, v5, v7, v8]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)."
      },
      "skills": [
        "Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits"
      ],
      "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."
    }
  },
  "givens": [
    {
      "id": "G001",
      "text": {
        "zh-Hant": "bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-exhaustive-minimizers；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v0\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use complete unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8] and loop-free duplicate-free undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v0]]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (vertices, edges, all_covers, minimum_size, minimum_covers, certificate)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [mincover-exhaustive-minimizers]",
      "after": "derived = [[vertices, edges, all_covers, minimum_size, minimum_covers, certificate]]",
      "relation": "implies",
      "rule_id": "source_ordered_fields_without_JS",
      "conditions": [],
      "explanation": {
        "zh-Hant": "本頁直接列完整來源欄序，不需JavaScript；contract的6/4/6/6與IVT的8/7/8由原mode選，不補欄也不由答案形狀猜。",
        "en": "List the full source field order without JavaScript.Original modes select contract6/4/6/6 and IVT8/7/8 roles,without padding or answer-shape inference."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "source roles = [[v0, v1, v2, v3, v4, v5, v6, v7, v8], [[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v0]], null]",
      "after": "derived = [[v0, v1, v2, v3, v4, v5, v6, v7, v8], [[v0, v1], [v0, v8], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8]], null]",
      "relation": "implies",
      "rule_id": "Mincover_complete_simple_graph_UTF16_English_and_optional_candidate",
      "conditions": [],
      "explanation": {
        "zh-Hant": "非空carrier至多12；端點UTF16排序後edge族English排序，loop/反向duplicate拒絕。未提供candidate時保持null，不擅補空candidate。",
        "en": "The carrier is nonempty with at most12 vertices;sort endpoints by UTF16 then edge families in English.Reject loops/reversed duplicates.An omitted candidate stays null,not an invented empty candidate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "source roles = [[v0, v1, v2, v3, v4, v5, v6, v7, v8], [[v0, v1], [v0, v8], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8]], 512]",
      "after": "derived = [[171, 173, 175, 181, 183, 187, 189, 191, 213, 215, 219, 221, 223, 235, 237, 239, 245, 247, 251, 253, 255, 341, 342, 343, 346, 347, 349, 350, 351, 362, 363, 365, 366, 367, 373, 374, 375, 378, 379, 381, 382, 383, 426, 427, 429, 430, 431, 437, 438, 439, 442, 443, 445, 446, 447, 469, 470, 471, 474, 475, 477, 478, 479, 490, 491, 493, 494, 495, 501, 502, 503, 506, 507, 509, 510, 511], [[0, 0], [1, 2], [2, 1], [3, 3], [4, 0], [5, 4], [6, 1], [7, 4], [8, 0], [9, 2], [10, 1], [11, 5], [12, 0], [13, 5], [14, 1], [15, 5], [16, 0], [17, 2], [18, 1], [19, 3], [20, 0], [21, 6], [22, 1], [23, 6], [24, 0], [25, 2], [26, 1], [27, 6], [28, 0], [29, 6], [30, 1], [31, 6], [32, 0], [33, 2], [34, 1], [35, 3], [36, 0], [37, 4], [38, 1], [39, 4], [40, 0], [41, 2], [42, 1], [43, 7], [44, 0], [45, 7], [46, 1], [47, 7], [48, 0], [49, 2], [50, 1], [51, 3], [52, 0], [53, 7], [54, 1], [55, 7], [56, 0], [57, 2], [58, 1], [59, 7], [60, 0], [61, 7], [62, 1], [63, 7], [64, 0], [65, 2], [66, 1], [67, 3], [68, 0], [69, 4], [70, 1], [71, 4], [72, 0], [73, 2], [74, 1], [75, 5], [76, 0], [77, 5], [78, 1], [79, 5], [80, 0], [81, 2], [82, 1], [83, 3], [84, 0], [85, 8], [86, 1], [87, 8], [88, 0], [89, 2], [90, 1], [91, 8], [92, 0], [93, 8], [94, 1], [95, 8], [96, 0], [97, 2], [98, 1], [99, 3], [100, 0], [101, 4], [102, 1], [103, 4], [104, 0], [105, 2], [106, 1], [107, 8], [108, 0], [109, 8], [110, 1], [111, 8], [112, 0], [113, 2], [114, 1], [115, 3], [116, 0], [117, 8], [118, 1], [119, 8], [120, 0], [121, 2], [122, 1], [123, 8], [124, 0], [125, 8], [126, 1], [127, 8], [128, 0], [129, 2], [130, 1], [131, 3], [132, 0], [133, 4], [134, 1], [135, 4], [136, 0], [137, 2], [138, 1], [139, 5], [140, 0], [141, 5], [142, 1], [143, 5], [144, 0], [145, 2], [146, 1], [147, 3], [148, 0], [149, 6], [150, 1], [151, 6], [152, 0], [153, 2], [154, 1], [155, 6], [156, 0], [157, 6], [158, 1], [159, 6], [160, 0], [161, 2], [162, 1], [163, 3], [164, 0], [165, 4], [166, 1], [167, 4], [168, 0], [169, 2], [170, 1], [172, 0], [174, 1], [176, 0], [177, 2], [178, 1], [179, 3], [180, 0], [182, 1], [184, 0], [185, 2], [186, 1], [188, 0], [190, 1], [192, 0], [193, 2], [194, 1], [195, 3], [196, 0], [197, 4], [198, 1], [199, 4], [200, 0], [201, 2], [202, 1], [203, 5], [204, 0], [205, 5], [206, 1], [207, 5], [208, 0], [209, 2], [210, 1], [211, 3], [212, 0], [214, 1], [216, 0], [217, 2], [218, 1], [220, 0], [222, 1], [224, 0], [225, 2], [226, 1], [227, 3], [228, 0], [229, 4], [230, 1], [231, 4], [232, 0], [233, 2], [234, 1], [236, 0], [238, 1], [240, 0], [241, 2], [242, 1], [243, 3], [244, 0], [246, 1], [248, 0], [249, 2], [250, 1], [252, 0], [254, 1], [256, 0], [257, 2], [258, 3], [259, 3], [260, 0], [261, 4], [262, 4], [263, 4], [264, 0], [265, 2], [266, 5], [267, 5], [268, 0], [269, 5], [270, 5], [271, 5], [272, 0], [273, 2], [274, 3], [275, 3], [276, 0], [277, 6], [278, 6], [279, 6], [280, 0], [281, 2], [282, 6], [283, 6], [284, 0], [285, 6], [286, 6], [287, 6], [288, 0], [289, 2], [290, 3], [291, 3], [292, 0], [293, 4], [294, 4], [295, 4], [296, 0], [297, 2], [298, 7], [299, 7], [300, 0], [301, 7], [302, 7], [303, 7], [304, 0], [305, 2], [306, 3], [307, 3], [308, 0], [309, 7], [310, 7], [311, 7], [312, 0], [313, 2], [314, 7], [315, 7], [316, 0], [317, 7], [318, 7], [319, 7], [320, 0], [321, 2], [322, 3], [323, 3], [324, 0], [325, 4], [326, 4], [327, 4], [328, 0], [329, 2], [330, 5], [331, 5], [332, 0], [333, 5], [334, 5], [335, 5], [336, 0], [337, 2], [338, 3], [339, 3], [340, 0], [344, 0], [345, 2], [348, 0], [352, 0], [353, 2], [354, 3], [355, 3], [356, 0], [357, 4], [358, 4], [359, 4], [360, 0], [361, 2], [364, 0], [368, 0], [369, 2], [370, 3], [371, 3], [372, 0], [376, 0], [377, 2], [380, 0], [384, 0], [385, 2], [386, 3], [387, 3], [388, 0], [389, 4], [390, 4], [391, 4], [392, 0], [393, 2], [394, 5], [395, 5], [396, 0], [397, 5], [398, 5], [399, 5], [400, 0], [401, 2], [402, 3], [403, 3], [404, 0], [405, 6], [406, 6], [407, 6], [408, 0], [409, 2], [410, 6], [411, 6], [412, 0], [413, 6], [414, 6], [415, 6], [416, 0], [417, 2], [418, 3], [419, 3], [420, 0], [421, 4], [422, 4], [423, 4], [424, 0], [425, 2], [428, 0], [432, 0], [433, 2], [434, 3], [435, 3], [436, 0], [440, 0], [441, 2], [444, 0], [448, 0], [449, 2], [450, 3], [451, 3], [452, 0], [453, 4], [454, 4], [455, 4], [456, 0], [457, 2], [458, 5], [459, 5], [460, 0], [461, 5], [462, 5], [463, 5], [464, 0], [465, 2], [466, 3], [467, 3], [468, 0], [472, 0], [473, 2], [476, 0], [480, 0], [481, 2], [482, 3], [483, 3], [484, 0], [485, 4], [486, 4], [487, 4], [488, 0], [489, 2], [492, 0], [496, 0], [497, 2], [498, 3], [499, 3], [500, 0], [504, 0], [505, 2], [508, 0]], [[v0, v1, v3, v5, v7], [v0, v2, v3, v5, v7], [v0, v1, v2, v3, v5, v7], [v0, v2, v4, v5, v7], [v0, v1, v2, v4, v5, v7], [v0, v1, v3, v4, v5, v7], [v0, v2, v3, v4, v5, v7], [v0, v1, v2, v3, v4, v5, v7], [v0, v2, v4, v6, v7], [v0, v1, v2, v4, v6, v7], [v0, v1, v3, v4, v6, v7], [v0, v2, v3, v4, v6, v7], [v0, v1, v2, v3, v4, v6, v7], [v0, v1, v3, v5, v6, v7], [v0, v2, v3, v5, v6, v7], [v0, v1, v2, v3, v5, v6, v7], [v0, v2, v4, v5, v6, v7], [v0, v1, v2, v4, v5, v6, v7], [v0, v1, v3, v4, v5, v6, v7], [v0, v2, v3, v4, v5, v6, v7], [v0, v1, v2, v3, v4, v5, v6, v7], [v0, v2, v4, v6, v8], [v1, v2, v4, v6, v8], [v0, v1, v2, v4, v6, v8], [v1, v3, v4, v6, v8], [v0, v1, v3, v4, v6, v8], [v0, v2, v3, v4, v6, v8], [v1, v2, v3, v4, v6, v8], [v0, v1, v2, v3, v4, v6, v8], [v1, v3, v5, v6, v8], [v0, v1, v3, v5, v6, v8], [v0, v2, v3, v5, v6, v8], [v1, v2, v3, v5, v6, v8], [v0, v1, v2, v3, v5, v6, v8], [v0, v2, v4, v5, v6, v8], [v1, v2, v4, v5, v6, v8], [v0, v1, v2, v4, v5, v6, v8], [v1, v3, v4, v5, v6, v8], [v0, v1, v3, v4, v5, v6, v8], [v0, v2, v3, v4, v5, v6, v8], [v1, v2, v3, v4, v5, v6, v8], [v0, v1, v2, v3, v4, v5, v6, v8], [v1, v3, v5, v7, v8], [v0, v1, v3, v5, v7, v8], [v0, v2, v3, v5, v7, v8], [v1, v2, v3, v5, v7, v8], [v0, v1, v2, v3, v5, v7, v8], [v0, v2, v4, v5, v7, v8], [v1, v2, v4, v5, v7, v8], [v0, v1, v2, v4, v5, v7, v8], [v1, v3, v4, v5, v7, v8], [v0, v1, v3, v4, v5, v7, v8], [v0, v2, v3, v4, v5, v7, v8], [v1, v2, v3, v4, v5, v7, v8], [v0, v1, v2, v3, v4, v5, v7, v8], [v0, v2, v4, v6, v7, v8], [v1, v2, v4, v6, v7, v8], [v0, v1, v2, v4, v6, v7, v8], [v1, v3, v4, v6, v7, v8], [v0, v1, v3, v4, v6, v7, v8], [v0, v2, v3, v4, v6, v7, v8], [v1, v2, v3, v4, v6, v7, v8], [v0, v1, v2, v3, v4, v6, v7, v8], [v1, v3, v5, v6, v7, v8], [v0, v1, v3, v5, v6, v7, v8], [v0, v2, v3, v5, v6, v7, v8], [v1, v2, v3, v5, v6, v7, v8], [v0, v1, v2, v3, v5, v6, v7, v8], [v0, v2, v4, v5, v6, v7, v8], [v1, v2, v4, v5, v6, v7, v8], [v0, v1, v2, v4, v5, v6, v7, v8], [v1, v3, v4, v5, v6, v7, v8], [v0, v1, v3, v4, v5, v6, v7, v8], [v0, v2, v3, v4, v5, v6, v7, v8], [v1, v2, v3, v4, v5, v6, v7, v8], [v0, v1, v2, v3, v4, v5, v6, v7, v8]], [0, 0, 0, 0, 0, 9, 30, 27, 9, 1], 5, [[v0, v1, v3, v5, v7], [v0, v2, v3, v5, v7], [v0, v2, v4, v5, v7], [v0, v2, v4, v6, v7], [v0, v2, v4, v6, v8], [v1, v2, v4, v6, v8], [v1, v3, v4, v6, v8], [v1, v3, v5, v6, v8], [v1, v3, v5, v7, v8]]]",
      "relation": "implies",
      "rule_id": "Mincover_ALL_masks_acceptance_refutations_ALLcovers_ALLminimizers",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐全部mask核所有edges；每個失敗mask保首uncovered edge index反例，通過者全列source順序。較小size計數0給下界，實際最小解給上界；全部ties依numeric mask保留，不改English/大小序。",
        "en": "Audit every mask against every edge,retaining the first uncovered-edge refutation for each failure and every accepted set in source order.Zero smaller-size counts give a lower bound and actual minimum covers attain it.Keep all ties in numeric-mask order,not English/size order."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      [
        "v0",
        "v1",
        "v2",
        "v3",
        "v4",
        "v5",
        "v6",
        "v7",
        "v8"
      ],
      [
        [
          "v0",
          "v1"
        ],
        [
          "v0",
          "v8"
        ],
        [
          "v1",
          "v2"
        ],
        [
          "v2",
          "v3"
        ],
        [
          "v3",
          "v4"
        ],
        [
          "v4",
          "v5"
        ],
        [
          "v5",
          "v6"
        ],
        [
          "v6",
          "v7"
        ],
        [
          "v7",
          "v8"
        ]
      ],
      [
        [
          "v0",
          "v1",
          "v3",
          "v5",
          "v7"
        ],
        [
          "v0",
          "v2",
          "v3",
          "v5",
          "v7"
        ],
        [
          "v0",
          "v1",
          "v2",
          "v3",
          "v5",
          "v7"
        ],
        [
          "v0",
          "v2",
          "v4",
          "v5",
          "v7"
        ],
        [
          "v0",
          "v1",
          "v2",
          "v4",
          "v5",
          "v7"
        ],
        [
          "v0",
          "v1",
          "v3",
          "v4",
          "v5",
          "v7"
        ],
        [
          "v0",
          "v2",
          "v3",
          "v4",
          "v5",
          "v7"
        ],
        [
          "v0",
          "v1",
          "v2",
          "v3",
          "v4",
          "v5",
          "v7"
        ],
        [
          "v0",
          "v2",
          "v4",
          "v6",
          "v7"
        ],
        [
          "v0",
          "v1",
          "v2",
          "v4",
          "v6",
          "v7"
        ],
        [
          "v0",
          "v1",
          "v3",
          "v4",
          "v6",
          "v7"
        ],
        [
          "v0",
          "v2",
          "v3",
          "v4",
          "v6",
          "v7"
        ],
        [
          "v0",
          "v1",
          "v2",
          "v3",
          "v4",
          "v6",
          "v7"
        ],
        [
          "v0",
          "v1",
          "v3",
          "v5",
          "v6",
          "v7"
        ],
        [
          "v0",
          "v2",
          "v3",
          "v5",
          "v6",
          "v7"
        ],
        [
          "v0",
          "v1",
          "v2",
          "v3",
          "v5",
          "v6",
          "v7"
        ],
        [
          "v0",
          "v2",
          "v4",
          "v5",
          "v6",
          "v7"
        ],
        [
          "v0",
          "v1",
          "v2",
          "v4",
          "v5",
          "v6",
          "v7"
        ],
        [
          "v0",
          "v1",
          "v3",
          "v4",
          "v5",
          "v6",
          "v7"
        ],
        [
          "v0",
          "v2",
          "v3",
          "v4",
          "v5",
          "v6",
          "v7"
        ],
        [
          "v0",
          "v1",
          "v2",
          "v3",
          "v4",
          "v5",
          "v6",
          "v7"
        ],
        [
          "v0",
          "v2",
          "v4",
          "v6",
          "v8"
        ],
        [
          "v1",
          "v2",
          "v4",
          "v6",
          "v8"
        ],
        [
          "v0",
          "v1",
          "v2",
          "v4",
          "v6",
          "v8"
        ],
        [
          "v1",
          "v3",
          "v4",
          "v6",
          "v8"
        ],
        [
          "v0",
          "v1",
          "v3",
          "v4",
          "v6",
          "v8"
        ],
        [
          "v0",
          "v2",
          "v3",
          "v4",
          "v6",
          "v8"
        ],
        [
          "v1",
          "v2",
          "v3",
          "v4",
          "v6",
          "v8"
        ],
        [
          "v0",
          "v1",
          "v2",
          "v3",
          "v4",
          "v6",
          "v8"
        ],
        [
          "v1",
          "v3",
          "v5",
          "v6",
          "v8"
        ],
        [
          "v0",
          "v1",
          "v3",
          "v5",
          "v6",
          "v8"
        ],
        [
          "v0",
          "v2",
          "v3",
          "v5",
          "v6",
          "v8"
        ],
        [
          "v1",
          "v2",
          "v3",
          "v5",
          "v6",
          "v8"
        ],
        [
          "v0",
          "v1",
          "v2",
          "v3",
          "v5",
          "v6",
          "v8"
        ],
        [
          "v0",
          "v2",
          "v4",
          "v5",
          "v6",
          "v8"
        ],
        [
          "v1",
          "v2",
          "v4",
          "v5",
          "v6",
          "v8"
        ],
        [
          "v0",
          "v1",
          "v2",
          "v4",
          "v5",
          "v6",
          "v8"
        ],
        [
          "v1",
          "v3",
          "v4",
          "v5",
          "v6",
          "v8"
        ],
        [
          "v0",
          "v1",
          "v3",
          "v4",
          "v5",
          "v6",
          "v8"
        ],
        [
          "v0",
          "v2",
          "v3",
          "v4",
          "v5",
          "v6",
          "v8"
        ],
        [
          "v1",
          "v2",
          "v3",
          "v4",
          "v5",
          "v6",
          "v8"
        ],
        [
          "v0",
          "v1",
          "v2",
          "v3",
          "v4",
          "v5",
          "v6",
          "v8"
        ],
        [
          "v1",
          "v3",
          "v5",
          "v7",
          "v8"
        ],
        [
          "v0",
          "v1",
          "v3",
          "v5",
          "v7",
          "v8"
        ],
        [
          "v0",
          "v2",
          "v3",
          "v5",
          "v7",
          "v8"
        ],
        [
          "v1",
          "v2",
          "v3",
          "v5",
          "v7",
          "v8"
        ],
        [
          "v0",
          "v1",
          "v2",
          "v3",
          "v5",
          "v7",
          "v8"
        ],
        [
          "v0",
          "v2",
          "v4",
          "v5",
          "v7",
          "v8"
        ],
        [
          "v1",
          "v2",
          "v4",
          "v5",
          "v7",
          "v8"
        ],
        [
          "v0",
          "v1",
          "v2",
          "v4",
          "v5",
          "v7",
          "v8"
        ],
        [
          "v1",
          "v3",
          "v4",
          "v5",
          "v7",
          "v8"
        ],
        [
          "v0",
          "v1",
          "v3",
          "v4",
          "v5",
          "v7",
          "v8"
        ],
        [
          "v0",
          "v2",
          "v3",
          "v4",
          "v5",
          "v7",
          "v8"
        ],
        [
          "v1",
          "v2",
          "v3",
          "v4",
          "v5",
          "v7",
          "v8"
        ],
        [
          "v0",
          "v1",
          "v2",
          "v3",
          "v4",
          "v5",
          "v7",
          "v8"
        ],
        [
          "v0",
          "v2",
          "v4",
          "v6",
          "v7",
          "v8"
        ],
        [
          "v1",
          "v2",
          "v4",
          "v6",
          "v7",
          "v8"
        ],
        [
          "v0",
          "v1",
          "v2",
          "v4",
          "v6",
          "v7",
          "v8"
        ],
        [
          "v1",
          "v3",
          "v4",
          "v6",
          "v7",
          "v8"
        ],
        [
          "v0",
          "v1",
          "v3",
          "v4",
          "v6",
          "v7",
          "v8"
        ],
        [
          "v0",
          "v2",
          "v3",
          "v4",
          "v6",
          "v7",
          "v8"
        ],
        [
          "v1",
          "v2",
          "v3",
          "v4",
          "v6",
          "v7",
          "v8"
        ],
        [
          "v0",
          "v1",
          "v2",
          "v3",
          "v4",
          "v6",
          "v7",
          "v8"
        ],
        [
          "v1",
          "v3",
          "v5",
          "v6",
          "v7",
          "v8"
        ],
        [
          "v0",
          "v1",
          "v3",
          "v5",
          "v6",
          "v7",
          "v8"
        ],
        [
          "v0",
          "v2",
          "v3",
          "v5",
          "v6",
          "v7",
          "v8"
        ],
        [
          "v1",
          "v2",
          "v3",
          "v5",
          "v6",
          "v7",
          "v8"
        ],
        [
          "v0",
          "v1",
          "v2",
          "v3",
          "v5",
          "v6",
          "v7",
          "v8"
        ],
        [
          "v0",
          "v2",
          "v4",
          "v5",
          "v6",
          "v7",
          "v8"
        ],
        [
          "v1",
          "v2",
          "v4",
          "v5",
          "v6",
          "v7",
          "v8"
        ],
        [
          "v0",
          "v1",
          "v2",
          "v4",
          "v5",
          "v6",
          "v7",
          "v8"
        ],
        [
          "v1",
          "v3",
          "v4",
          "v5",
          "v6",
          "v7",
          "v8"
        ],
        [
          "v0",
          "v1",
          "v3",
          "v4",
          "v5",
          "v6",
          "v7",
          "v8"
        ],
        [
          "v0",
          "v2",
          "v3",
          "v4",
          "v5",
          "v6",
          "v7",
          "v8"
        ],
        [
          "v1",
          "v2",
          "v3",
          "v4",
          "v5",
          "v6",
          "v7",
          "v8"
        ],
        [
          "v0",
          "v1",
          "v2",
          "v3",
          "v4",
          "v5",
          "v6",
          "v7",
          "v8"
        ]
      ],
      5,
      [
        [
          "v0",
          "v1",
          "v3",
          "v5",
          "v7"
        ],
        [
          "v0",
          "v2",
          "v3",
          "v5",
          "v7"
        ],
        [
          "v0",
          "v2",
          "v4",
          "v5",
          "v7"
        ],
        [
          "v0",
          "v2",
          "v4",
          "v6",
          "v7"
        ],
        [
          "v0",
          "v2",
          "v4",
          "v6",
          "v8"
        ],
        [
          "v1",
          "v2",
          "v4",
          "v6",
          "v8"
        ],
        [
          "v1",
          "v3",
          "v4",
          "v6",
          "v8"
        ],
        [
          "v1",
          "v3",
          "v5",
          "v6",
          "v8"
        ],
        [
          "v1",
          "v3",
          "v5",
          "v7",
          "v8"
        ]
      ],
      "all_vertex_subsets_are_audited_before_selecting_all_minimizers"
    ]
  },
  "conclusion": {
    "canonical_value": "([v0, v1, v2, v3, v4, v5, v6, v7, v8], [[v0, v1], [v0, v8], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8]], [[v0, v1, v3, v5, v7], [v0, v2, v3, v5, v7], [v0, v1, v2, v3, v5, v7], [v0, v2, v4, v5, v7], [v0, v1, v2, v4, v5, v7], [v0, v1, v3, v4, v5, v7], [v0, v2, v3, v4, v5, v7], [v0, v1, v2, v3, v4, v5, v7], [v0, v2, v4, v6, v7], [v0, v1, v2, v4, v6, v7], [v0, v1, v3, v4, v6, v7], [v0, v2, v3, v4, v6, v7], [v0, v1, v2, v3, v4, v6, v7], [v0, v1, v3, v5, v6, v7], [v0, v2, v3, v5, v6, v7], [v0, v1, v2, v3, v5, v6, v7], [v0, v2, v4, v5, v6, v7], [v0, v1, v2, v4, v5, v6, v7], [v0, v1, v3, v4, v5, v6, v7], [v0, v2, v3, v4, v5, v6, v7], [v0, v1, v2, v3, v4, v5, v6, v7], [v0, v2, v4, v6, v8], [v1, v2, v4, v6, v8], [v0, v1, v2, v4, v6, v8], [v1, v3, v4, v6, v8], [v0, v1, v3, v4, v6, v8], [v0, v2, v3, v4, v6, v8], [v1, v2, v3, v4, v6, v8], [v0, v1, v2, v3, v4, v6, v8], [v1, v3, v5, v6, v8], [v0, v1, v3, v5, v6, v8], [v0, v2, v3, v5, v6, v8], [v1, v2, v3, v5, v6, v8], [v0, v1, v2, v3, v5, v6, v8], [v0, v2, v4, v5, v6, v8], [v1, v2, v4, v5, v6, v8], [v0, v1, v2, v4, v5, v6, v8], [v1, v3, v4, v5, v6, v8], [v0, v1, v3, v4, v5, v6, v8], [v0, v2, v3, v4, v5, v6, v8], [v1, v2, v3, v4, v5, v6, v8], [v0, v1, v2, v3, v4, v5, v6, v8], [v1, v3, v5, v7, v8], [v0, v1, v3, v5, v7, v8], [v0, v2, v3, v5, v7, v8], [v1, v2, v3, v5, v7, v8], [v0, v1, v2, v3, v5, v7, v8], [v0, v2, v4, v5, v7, v8], [v1, v2, v4, v5, v7, v8], [v0, v1, v2, v4, v5, v7, v8], [v1, v3, v4, v5, v7, v8], [v0, v1, v3, v4, v5, v7, v8], [v0, v2, v3, v4, v5, v7, v8], [v1, v2, v3, v4, v5, v7, v8], [v0, v1, v2, v3, v4, v5, v7, v8], [v0, v2, v4, v6, v7, v8], [v1, v2, v4, v6, v7, v8], [v0, v1, v2, v4, v6, v7, v8], [v1, v3, v4, v6, v7, v8], [v0, v1, v3, v4, v6, v7, v8], [v0, v2, v3, v4, v6, v7, v8], [v1, v2, v3, v4, v6, v7, v8], [v0, v1, v2, v3, v4, v6, v7, v8], [v1, v3, v5, v6, v7, v8], [v0, v1, v3, v5, v6, v7, v8], [v0, v2, v3, v5, v6, v7, v8], [v1, v2, v3, v5, v6, v7, v8], [v0, v1, v2, v3, v5, v6, v7, v8], [v0, v2, v4, v5, v6, v7, v8], [v1, v2, v4, v5, v6, v7, v8], [v0, v1, v2, v4, v5, v6, v7, v8], [v1, v3, v4, v5, v6, v7, v8], [v0, v1, v3, v4, v5, v6, v7, v8], [v0, v2, v3, v4, v5, v6, v7, v8], [v1, v2, v3, v4, v5, v6, v7, v8], [v0, v1, v2, v3, v4, v5, v6, v7, v8]], 5, [[v0, v1, v3, v5, v7], [v0, v2, v3, v5, v7], [v0, v2, v4, v5, v7], [v0, v2, v4, v6, v7], [v0, v2, v4, v6, v8], [v1, v2, v4, v6, v8], [v1, v3, v4, v6, v8], [v1, v3, v5, v6, v8], [v1, v3, v5, v7, v8]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)",
    "explanation": {
      "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。",
      "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged."
    }
  },
  "definition_note": {
    "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。",
    "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate."
  },
  "source": {
    "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db",
    "question_binding": {
      "batch_id": "batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations",
      "batch_path": "public/data/questions/batches/batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations/questions.jsonl",
      "batch_sha256": "a358c588de9908cad99f8441c00419458fa8706a1d43ab7d5362e6fdc3b1704f",
      "original_line": 15,
      "original_line_sha256": "0683fa9263d3fbb68fbd99d0fb39bedbf5be0a81305d81ea9161eac0c0797ae7",
      "question_compact_sha256": "753bcbc17c630fbe5968b00ce8d6db3c67b2238e13fdeaf6e2cdd2a4cd4a4984",
      "english_row_compact_sha256": "5672eea51f46a01fbc475f1c95bad841c1a0545e2e6b2831f8e524e4484f90cb",
      "english_manifest_path": "public/data/i18n/editions/batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-minimum-vertex-cover": {
        "sha256": "4f8d510264312335a0f27c32a69f66f31350aede08398538e94a015583d8250b",
        "english_sha256": "0ec533e9f581c06215d447562ddc54c5249c8bf96d6f057f82089a97614d035c",
        "original_path": "public/data/mko/minimum-vertex-cover.json",
        "english_path": "public/data/i18n/editions/mko-minimum-vertex-cover.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "74f6f34e0a695e06f9d836b5f5c677f9f716d2291c75c8f2f5b809ff0935bcff",
    "original_candidate_sha256": "02e6b51de7b2d886fc5b67ef21ce706724dbebdbd04149fa756a205ca417cf1e",
    "original_candidate_line_sha256": "02e6b51de7b2d886fc5b67ef21ce706724dbebdbd04149fa756a205ca417cf1e"
  },
  "historical_source_state": {
    "review_state": {
      "derivation": "candidate_authored_and_locally_checked",
      "source_question_state": "candidate",
      "independent_AI_or_human_review": "NOT_PERFORMED",
      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
    },
    "scope_note": {
      "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。",
      "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged."
    },
    "label": "Frozen producer metadata; current per-question review is separate below."
  },
  "revision": {
    "id": "review-20261026-010-p52",
    "review_date": "2026-10-26",
    "owner_authorized_reader_publication": true,
    "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.",
    "review_receipts": {
      "semantic_review_sha256": "90df1e889242e9afdf01e068788ee8500ef366ffcfc0496e30a6800de32631c3",
      "question_first_witness_sha256": "ae1e849608590ca856c9f3b6911ba58e3bf53c608eb7369606192a770f48bd5c"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 3,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "32456c40f9ec15a9e271d79c442fa99da7bab3530c45e0a1a2b206a1ec6ed469",
      "arithmetic_steps": 0
    },
    "uses_answer_as_premise": false,
    "human_review": "NOT_PERFORMED",
    "formalization": "not_formalized",
    "Lean": "NOT_RUN",
    "Canonical_truth_promotion": false,
    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
  }
}