{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261015-advance-20261115-r06-minimum-vertex-cover-0023",
  "pilot_number": "R0051-26048",
  "number": 26048,
  "family_id": "qf-mod46-mincover-exhaustive-minimizers",
  "title": {
    "zh-Hant": "第 26048 題：最小頂點覆蓋",
    "en": "Question 26048: 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-0023",
    "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\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"],[\"v0\",\"v7\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([v0, v1, v2, v3, v4, v5, v6, v7], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7]], [[v0], [v0, v1], [v0, v2], [v0, v1, v2], [v0, v3], [v0, v1, v3], [v0, v2, v3], [v0, v1, v2, v3], [v0, v4], [v0, v1, v4], [v0, v2, v4], [v0, v1, v2, v4], [v0, v3, v4], [v0, v1, v3, v4], [v0, v2, v3, v4], [v0, v1, v2, v3, v4], [v0, v5], [v0, v1, v5], [v0, v2, v5], [v0, v1, v2, v5], [v0, v3, v5], [v0, v1, v3, v5], [v0, v2, v3, v5], [v0, v1, v2, v3, v5], [v0, v4, v5], [v0, v1, v4, v5], [v0, v2, v4, v5], [v0, v1, v2, v4, v5], [v0, v3, v4, v5], [v0, v1, v3, v4, v5], [v0, v2, v3, v4, v5], [v0, v1, v2, v3, v4, v5], [v0, v6], [v0, v1, v6], [v0, v2, v6], [v0, v1, v2, v6], [v0, v3, v6], [v0, v1, v3, v6], [v0, v2, v3, v6], [v0, v1, v2, v3, v6], [v0, v4, v6], [v0, v1, v4, v6], [v0, v2, v4, v6], [v0, v1, v2, v4, v6], [v0, v3, v4, v6], [v0, v1, v3, v4, v6], [v0, v2, v3, v4, v6], [v0, v1, v2, v3, v4, v6], [v0, v5, v6], [v0, v1, v5, v6], [v0, v2, v5, v6], [v0, v1, v2, v5, v6], [v0, v3, v5, v6], [v0, v1, v3, v5, v6], [v0, v2, v3, v5, v6], [v0, v1, v2, v3, v5, v6], [v0, v4, v5, v6], [v0, v1, v4, v5, v6], [v0, v2, v4, v5, v6], [v0, v1, v2, v4, v5, v6], [v0, v3, v4, v5, v6], [v0, v1, v3, v4, v5, v6], [v0, v2, v3, v4, v5, v6], [v0, v1, v2, v3, v4, v5, v6], [v0, v7], [v0, v1, v7], [v0, v2, v7], [v0, v1, v2, v7], [v0, v3, v7], [v0, v1, v3, v7], [v0, v2, v3, v7], [v0, v1, v2, v3, v7], [v0, v4, v7], [v0, v1, v4, v7], [v0, v2, v4, v7], [v0, v1, v2, v4, v7], [v0, v3, v4, v7], [v0, v1, v3, v4, v7], [v0, v2, v3, v4, v7], [v0, v1, v2, v3, v4, v7], [v0, v5, v7], [v0, v1, v5, v7], [v0, v2, v5, v7], [v0, v1, v2, v5, v7], [v0, v3, v5, v7], [v0, v1, v3, v5, v7], [v0, v2, v3, v5, v7], [v0, v1, v2, v3, v5, v7], [v0, v4, v5, v7], [v0, v1, v4, v5, v7], [v0, v2, v4, v5, v7], [v0, v1, v2, v4, v5, v7], [v0, v3, v4, v5, v7], [v0, v1, v3, v4, v5, v7], [v0, v2, v3, v4, v5, v7], [v0, v1, v2, v3, v4, v5, v7], [v0, v6, v7], [v0, v1, v6, v7], [v0, v2, v6, v7], [v0, v1, v2, v6, v7], [v0, v3, v6, v7], [v0, v1, v3, v6, v7], [v0, v2, v3, v6, v7], [v0, v1, v2, v3, v6, v7], [v0, v4, v6, v7], [v0, v1, v4, v6, v7], [v0, v2, v4, v6, v7], [v0, v1, v2, v4, v6, v7], [v0, v3, v4, v6, v7], [v0, v1, v3, v4, v6, v7], [v0, v2, v3, v4, v6, v7], [v0, v1, v2, v3, v4, v6, v7], [v0, v5, v6, v7], [v0, v1, v5, v6, v7], [v0, v2, v5, v6, v7], [v0, v1, v2, v5, v6, v7], [v0, v3, v5, v6, v7], [v0, v1, v3, v5, v6, v7], [v0, v2, v3, v5, v6, v7], [v0, v1, v2, v3, v5, v6, v7], [v0, v4, v5, v6, v7], [v0, v1, v4, v5, v6, v7], [v0, v2, v4, v5, v6, v7], [v0, v1, v2, v4, v5, v6, v7], [v0, v3, v4, v5, v6, v7], [v0, v1, v3, v4, v5, v6, v7], [v0, v2, v3, v4, v5, v6, v7], [v1, v2, v3, v4, v5, v6, v7], [v0, v1, v2, v3, v4, v5, v6, v7]], 1, [[v0]], 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], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7]], [[v0], [v0, v1], [v0, v2], [v0, v1, v2], [v0, v3], [v0, v1, v3], [v0, v2, v3], [v0, v1, v2, v3], [v0, v4], [v0, v1, v4], [v0, v2, v4], [v0, v1, v2, v4], [v0, v3, v4], [v0, v1, v3, v4], [v0, v2, v3, v4], [v0, v1, v2, v3, v4], [v0, v5], [v0, v1, v5], [v0, v2, v5], [v0, v1, v2, v5], [v0, v3, v5], [v0, v1, v3, v5], [v0, v2, v3, v5], [v0, v1, v2, v3, v5], [v0, v4, v5], [v0, v1, v4, v5], [v0, v2, v4, v5], [v0, v1, v2, v4, v5], [v0, v3, v4, v5], [v0, v1, v3, v4, v5], [v0, v2, v3, v4, v5], [v0, v1, v2, v3, v4, v5], [v0, v6], [v0, v1, v6], [v0, v2, v6], [v0, v1, v2, v6], [v0, v3, v6], [v0, v1, v3, v6], [v0, v2, v3, v6], [v0, v1, v2, v3, v6], [v0, v4, v6], [v0, v1, v4, v6], [v0, v2, v4, v6], [v0, v1, v2, v4, v6], [v0, v3, v4, v6], [v0, v1, v3, v4, v6], [v0, v2, v3, v4, v6], [v0, v1, v2, v3, v4, v6], [v0, v5, v6], [v0, v1, v5, v6], [v0, v2, v5, v6], [v0, v1, v2, v5, v6], [v0, v3, v5, v6], [v0, v1, v3, v5, v6], [v0, v2, v3, v5, v6], [v0, v1, v2, v3, v5, v6], [v0, v4, v5, v6], [v0, v1, v4, v5, v6], [v0, v2, v4, v5, v6], [v0, v1, v2, v4, v5, v6], [v0, v3, v4, v5, v6], [v0, v1, v3, v4, v5, v6], [v0, v2, v3, v4, v5, v6], [v0, v1, v2, v3, v4, v5, v6], [v0, v7], [v0, v1, v7], [v0, v2, v7], [v0, v1, v2, v7], [v0, v3, v7], [v0, v1, v3, v7], [v0, v2, v3, v7], [v0, v1, v2, v3, v7], [v0, v4, v7], [v0, v1, v4, v7], [v0, v2, v4, v7], [v0, v1, v2, v4, v7], [v0, v3, v4, v7], [v0, v1, v3, v4, v7], [v0, v2, v3, v4, v7], [v0, v1, v2, v3, v4, v7], [v0, v5, v7], [v0, v1, v5, v7], [v0, v2, v5, v7], [v0, v1, v2, v5, v7], [v0, v3, v5, v7], [v0, v1, v3, v5, v7], [v0, v2, v3, v5, v7], [v0, v1, v2, v3, v5, v7], [v0, v4, v5, v7], [v0, v1, v4, v5, v7], [v0, v2, v4, v5, v7], [v0, v1, v2, v4, v5, v7], [v0, v3, v4, v5, v7], [v0, v1, v3, v4, v5, v7], [v0, v2, v3, v4, v5, v7], [v0, v1, v2, v3, v4, v5, v7], [v0, v6, v7], [v0, v1, v6, v7], [v0, v2, v6, v7], [v0, v1, v2, v6, v7], [v0, v3, v6, v7], [v0, v1, v3, v6, v7], [v0, v2, v3, v6, v7], [v0, v1, v2, v3, v6, v7], [v0, v4, v6, v7], [v0, v1, v4, v6, v7], [v0, v2, v4, v6, v7], [v0, v1, v2, v4, v6, v7], [v0, v3, v4, v6, v7], [v0, v1, v3, v4, v6, v7], [v0, v2, v3, v4, v6, v7], [v0, v1, v2, v3, v4, v6, v7], [v0, v5, v6, v7], [v0, v1, v5, v6, v7], [v0, v2, v5, v6, v7], [v0, v1, v2, v5, v6, v7], [v0, v3, v5, v6, v7], [v0, v1, v3, v5, v6, v7], [v0, v2, v3, v5, v6, v7], [v0, v1, v2, v3, v5, v6, v7], [v0, v4, v5, v6, v7], [v0, v1, v4, v5, v6, v7], [v0, v2, v4, v5, v6, v7], [v0, v1, v2, v4, v5, v6, v7], [v0, v3, v4, v5, v6, v7], [v0, v1, v3, v4, v5, v6, v7], [v0, v2, v3, v4, v5, v6, v7], [v1, v2, v3, v4, v5, v6, v7], [v0, v1, v2, v3, v4, v5, v6, v7]], 1, [[v0]], 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": 3,
      "skills": [
        "最小頂點覆蓋"
      ]
    },
    "variant": {
      "family_id": "qf-mod46-mincover-exhaustive-minimizers",
      "seed": 202611156023,
      "parameters": {
        "vertices": [
          "v0",
          "v1",
          "v2",
          "v3",
          "v4",
          "v5",
          "v6",
          "v7"
        ],
        "edges": [
          [
            "v0",
            "v1"
          ],
          [
            "v0",
            "v2"
          ],
          [
            "v0",
            "v3"
          ],
          [
            "v0",
            "v4"
          ],
          [
            "v0",
            "v5"
          ],
          [
            "v0",
            "v6"
          ],
          [
            "v0",
            "v7"
          ]
        ],
        "domain": "Complete_finite_minimum_vertex_cover",
        "operation": "mincover-exhaustive-minimizers",
        "semantic_template": "qf-mod46-mincover-exhaustive-minimizers/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"],[\"v0\",\"v7\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"]}"
      }
    },
    "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-0023",
    "batch_id": "batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "7dc7e10dd54f77b41a640bc810777900d0bf85f1b858b5647b9a198f2ed85c1d",
      "family_id": "qf-mod46-mincover-exhaustive-minimizers",
      "seed": 202611156023,
      "semantic_template": "qf-mod46-mincover-exhaustive-minimizers/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"],[\"v0\",\"v7\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"]}"
    },
    "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] and loop-free duplicate-free undirected edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7]]. 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], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7]], [[v0], [v0, v1], [v0, v2], [v0, v1, v2], [v0, v3], [v0, v1, v3], [v0, v2, v3], [v0, v1, v2, v3], [v0, v4], [v0, v1, v4], [v0, v2, v4], [v0, v1, v2, v4], [v0, v3, v4], [v0, v1, v3, v4], [v0, v2, v3, v4], [v0, v1, v2, v3, v4], [v0, v5], [v0, v1, v5], [v0, v2, v5], [v0, v1, v2, v5], [v0, v3, v5], [v0, v1, v3, v5], [v0, v2, v3, v5], [v0, v1, v2, v3, v5], [v0, v4, v5], [v0, v1, v4, v5], [v0, v2, v4, v5], [v0, v1, v2, v4, v5], [v0, v3, v4, v5], [v0, v1, v3, v4, v5], [v0, v2, v3, v4, v5], [v0, v1, v2, v3, v4, v5], [v0, v6], [v0, v1, v6], [v0, v2, v6], [v0, v1, v2, v6], [v0, v3, v6], [v0, v1, v3, v6], [v0, v2, v3, v6], [v0, v1, v2, v3, v6], [v0, v4, v6], [v0, v1, v4, v6], [v0, v2, v4, v6], [v0, v1, v2, v4, v6], [v0, v3, v4, v6], [v0, v1, v3, v4, v6], [v0, v2, v3, v4, v6], [v0, v1, v2, v3, v4, v6], [v0, v5, v6], [v0, v1, v5, v6], [v0, v2, v5, v6], [v0, v1, v2, v5, v6], [v0, v3, v5, v6], [v0, v1, v3, v5, v6], [v0, v2, v3, v5, v6], [v0, v1, v2, v3, v5, v6], [v0, v4, v5, v6], [v0, v1, v4, v5, v6], [v0, v2, v4, v5, v6], [v0, v1, v2, v4, v5, v6], [v0, v3, v4, v5, v6], [v0, v1, v3, v4, v5, v6], [v0, v2, v3, v4, v5, v6], [v0, v1, v2, v3, v4, v5, v6], [v0, v7], [v0, v1, v7], [v0, v2, v7], [v0, v1, v2, v7], [v0, v3, v7], [v0, v1, v3, v7], [v0, v2, v3, v7], [v0, v1, v2, v3, v7], [v0, v4, v7], [v0, v1, v4, v7], [v0, v2, v4, v7], [v0, v1, v2, v4, v7], [v0, v3, v4, v7], [v0, v1, v3, v4, v7], [v0, v2, v3, v4, v7], [v0, v1, v2, v3, v4, v7], [v0, v5, v7], [v0, v1, v5, v7], [v0, v2, v5, v7], [v0, v1, v2, v5, v7], [v0, v3, v5, v7], [v0, v1, v3, v5, v7], [v0, v2, v3, v5, v7], [v0, v1, v2, v3, v5, v7], [v0, v4, v5, v7], [v0, v1, v4, v5, v7], [v0, v2, v4, v5, v7], [v0, v1, v2, v4, v5, v7], [v0, v3, v4, v5, v7], [v0, v1, v3, v4, v5, v7], [v0, v2, v3, v4, v5, v7], [v0, v1, v2, v3, v4, v5, v7], [v0, v6, v7], [v0, v1, v6, v7], [v0, v2, v6, v7], [v0, v1, v2, v6, v7], [v0, v3, v6, v7], [v0, v1, v3, v6, v7], [v0, v2, v3, v6, v7], [v0, v1, v2, v3, v6, v7], [v0, v4, v6, v7], [v0, v1, v4, v6, v7], [v0, v2, v4, v6, v7], [v0, v1, v2, v4, v6, v7], [v0, v3, v4, v6, v7], [v0, v1, v3, v4, v6, v7], [v0, v2, v3, v4, v6, v7], [v0, v1, v2, v3, v4, v6, v7], [v0, v5, v6, v7], [v0, v1, v5, v6, v7], [v0, v2, v5, v6, v7], [v0, v1, v2, v5, v6, v7], [v0, v3, v5, v6, v7], [v0, v1, v3, v5, v6, v7], [v0, v2, v3, v5, v6, v7], [v0, v1, v2, v3, v5, v6, v7], [v0, v4, v5, v6, v7], [v0, v1, v4, v5, v6, v7], [v0, v2, v4, v5, v6, v7], [v0, v1, v2, v4, v5, v6, v7], [v0, v3, v4, v5, v6, v7], [v0, v1, v3, v4, v5, v6, v7], [v0, v2, v3, v4, v5, v6, v7], [v1, v2, v3, v4, v5, v6, v7], [v0, v1, v2, v3, v4, v5, v6, v7]], 1, [[v0]], 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], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7]], [[v0], [v0, v1], [v0, v2], [v0, v1, v2], [v0, v3], [v0, v1, v3], [v0, v2, v3], [v0, v1, v2, v3], [v0, v4], [v0, v1, v4], [v0, v2, v4], [v0, v1, v2, v4], [v0, v3, v4], [v0, v1, v3, v4], [v0, v2, v3, v4], [v0, v1, v2, v3, v4], [v0, v5], [v0, v1, v5], [v0, v2, v5], [v0, v1, v2, v5], [v0, v3, v5], [v0, v1, v3, v5], [v0, v2, v3, v5], [v0, v1, v2, v3, v5], [v0, v4, v5], [v0, v1, v4, v5], [v0, v2, v4, v5], [v0, v1, v2, v4, v5], [v0, v3, v4, v5], [v0, v1, v3, v4, v5], [v0, v2, v3, v4, v5], [v0, v1, v2, v3, v4, v5], [v0, v6], [v0, v1, v6], [v0, v2, v6], [v0, v1, v2, v6], [v0, v3, v6], [v0, v1, v3, v6], [v0, v2, v3, v6], [v0, v1, v2, v3, v6], [v0, v4, v6], [v0, v1, v4, v6], [v0, v2, v4, v6], [v0, v1, v2, v4, v6], [v0, v3, v4, v6], [v0, v1, v3, v4, v6], [v0, v2, v3, v4, v6], [v0, v1, v2, v3, v4, v6], [v0, v5, v6], [v0, v1, v5, v6], [v0, v2, v5, v6], [v0, v1, v2, v5, v6], [v0, v3, v5, v6], [v0, v1, v3, v5, v6], [v0, v2, v3, v5, v6], [v0, v1, v2, v3, v5, v6], [v0, v4, v5, v6], [v0, v1, v4, v5, v6], [v0, v2, v4, v5, v6], [v0, v1, v2, v4, v5, v6], [v0, v3, v4, v5, v6], [v0, v1, v3, v4, v5, v6], [v0, v2, v3, v4, v5, v6], [v0, v1, v2, v3, v4, v5, v6], [v0, v7], [v0, v1, v7], [v0, v2, v7], [v0, v1, v2, v7], [v0, v3, v7], [v0, v1, v3, v7], [v0, v2, v3, v7], [v0, v1, v2, v3, v7], [v0, v4, v7], [v0, v1, v4, v7], [v0, v2, v4, v7], [v0, v1, v2, v4, v7], [v0, v3, v4, v7], [v0, v1, v3, v4, v7], [v0, v2, v3, v4, v7], [v0, v1, v2, v3, v4, v7], [v0, v5, v7], [v0, v1, v5, v7], [v0, v2, v5, v7], [v0, v1, v2, v5, v7], [v0, v3, v5, v7], [v0, v1, v3, v5, v7], [v0, v2, v3, v5, v7], [v0, v1, v2, v3, v5, v7], [v0, v4, v5, v7], [v0, v1, v4, v5, v7], [v0, v2, v4, v5, v7], [v0, v1, v2, v4, v5, v7], [v0, v3, v4, v5, v7], [v0, v1, v3, v4, v5, v7], [v0, v2, v3, v4, v5, v7], [v0, v1, v2, v3, v4, v5, v7], [v0, v6, v7], [v0, v1, v6, v7], [v0, v2, v6, v7], [v0, v1, v2, v6, v7], [v0, v3, v6, v7], [v0, v1, v3, v6, v7], [v0, v2, v3, v6, v7], [v0, v1, v2, v3, v6, v7], [v0, v4, v6, v7], [v0, v1, v4, v6, v7], [v0, v2, v4, v6, v7], [v0, v1, v2, v4, v6, v7], [v0, v3, v4, v6, v7], [v0, v1, v3, v4, v6, v7], [v0, v2, v3, v4, v6, v7], [v0, v1, v2, v3, v4, v6, v7], [v0, v5, v6, v7], [v0, v1, v5, v6, v7], [v0, v2, v5, v6, v7], [v0, v1, v2, v5, v6, v7], [v0, v3, v5, v6, v7], [v0, v1, v3, v5, v6, v7], [v0, v2, v3, v5, v6, v7], [v0, v1, v2, v3, v5, v6, v7], [v0, v4, v5, v6, v7], [v0, v1, v4, v5, v6, v7], [v0, v2, v4, v5, v6, v7], [v0, v1, v2, v4, v5, v6, v7], [v0, v3, v4, v5, v6, v7], [v0, v1, v3, v4, v5, v6, v7], [v0, v2, v3, v4, v5, v6, v7], [v1, v2, v3, v4, v5, v6, v7], [v0, v1, v2, v3, v4, v5, v6, v7]], 1, [[v0]], 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\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"],[\"v0\",\"v7\"]]}。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] and loop-free duplicate-free undirected edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7]]. 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], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7]], null]",
      "after": "derived = [[v0, v1, v2, v3, v4, v5, v6, v7], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7]], 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], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7]], 256]",
      "after": "derived = [[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131, 133, 135, 137, 139, 141, 143, 145, 147, 149, 151, 153, 155, 157, 159, 161, 163, 165, 167, 169, 171, 173, 175, 177, 179, 181, 183, 185, 187, 189, 191, 193, 195, 197, 199, 201, 203, 205, 207, 209, 211, 213, 215, 217, 219, 221, 223, 225, 227, 229, 231, 233, 235, 237, 239, 241, 243, 245, 247, 249, 251, 253, 254, 255], [[0, 0], [2, 1], [4, 0], [6, 2], [8, 0], [10, 1], [12, 0], [14, 3], [16, 0], [18, 1], [20, 0], [22, 2], [24, 0], [26, 1], [28, 0], [30, 4], [32, 0], [34, 1], [36, 0], [38, 2], [40, 0], [42, 1], [44, 0], [46, 3], [48, 0], [50, 1], [52, 0], [54, 2], [56, 0], [58, 1], [60, 0], [62, 5], [64, 0], [66, 1], [68, 0], [70, 2], [72, 0], [74, 1], [76, 0], [78, 3], [80, 0], [82, 1], [84, 0], [86, 2], [88, 0], [90, 1], [92, 0], [94, 4], [96, 0], [98, 1], [100, 0], [102, 2], [104, 0], [106, 1], [108, 0], [110, 3], [112, 0], [114, 1], [116, 0], [118, 2], [120, 0], [122, 1], [124, 0], [126, 6], [128, 0], [130, 1], [132, 0], [134, 2], [136, 0], [138, 1], [140, 0], [142, 3], [144, 0], [146, 1], [148, 0], [150, 2], [152, 0], [154, 1], [156, 0], [158, 4], [160, 0], [162, 1], [164, 0], [166, 2], [168, 0], [170, 1], [172, 0], [174, 3], [176, 0], [178, 1], [180, 0], [182, 2], [184, 0], [186, 1], [188, 0], [190, 5], [192, 0], [194, 1], [196, 0], [198, 2], [200, 0], [202, 1], [204, 0], [206, 3], [208, 0], [210, 1], [212, 0], [214, 2], [216, 0], [218, 1], [220, 0], [222, 4], [224, 0], [226, 1], [228, 0], [230, 2], [232, 0], [234, 1], [236, 0], [238, 3], [240, 0], [242, 1], [244, 0], [246, 2], [248, 0], [250, 1], [252, 0]], [[v0], [v0, v1], [v0, v2], [v0, v1, v2], [v0, v3], [v0, v1, v3], [v0, v2, v3], [v0, v1, v2, v3], [v0, v4], [v0, v1, v4], [v0, v2, v4], [v0, v1, v2, v4], [v0, v3, v4], [v0, v1, v3, v4], [v0, v2, v3, v4], [v0, v1, v2, v3, v4], [v0, v5], [v0, v1, v5], [v0, v2, v5], [v0, v1, v2, v5], [v0, v3, v5], [v0, v1, v3, v5], [v0, v2, v3, v5], [v0, v1, v2, v3, v5], [v0, v4, v5], [v0, v1, v4, v5], [v0, v2, v4, v5], [v0, v1, v2, v4, v5], [v0, v3, v4, v5], [v0, v1, v3, v4, v5], [v0, v2, v3, v4, v5], [v0, v1, v2, v3, v4, v5], [v0, v6], [v0, v1, v6], [v0, v2, v6], [v0, v1, v2, v6], [v0, v3, v6], [v0, v1, v3, v6], [v0, v2, v3, v6], [v0, v1, v2, v3, v6], [v0, v4, v6], [v0, v1, v4, v6], [v0, v2, v4, v6], [v0, v1, v2, v4, v6], [v0, v3, v4, v6], [v0, v1, v3, v4, v6], [v0, v2, v3, v4, v6], [v0, v1, v2, v3, v4, v6], [v0, v5, v6], [v0, v1, v5, v6], [v0, v2, v5, v6], [v0, v1, v2, v5, v6], [v0, v3, v5, v6], [v0, v1, v3, v5, v6], [v0, v2, v3, v5, v6], [v0, v1, v2, v3, v5, v6], [v0, v4, v5, v6], [v0, v1, v4, v5, v6], [v0, v2, v4, v5, v6], [v0, v1, v2, v4, v5, v6], [v0, v3, v4, v5, v6], [v0, v1, v3, v4, v5, v6], [v0, v2, v3, v4, v5, v6], [v0, v1, v2, v3, v4, v5, v6], [v0, v7], [v0, v1, v7], [v0, v2, v7], [v0, v1, v2, v7], [v0, v3, v7], [v0, v1, v3, v7], [v0, v2, v3, v7], [v0, v1, v2, v3, v7], [v0, v4, v7], [v0, v1, v4, v7], [v0, v2, v4, v7], [v0, v1, v2, v4, v7], [v0, v3, v4, v7], [v0, v1, v3, v4, v7], [v0, v2, v3, v4, v7], [v0, v1, v2, v3, v4, v7], [v0, v5, v7], [v0, v1, v5, v7], [v0, v2, v5, v7], [v0, v1, v2, v5, v7], [v0, v3, v5, v7], [v0, v1, v3, v5, v7], [v0, v2, v3, v5, v7], [v0, v1, v2, v3, v5, v7], [v0, v4, v5, v7], [v0, v1, v4, v5, v7], [v0, v2, v4, v5, v7], [v0, v1, v2, v4, v5, v7], [v0, v3, v4, v5, v7], [v0, v1, v3, v4, v5, v7], [v0, v2, v3, v4, v5, v7], [v0, v1, v2, v3, v4, v5, v7], [v0, v6, v7], [v0, v1, v6, v7], [v0, v2, v6, v7], [v0, v1, v2, v6, v7], [v0, v3, v6, v7], [v0, v1, v3, v6, v7], [v0, v2, v3, v6, v7], [v0, v1, v2, v3, v6, v7], [v0, v4, v6, v7], [v0, v1, v4, v6, v7], [v0, v2, v4, v6, v7], [v0, v1, v2, v4, v6, v7], [v0, v3, v4, v6, v7], [v0, v1, v3, v4, v6, v7], [v0, v2, v3, v4, v6, v7], [v0, v1, v2, v3, v4, v6, v7], [v0, v5, v6, v7], [v0, v1, v5, v6, v7], [v0, v2, v5, v6, v7], [v0, v1, v2, v5, v6, v7], [v0, v3, v5, v6, v7], [v0, v1, v3, v5, v6, v7], [v0, v2, v3, v5, v6, v7], [v0, v1, v2, v3, v5, v6, v7], [v0, v4, v5, v6, v7], [v0, v1, v4, v5, v6, v7], [v0, v2, v4, v5, v6, v7], [v0, v1, v2, v4, v5, v6, v7], [v0, v3, v4, v5, v6, v7], [v0, v1, v3, v4, v5, v6, v7], [v0, v2, v3, v4, v5, v6, v7], [v1, v2, v3, v4, v5, v6, v7], [v0, v1, v2, v3, v4, v5, v6, v7]], [0, 1, 7, 21, 35, 35, 21, 8, 1], 1, [[v0]]]",
      "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"
      ],
      [
        [
          "v0",
          "v1"
        ],
        [
          "v0",
          "v2"
        ],
        [
          "v0",
          "v3"
        ],
        [
          "v0",
          "v4"
        ],
        [
          "v0",
          "v5"
        ],
        [
          "v0",
          "v6"
        ],
        [
          "v0",
          "v7"
        ]
      ],
      [
        [
          "v0"
        ],
        [
          "v0",
          "v1"
        ],
        [
          "v0",
          "v2"
        ],
        [
          "v0",
          "v1",
          "v2"
        ],
        [
          "v0",
          "v3"
        ],
        [
          "v0",
          "v1",
          "v3"
        ],
        [
          "v0",
          "v2",
          "v3"
        ],
        [
          "v0",
          "v1",
          "v2",
          "v3"
        ],
        [
          "v0",
          "v4"
        ],
        [
          "v0",
          "v1",
          "v4"
        ],
        [
          "v0",
          "v2",
          "v4"
        ],
        [
          "v0",
          "v1",
          "v2",
          "v4"
        ],
        [
          "v0",
          "v3",
          "v4"
        ],
        [
          "v0",
          "v1",
          "v3",
          "v4"
        ],
        [
          "v0",
          "v2",
          "v3",
          "v4"
        ],
        [
          "v0",
          "v1",
          "v2",
          "v3",
          "v4"
        ],
        [
          "v0",
          "v5"
        ],
        [
          "v0",
          "v1",
          "v5"
        ],
        [
          "v0",
          "v2",
          "v5"
        ],
        [
          "v0",
          "v1",
          "v2",
          "v5"
        ],
        [
          "v0",
          "v3",
          "v5"
        ],
        [
          "v0",
          "v1",
          "v3",
          "v5"
        ],
        [
          "v0",
          "v2",
          "v3",
          "v5"
        ],
        [
          "v0",
          "v1",
          "v2",
          "v3",
          "v5"
        ],
        [
          "v0",
          "v4",
          "v5"
        ],
        [
          "v0",
          "v1",
          "v4",
          "v5"
        ],
        [
          "v0",
          "v2",
          "v4",
          "v5"
        ],
        [
          "v0",
          "v1",
          "v2",
          "v4",
          "v5"
        ],
        [
          "v0",
          "v3",
          "v4",
          "v5"
        ],
        [
          "v0",
          "v1",
          "v3",
          "v4",
          "v5"
        ],
        [
          "v0",
          "v2",
          "v3",
          "v4",
          "v5"
        ],
        [
          "v0",
          "v1",
          "v2",
          "v3",
          "v4",
          "v5"
        ],
        [
          "v0",
          "v6"
        ],
        [
          "v0",
          "v1",
          "v6"
        ],
        [
          "v0",
          "v2",
          "v6"
        ],
        [
          "v0",
          "v1",
          "v2",
          "v6"
        ],
        [
          "v0",
          "v3",
          "v6"
        ],
        [
          "v0",
          "v1",
          "v3",
          "v6"
        ],
        [
          "v0",
          "v2",
          "v3",
          "v6"
        ],
        [
          "v0",
          "v1",
          "v2",
          "v3",
          "v6"
        ],
        [
          "v0",
          "v4",
          "v6"
        ],
        [
          "v0",
          "v1",
          "v4",
          "v6"
        ],
        [
          "v0",
          "v2",
          "v4",
          "v6"
        ],
        [
          "v0",
          "v1",
          "v2",
          "v4",
          "v6"
        ],
        [
          "v0",
          "v3",
          "v4",
          "v6"
        ],
        [
          "v0",
          "v1",
          "v3",
          "v4",
          "v6"
        ],
        [
          "v0",
          "v2",
          "v3",
          "v4",
          "v6"
        ],
        [
          "v0",
          "v1",
          "v2",
          "v3",
          "v4",
          "v6"
        ],
        [
          "v0",
          "v5",
          "v6"
        ],
        [
          "v0",
          "v1",
          "v5",
          "v6"
        ],
        [
          "v0",
          "v2",
          "v5",
          "v6"
        ],
        [
          "v0",
          "v1",
          "v2",
          "v5",
          "v6"
        ],
        [
          "v0",
          "v3",
          "v5",
          "v6"
        ],
        [
          "v0",
          "v1",
          "v3",
          "v5",
          "v6"
        ],
        [
          "v0",
          "v2",
          "v3",
          "v5",
          "v6"
        ],
        [
          "v0",
          "v1",
          "v2",
          "v3",
          "v5",
          "v6"
        ],
        [
          "v0",
          "v4",
          "v5",
          "v6"
        ],
        [
          "v0",
          "v1",
          "v4",
          "v5",
          "v6"
        ],
        [
          "v0",
          "v2",
          "v4",
          "v5",
          "v6"
        ],
        [
          "v0",
          "v1",
          "v2",
          "v4",
          "v5",
          "v6"
        ],
        [
          "v0",
          "v3",
          "v4",
          "v5",
          "v6"
        ],
        [
          "v0",
          "v1",
          "v3",
          "v4",
          "v5",
          "v6"
        ],
        [
          "v0",
          "v2",
          "v3",
          "v4",
          "v5",
          "v6"
        ],
        [
          "v0",
          "v1",
          "v2",
          "v3",
          "v4",
          "v5",
          "v6"
        ],
        [
          "v0",
          "v7"
        ],
        [
          "v0",
          "v1",
          "v7"
        ],
        [
          "v0",
          "v2",
          "v7"
        ],
        [
          "v0",
          "v1",
          "v2",
          "v7"
        ],
        [
          "v0",
          "v3",
          "v7"
        ],
        [
          "v0",
          "v1",
          "v3",
          "v7"
        ],
        [
          "v0",
          "v2",
          "v3",
          "v7"
        ],
        [
          "v0",
          "v1",
          "v2",
          "v3",
          "v7"
        ],
        [
          "v0",
          "v4",
          "v7"
        ],
        [
          "v0",
          "v1",
          "v4",
          "v7"
        ],
        [
          "v0",
          "v2",
          "v4",
          "v7"
        ],
        [
          "v0",
          "v1",
          "v2",
          "v4",
          "v7"
        ],
        [
          "v0",
          "v3",
          "v4",
          "v7"
        ],
        [
          "v0",
          "v1",
          "v3",
          "v4",
          "v7"
        ],
        [
          "v0",
          "v2",
          "v3",
          "v4",
          "v7"
        ],
        [
          "v0",
          "v1",
          "v2",
          "v3",
          "v4",
          "v7"
        ],
        [
          "v0",
          "v5",
          "v7"
        ],
        [
          "v0",
          "v1",
          "v5",
          "v7"
        ],
        [
          "v0",
          "v2",
          "v5",
          "v7"
        ],
        [
          "v0",
          "v1",
          "v2",
          "v5",
          "v7"
        ],
        [
          "v0",
          "v3",
          "v5",
          "v7"
        ],
        [
          "v0",
          "v1",
          "v3",
          "v5",
          "v7"
        ],
        [
          "v0",
          "v2",
          "v3",
          "v5",
          "v7"
        ],
        [
          "v0",
          "v1",
          "v2",
          "v3",
          "v5",
          "v7"
        ],
        [
          "v0",
          "v4",
          "v5",
          "v7"
        ],
        [
          "v0",
          "v1",
          "v4",
          "v5",
          "v7"
        ],
        [
          "v0",
          "v2",
          "v4",
          "v5",
          "v7"
        ],
        [
          "v0",
          "v1",
          "v2",
          "v4",
          "v5",
          "v7"
        ],
        [
          "v0",
          "v3",
          "v4",
          "v5",
          "v7"
        ],
        [
          "v0",
          "v1",
          "v3",
          "v4",
          "v5",
          "v7"
        ],
        [
          "v0",
          "v2",
          "v3",
          "v4",
          "v5",
          "v7"
        ],
        [
          "v0",
          "v1",
          "v2",
          "v3",
          "v4",
          "v5",
          "v7"
        ],
        [
          "v0",
          "v6",
          "v7"
        ],
        [
          "v0",
          "v1",
          "v6",
          "v7"
        ],
        [
          "v0",
          "v2",
          "v6",
          "v7"
        ],
        [
          "v0",
          "v1",
          "v2",
          "v6",
          "v7"
        ],
        [
          "v0",
          "v3",
          "v6",
          "v7"
        ],
        [
          "v0",
          "v1",
          "v3",
          "v6",
          "v7"
        ],
        [
          "v0",
          "v2",
          "v3",
          "v6",
          "v7"
        ],
        [
          "v0",
          "v1",
          "v2",
          "v3",
          "v6",
          "v7"
        ],
        [
          "v0",
          "v4",
          "v6",
          "v7"
        ],
        [
          "v0",
          "v1",
          "v4",
          "v6",
          "v7"
        ],
        [
          "v0",
          "v2",
          "v4",
          "v6",
          "v7"
        ],
        [
          "v0",
          "v1",
          "v2",
          "v4",
          "v6",
          "v7"
        ],
        [
          "v0",
          "v3",
          "v4",
          "v6",
          "v7"
        ],
        [
          "v0",
          "v1",
          "v3",
          "v4",
          "v6",
          "v7"
        ],
        [
          "v0",
          "v2",
          "v3",
          "v4",
          "v6",
          "v7"
        ],
        [
          "v0",
          "v1",
          "v2",
          "v3",
          "v4",
          "v6",
          "v7"
        ],
        [
          "v0",
          "v5",
          "v6",
          "v7"
        ],
        [
          "v0",
          "v1",
          "v5",
          "v6",
          "v7"
        ],
        [
          "v0",
          "v2",
          "v5",
          "v6",
          "v7"
        ],
        [
          "v0",
          "v1",
          "v2",
          "v5",
          "v6",
          "v7"
        ],
        [
          "v0",
          "v3",
          "v5",
          "v6",
          "v7"
        ],
        [
          "v0",
          "v1",
          "v3",
          "v5",
          "v6",
          "v7"
        ],
        [
          "v0",
          "v2",
          "v3",
          "v5",
          "v6",
          "v7"
        ],
        [
          "v0",
          "v1",
          "v2",
          "v3",
          "v5",
          "v6",
          "v7"
        ],
        [
          "v0",
          "v4",
          "v5",
          "v6",
          "v7"
        ],
        [
          "v0",
          "v1",
          "v4",
          "v5",
          "v6",
          "v7"
        ],
        [
          "v0",
          "v2",
          "v4",
          "v5",
          "v6",
          "v7"
        ],
        [
          "v0",
          "v1",
          "v2",
          "v4",
          "v5",
          "v6",
          "v7"
        ],
        [
          "v0",
          "v3",
          "v4",
          "v5",
          "v6",
          "v7"
        ],
        [
          "v0",
          "v1",
          "v3",
          "v4",
          "v5",
          "v6",
          "v7"
        ],
        [
          "v0",
          "v2",
          "v3",
          "v4",
          "v5",
          "v6",
          "v7"
        ],
        [
          "v1",
          "v2",
          "v3",
          "v4",
          "v5",
          "v6",
          "v7"
        ],
        [
          "v0",
          "v1",
          "v2",
          "v3",
          "v4",
          "v5",
          "v6",
          "v7"
        ]
      ],
      1,
      [
        [
          "v0"
        ]
      ],
      "all_vertex_subsets_are_audited_before_selecting_all_minimizers"
    ]
  },
  "conclusion": {
    "canonical_value": "([v0, v1, v2, v3, v4, v5, v6, v7], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7]], [[v0], [v0, v1], [v0, v2], [v0, v1, v2], [v0, v3], [v0, v1, v3], [v0, v2, v3], [v0, v1, v2, v3], [v0, v4], [v0, v1, v4], [v0, v2, v4], [v0, v1, v2, v4], [v0, v3, v4], [v0, v1, v3, v4], [v0, v2, v3, v4], [v0, v1, v2, v3, v4], [v0, v5], [v0, v1, v5], [v0, v2, v5], [v0, v1, v2, v5], [v0, v3, v5], [v0, v1, v3, v5], [v0, v2, v3, v5], [v0, v1, v2, v3, v5], [v0, v4, v5], [v0, v1, v4, v5], [v0, v2, v4, v5], [v0, v1, v2, v4, v5], [v0, v3, v4, v5], [v0, v1, v3, v4, v5], [v0, v2, v3, v4, v5], [v0, v1, v2, v3, v4, v5], [v0, v6], [v0, v1, v6], [v0, v2, v6], [v0, v1, v2, v6], [v0, v3, v6], [v0, v1, v3, v6], [v0, v2, v3, v6], [v0, v1, v2, v3, v6], [v0, v4, v6], [v0, v1, v4, v6], [v0, v2, v4, v6], [v0, v1, v2, v4, v6], [v0, v3, v4, v6], [v0, v1, v3, v4, v6], [v0, v2, v3, v4, v6], [v0, v1, v2, v3, v4, v6], [v0, v5, v6], [v0, v1, v5, v6], [v0, v2, v5, v6], [v0, v1, v2, v5, v6], [v0, v3, v5, v6], [v0, v1, v3, v5, v6], [v0, v2, v3, v5, v6], [v0, v1, v2, v3, v5, v6], [v0, v4, v5, v6], [v0, v1, v4, v5, v6], [v0, v2, v4, v5, v6], [v0, v1, v2, v4, v5, v6], [v0, v3, v4, v5, v6], [v0, v1, v3, v4, v5, v6], [v0, v2, v3, v4, v5, v6], [v0, v1, v2, v3, v4, v5, v6], [v0, v7], [v0, v1, v7], [v0, v2, v7], [v0, v1, v2, v7], [v0, v3, v7], [v0, v1, v3, v7], [v0, v2, v3, v7], [v0, v1, v2, v3, v7], [v0, v4, v7], [v0, v1, v4, v7], [v0, v2, v4, v7], [v0, v1, v2, v4, v7], [v0, v3, v4, v7], [v0, v1, v3, v4, v7], [v0, v2, v3, v4, v7], [v0, v1, v2, v3, v4, v7], [v0, v5, v7], [v0, v1, v5, v7], [v0, v2, v5, v7], [v0, v1, v2, v5, v7], [v0, v3, v5, v7], [v0, v1, v3, v5, v7], [v0, v2, v3, v5, v7], [v0, v1, v2, v3, v5, v7], [v0, v4, v5, v7], [v0, v1, v4, v5, v7], [v0, v2, v4, v5, v7], [v0, v1, v2, v4, v5, v7], [v0, v3, v4, v5, v7], [v0, v1, v3, v4, v5, v7], [v0, v2, v3, v4, v5, v7], [v0, v1, v2, v3, v4, v5, v7], [v0, v6, v7], [v0, v1, v6, v7], [v0, v2, v6, v7], [v0, v1, v2, v6, v7], [v0, v3, v6, v7], [v0, v1, v3, v6, v7], [v0, v2, v3, v6, v7], [v0, v1, v2, v3, v6, v7], [v0, v4, v6, v7], [v0, v1, v4, v6, v7], [v0, v2, v4, v6, v7], [v0, v1, v2, v4, v6, v7], [v0, v3, v4, v6, v7], [v0, v1, v3, v4, v6, v7], [v0, v2, v3, v4, v6, v7], [v0, v1, v2, v3, v4, v6, v7], [v0, v5, v6, v7], [v0, v1, v5, v6, v7], [v0, v2, v5, v6, v7], [v0, v1, v2, v5, v6, v7], [v0, v3, v5, v6, v7], [v0, v1, v3, v5, v6, v7], [v0, v2, v3, v5, v6, v7], [v0, v1, v2, v3, v5, v6, v7], [v0, v4, v5, v6, v7], [v0, v1, v4, v5, v6, v7], [v0, v2, v4, v5, v6, v7], [v0, v1, v2, v4, v5, v6, v7], [v0, v3, v4, v5, v6, v7], [v0, v1, v3, v4, v5, v6, v7], [v0, v2, v3, v4, v5, v6, v7], [v1, v2, v3, v4, v5, v6, v7], [v0, v1, v2, v3, v4, v5, v6, v7]], 1, [[v0]], 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": 23,
      "original_line_sha256": "7c8d426134a34d99aef013fd5bd5fe87105b030db95facfa1cfd62f5ec551d37",
      "question_compact_sha256": "7dc7e10dd54f77b41a640bc810777900d0bf85f1b858b5647b9a198f2ed85c1d",
      "english_row_compact_sha256": "451dc9f2f24f9b50f6411f9546d1af4c21efe1f186c7383b002855f6146d95e7",
      "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": "1fce985aded14d596df52d6ae452e3d566c807331336c7a303cf4e47f1703edc",
    "original_candidate_line_sha256": "1fce985aded14d596df52d6ae452e3d566c807331336c7a303cf4e47f1703edc"
  },
  "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": "2d4479151b6ed18aeb242f588e6ba12974377f6bdd672a8ffa4fe2602590ec82",
      "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."
  }
}