{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261015-advance-20261115-r06-minimum-vertex-cover-0066",
  "pilot_number": "R0051-26066",
  "number": 26066,
  "family_id": "qf-mod46-mincover-minimal-not-minimum",
  "title": {
    "zh-Hant": "第 26066 題：最小頂點覆蓋",
    "en": "Question 26066: A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261015-advance-20261115-r06-minimum-vertex-cover-0066",
    "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-minimal-not-minimum；資料={\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"c1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\",\"l1_6\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"],[\"c1\",\"l1_3\"],[\"c1\",\"l1_4\"],[\"c1\",\"l1_5\"],[\"c1\",\"l1_6\"]],\"candidate\":[\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\",\"l1_6\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], [[l0_0, [l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], 否], [l0_1, [l0_0, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], 否], [l1_0, [l0_0, l0_1, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], 否], [l1_1, [l0_0, l0_1, l1_0, l1_2, l1_3, l1_4, l1_5, l1_6], 否], [l1_2, [l0_0, l0_1, l1_0, l1_1, l1_3, l1_4, l1_5, l1_6], 否], [l1_3, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_4, l1_5, l1_6], 否], [l1_4, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_5, l1_6], 否], [l1_5, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_6], 否], [l1_6, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], 否]], 是, 2, [c0, c1], 否, inclusion_minimal_is_not_minimum)",
      "explanation_zh": "Wave6第九層核對=([l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], [[l0_0, [l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], 否], [l0_1, [l0_0, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], 否], [l1_0, [l0_0, l0_1, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], 否], [l1_1, [l0_0, l0_1, l1_0, l1_2, l1_3, l1_4, l1_5, l1_6], 否], [l1_2, [l0_0, l0_1, l1_0, l1_1, l1_3, l1_4, l1_5, l1_6], 否], [l1_3, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_4, l1_5, l1_6], 否], [l1_4, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_5, l1_6], 否], [l1_5, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_6], 否], [l1_6, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], 否]], 是, 2, [c0, c1], 否, inclusion_minimal_is_not_minimum)；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-minimal-not-minimum",
      "seed": 202611156066,
      "parameters": {
        "vertices": [
          "c0",
          "l0_0",
          "l0_1",
          "c1",
          "l1_0",
          "l1_1",
          "l1_2",
          "l1_3",
          "l1_4",
          "l1_5",
          "l1_6"
        ],
        "edges": [
          [
            "c0",
            "l0_0"
          ],
          [
            "c0",
            "l0_1"
          ],
          [
            "c1",
            "l1_0"
          ],
          [
            "c1",
            "l1_1"
          ],
          [
            "c1",
            "l1_2"
          ],
          [
            "c1",
            "l1_3"
          ],
          [
            "c1",
            "l1_4"
          ],
          [
            "c1",
            "l1_5"
          ],
          [
            "c1",
            "l1_6"
          ]
        ],
        "candidate": [
          "l0_0",
          "l0_1",
          "l1_0",
          "l1_1",
          "l1_2",
          "l1_3",
          "l1_4",
          "l1_5",
          "l1_6"
        ],
        "domain": "Complete_finite_minimum_vertex_cover",
        "operation": "mincover-minimal-not-minimum",
        "semantic_template": "qf-mod46-mincover-minimal-not-minimum/{\"candidate\":[\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\",\"l1_6\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"],[\"c1\",\"l1_3\"],[\"c1\",\"l1_4\"],[\"c1\",\"l1_5\"],[\"c1\",\"l1_6\"]],\"vertices\":[\"c0\",\"c1\",\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\",\"l1_6\"]}"
      }
    },
    "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-0066",
    "batch_id": "batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "d8fec2b81f4a1ca2bb032866cac66d3d66defbdb01cdf2d608407a084215f506",
      "family_id": "qf-mod46-mincover-minimal-not-minimum",
      "seed": 202611156066,
      "semantic_template": "qf-mod46-mincover-minimal-not-minimum/{\"candidate\":[\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\",\"l1_6\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"],[\"c1\",\"l1_3\"],[\"c1\",\"l1_4\"],[\"c1\",\"l1_5\"],[\"c1\",\"l1_6\"]],\"vertices\":[\"c0\",\"c1\",\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\",\"l1_6\"]}"
    },
    "fields": {
      "title": "A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover",
      "stem": "Use complete unique vertices=[c0, l0_0, l0_1, c1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6] and loop-free duplicate-free undirected edges=[[c0, l0_0], [c0, l0_1], [c1, l1_0], [c1, l1_1], [c1, l1_2], [c1, l1_3], [c1, l1_4], [c1, l1_5], [c1, l1_6]], candidate cover=[l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, smaller_cover, minimum, certificate).",
      "answer": {
        "kind": "expression",
        "value": "([l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], [[l0_0, [l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], No], [l0_1, [l0_0, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], No], [l1_0, [l0_0, l0_1, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], No], [l1_1, [l0_0, l0_1, l1_0, l1_2, l1_3, l1_4, l1_5, l1_6], No], [l1_2, [l0_0, l0_1, l1_0, l1_1, l1_3, l1_4, l1_5, l1_6], No], [l1_3, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_4, l1_5, l1_6], No], [l1_4, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_5, l1_6], No], [l1_5, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_6], No], [l1_6, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], No]], Yes, 2, [c0, c1], No, inclusion_minimal_is_not_minimum)",
        "explanation": "Every single-vertex deletion must lose coverage, while another complete cover is strictly smaller. Retain deletion rows, minimum size and a smaller witness. Inclusion-minimal, minimum and maximal are different notions. The exact result, retaining the requested order, is ([l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], [[l0_0, [l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], No], [l0_1, [l0_0, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], No], [l1_0, [l0_0, l0_1, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], No], [l1_1, [l0_0, l0_1, l1_0, l1_2, l1_3, l1_4, l1_5, l1_6], No], [l1_2, [l0_0, l0_1, l1_0, l1_1, l1_3, l1_4, l1_5, l1_6], No], [l1_3, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_4, l1_5, l1_6], No], [l1_4, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_5, l1_6], No], [l1_5, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_6], No], [l1_6, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], No]], Yes, 2, [c0, c1], No, inclusion_minimal_is_not_minimum)."
      },
      "skills": [
        "A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover"
      ],
      "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-minimal-not-minimum；資料={\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"c1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\",\"l1_6\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"],[\"c1\",\"l1_3\"],[\"c1\",\"l1_4\"],[\"c1\",\"l1_5\"],[\"c1\",\"l1_6\"]],\"candidate\":[\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\",\"l1_6\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use complete unique vertices=[c0, l0_0, l0_1, c1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6] and loop-free duplicate-free undirected edges=[[c0, l0_0], [c0, l0_1], [c1, l1_0], [c1, l1_1], [c1, l1_2], [c1, l1_3], [c1, l1_4], [c1, l1_5], [c1, l1_6]], candidate cover=[l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, smaller_cover, minimum, certificate)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [mincover-minimal-not-minimum]",
      "after": "derived = [[candidate, deletion_rows, inclusion_minimal, minimum_size, smaller_cover, minimum, 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 = [[c0, l0_0, l0_1, c1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], [[c0, l0_0], [c0, l0_1], [c1, l1_0], [c1, l1_1], [c1, l1_2], [c1, l1_3], [c1, l1_4], [c1, l1_5], [c1, l1_6]], [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6]]",
      "after": "derived = [[c0, c1, l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], [[c0, l0_0], [c0, l0_1], [c1, l1_0], [c1, l1_1], [c1, l1_2], [c1, l1_3], [c1, l1_4], [c1, l1_5], [c1, l1_6]], [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6]]",
      "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 = [[c0, c1, l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], [[c0, l0_0], [c0, l0_1], [c1, l1_0], [c1, l1_1], [c1, l1_2], [c1, l1_3], [c1, l1_4], [c1, l1_5], [c1, l1_6]], 2048]",
      "after": "derived = [[3, 7, 11, 14, 15, 19, 23, 27, 30, 31, 35, 39, 43, 46, 47, 51, 55, 59, 62, 63, 67, 71, 75, 78, 79, 83, 87, 91, 94, 95, 99, 103, 107, 110, 111, 115, 119, 123, 126, 127, 131, 135, 139, 142, 143, 147, 151, 155, 158, 159, 163, 167, 171, 174, 175, 179, 183, 187, 190, 191, 195, 199, 203, 206, 207, 211, 215, 219, 222, 223, 227, 231, 235, 238, 239, 243, 247, 251, 254, 255, 259, 263, 267, 270, 271, 275, 279, 283, 286, 287, 291, 295, 299, 302, 303, 307, 311, 315, 318, 319, 323, 327, 331, 334, 335, 339, 343, 347, 350, 351, 355, 359, 363, 366, 367, 371, 375, 379, 382, 383, 387, 391, 395, 398, 399, 403, 407, 411, 414, 415, 419, 423, 427, 430, 431, 435, 439, 443, 446, 447, 451, 455, 459, 462, 463, 467, 471, 475, 478, 479, 483, 487, 491, 494, 495, 499, 503, 507, 510, 511, 515, 519, 523, 526, 527, 531, 535, 539, 542, 543, 547, 551, 555, 558, 559, 563, 567, 571, 574, 575, 579, 583, 587, 590, 591, 595, 599, 603, 606, 607, 611, 615, 619, 622, 623, 627, 631, 635, 638, 639, 643, 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[c0, c1, l1_0, l1_1, l1_3, l1_5], [c0, c1, l0_0, l1_0, l1_1, l1_3, l1_5], [c0, c1, l0_1, l1_0, l1_1, l1_3, l1_5], [c1, l0_0, l0_1, l1_0, l1_1, l1_3, l1_5], [c0, c1, l0_0, l0_1, l1_0, l1_1, l1_3, l1_5], [c0, c1, l1_2, l1_3, l1_5], [c0, c1, l0_0, l1_2, l1_3, l1_5], [c0, c1, l0_1, l1_2, l1_3, l1_5], [c1, l0_0, l0_1, l1_2, l1_3, l1_5], [c0, c1, l0_0, l0_1, l1_2, l1_3, l1_5], [c0, c1, l1_0, l1_2, l1_3, l1_5], [c0, c1, l0_0, l1_0, l1_2, l1_3, l1_5], [c0, c1, l0_1, l1_0, l1_2, l1_3, l1_5], [c1, l0_0, l0_1, l1_0, l1_2, l1_3, l1_5], [c0, c1, l0_0, l0_1, l1_0, l1_2, l1_3, l1_5], [c0, c1, l1_1, l1_2, l1_3, l1_5], [c0, c1, l0_0, l1_1, l1_2, l1_3, l1_5], [c0, c1, l0_1, l1_1, l1_2, l1_3, l1_5], [c1, l0_0, l0_1, l1_1, l1_2, l1_3, l1_5], [c0, c1, l0_0, l0_1, l1_1, l1_2, l1_3, l1_5], [c0, c1, l1_0, l1_1, l1_2, l1_3, l1_5], [c0, c1, l0_0, l1_0, l1_1, l1_2, l1_3, l1_5], [c0, c1, l0_1, l1_0, l1_1, l1_2, l1_3, l1_5], [c1, l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_5], [c0, c1, l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_5], [c0, c1, l1_4, l1_5], [c0, c1, l0_0, l1_4, l1_5], [c0, c1, l0_1, l1_4, l1_5], [c1, l0_0, l0_1, l1_4, l1_5], [c0, c1, l0_0, l0_1, l1_4, l1_5], [c0, c1, l1_0, l1_4, l1_5], [c0, c1, l0_0, l1_0, l1_4, l1_5], [c0, c1, l0_1, l1_0, l1_4, l1_5], [c1, l0_0, l0_1, l1_0, l1_4, l1_5], [c0, c1, l0_0, l0_1, l1_0, l1_4, l1_5], [c0, c1, l1_1, l1_4, l1_5], [c0, c1, l0_0, l1_1, l1_4, l1_5], [c0, c1, l0_1, l1_1, l1_4, l1_5], [c1, l0_0, l0_1, l1_1, l1_4, l1_5], [c0, c1, l0_0, l0_1, l1_1, l1_4, l1_5], [c0, c1, l1_0, l1_1, l1_4, l1_5], [c0, c1, l0_0, l1_0, l1_1, l1_4, l1_5], [c0, c1, l0_1, l1_0, l1_1, l1_4, l1_5], [c1, l0_0, l0_1, l1_0, l1_1, l1_4, l1_5], [c0, c1, l0_0, l0_1, l1_0, l1_1, l1_4, l1_5], [c0, c1, l1_2, l1_4, l1_5], [c0, c1, l0_0, l1_2, l1_4, l1_5], [c0, c1, l0_1, l1_2, l1_4, l1_5], [c1, l0_0, l0_1, l1_2, l1_4, l1_5], [c0, c1, l0_0, l0_1, l1_2, l1_4, l1_5], [c0, c1, l1_0, l1_2, l1_4, l1_5], [c0, c1, l0_0, l1_0, l1_2, l1_4, l1_5], [c0, c1, l0_1, l1_0, l1_2, l1_4, l1_5], [c1, l0_0, l0_1, l1_0, l1_2, l1_4, l1_5], [c0, c1, l0_0, l0_1, l1_0, l1_2, l1_4, l1_5], [c0, c1, l1_1, l1_2, l1_4, l1_5], [c0, c1, l0_0, l1_1, l1_2, l1_4, l1_5], [c0, c1, l0_1, l1_1, l1_2, l1_4, l1_5], [c1, l0_0, l0_1, l1_1, l1_2, l1_4, l1_5], [c0, c1, l0_0, l0_1, l1_1, l1_2, l1_4, l1_5], [c0, c1, l1_0, l1_1, l1_2, l1_4, l1_5], [c0, c1, l0_0, l1_0, l1_1, l1_2, l1_4, l1_5], [c0, c1, l0_1, l1_0, l1_1, l1_2, l1_4, l1_5], [c1, l0_0, l0_1, l1_0, l1_1, l1_2, l1_4, l1_5], [c0, c1, l0_0, l0_1, l1_0, l1_1, l1_2, l1_4, l1_5], [c0, c1, l1_3, l1_4, l1_5], [c0, c1, l0_0, l1_3, l1_4, l1_5], [c0, c1, l0_1, l1_3, l1_4, l1_5], [c1, l0_0, l0_1, l1_3, l1_4, l1_5], [c0, c1, l0_0, l0_1, l1_3, l1_4, l1_5], [c0, c1, l1_0, l1_3, l1_4, l1_5], [c0, c1, l0_0, l1_0, l1_3, l1_4, l1_5], [c0, c1, l0_1, l1_0, l1_3, l1_4, l1_5], [c1, l0_0, l0_1, l1_0, l1_3, l1_4, l1_5], [c0, c1, l0_0, l0_1, l1_0, l1_3, l1_4, l1_5], [c0, c1, l1_1, l1_3, l1_4, l1_5], [c0, c1, l0_0, l1_1, l1_3, l1_4, l1_5], [c0, c1, l0_1, l1_1, l1_3, l1_4, l1_5], [c1, l0_0, l0_1, l1_1, l1_3, l1_4, l1_5], [c0, c1, l0_0, l0_1, l1_1, l1_3, l1_4, l1_5], [c0, c1, l1_0, l1_1, l1_3, l1_4, l1_5], [c0, c1, l0_0, l1_0, l1_1, l1_3, l1_4, l1_5], [c0, c1, l0_1, l1_0, l1_1, l1_3, l1_4, l1_5], [c1, l0_0, l0_1, l1_0, l1_1, l1_3, l1_4, l1_5], [c0, c1, l0_0, l0_1, l1_0, l1_1, l1_3, l1_4, l1_5], [c0, c1, l1_2, l1_3, l1_4, l1_5], [c0, c1, l0_0, l1_2, l1_3, l1_4, l1_5], [c0, c1, l0_1, l1_2, l1_3, l1_4, l1_5], [c1, l0_0, l0_1, l1_2, l1_3, l1_4, l1_5], [c0, c1, l0_0, l0_1, l1_2, l1_3, l1_4, l1_5], [c0, c1, l1_0, l1_2, l1_3, l1_4, l1_5], [c0, c1, l0_0, l1_0, l1_2, l1_3, l1_4, l1_5], [c0, c1, l0_1, l1_0, l1_2, l1_3, l1_4, l1_5], [c1, l0_0, l0_1, l1_0, l1_2, l1_3, l1_4, l1_5], [c0, c1, l0_0, l0_1, l1_0, l1_2, l1_3, l1_4, l1_5], [c0, c1, l1_1, l1_2, l1_3, l1_4, l1_5], [c0, c1, l0_0, l1_1, l1_2, l1_3, l1_4, l1_5], [c0, c1, l0_1, l1_1, l1_2, l1_3, l1_4, l1_5], [c1, l0_0, l0_1, l1_1, l1_2, l1_3, l1_4, l1_5], [c0, c1, l0_0, l0_1, l1_1, l1_2, l1_3, l1_4, l1_5], [c0, c1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], [c0, c1, l0_0, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], [c0, c1, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], [c1, l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], [c0, c1, l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], [c0, c1, l1_6], [c0, c1, l0_0, l1_6], [c0, c1, l0_1, l1_6], [c1, l0_0, l0_1, l1_6], [c0, c1, l0_0, l0_1, l1_6], [c0, c1, l1_0, l1_6], [c0, c1, l0_0, l1_0, l1_6], [c0, c1, l0_1, l1_0, l1_6], [c1, l0_0, l0_1, l1_0, l1_6], [c0, c1, l0_0, l0_1, l1_0, l1_6], [c0, c1, l1_1, l1_6], [c0, c1, l0_0, l1_1, l1_6], [c0, c1, l0_1, l1_1, l1_6], [c1, l0_0, l0_1, l1_1, l1_6], [c0, c1, l0_0, l0_1, l1_1, l1_6], [c0, c1, l1_0, l1_1, l1_6], [c0, c1, l0_0, l1_0, l1_1, l1_6], [c0, c1, l0_1, l1_0, l1_1, l1_6], [c1, l0_0, l0_1, l1_0, l1_1, l1_6], [c0, c1, l0_0, l0_1, l1_0, l1_1, l1_6], [c0, c1, l1_2, l1_6], [c0, c1, l0_0, l1_2, l1_6], [c0, c1, l0_1, l1_2, l1_6], [c1, l0_0, l0_1, l1_2, l1_6], [c0, c1, l0_0, l0_1, l1_2, l1_6], [c0, c1, l1_0, l1_2, l1_6], [c0, c1, l0_0, l1_0, l1_2, l1_6], [c0, c1, l0_1, l1_0, l1_2, l1_6], [c1, l0_0, l0_1, l1_0, l1_2, l1_6], [c0, c1, l0_0, l0_1, l1_0, l1_2, l1_6], [c0, c1, l1_1, l1_2, l1_6], [c0, c1, l0_0, l1_1, l1_2, l1_6], [c0, c1, l0_1, l1_1, l1_2, l1_6], [c1, l0_0, l0_1, l1_1, l1_2, l1_6], [c0, c1, l0_0, l0_1, l1_1, l1_2, l1_6], [c0, c1, l1_0, l1_1, l1_2, l1_6], [c0, c1, l0_0, l1_0, l1_1, l1_2, l1_6], [c0, c1, l0_1, l1_0, l1_1, l1_2, l1_6], [c1, l0_0, l0_1, l1_0, l1_1, l1_2, l1_6], [c0, c1, l0_0, l0_1, l1_0, l1_1, l1_2, l1_6], [c0, c1, l1_3, l1_6], [c0, c1, l0_0, l1_3, l1_6], [c0, c1, l0_1, l1_3, l1_6], [c1, l0_0, l0_1, l1_3, l1_6], [c0, c1, l0_0, l0_1, l1_3, l1_6], [c0, c1, l1_0, l1_3, l1_6], [c0, c1, l0_0, l1_0, l1_3, l1_6], [c0, c1, l0_1, l1_0, l1_3, l1_6], [c1, l0_0, l0_1, l1_0, l1_3, l1_6], [c0, c1, l0_0, l0_1, l1_0, l1_3, l1_6], [c0, c1, l1_1, l1_3, l1_6], [c0, c1, l0_0, l1_1, l1_3, l1_6], [c0, c1, l0_1, l1_1, l1_3, l1_6], [c1, l0_0, l0_1, l1_1, l1_3, l1_6], [c0, c1, l0_0, l0_1, l1_1, l1_3, l1_6], [c0, c1, l1_0, l1_1, l1_3, l1_6], [c0, c1, l0_0, l1_0, l1_1, l1_3, l1_6], [c0, c1, l0_1, l1_0, l1_1, l1_3, l1_6], [c1, l0_0, l0_1, l1_0, l1_1, l1_3, l1_6], [c0, c1, l0_0, l0_1, l1_0, l1_1, l1_3, l1_6], [c0, c1, l1_2, l1_3, l1_6], [c0, c1, l0_0, l1_2, l1_3, l1_6], [c0, c1, l0_1, l1_2, l1_3, l1_6], [c1, l0_0, l0_1, l1_2, l1_3, l1_6], [c0, c1, l0_0, l0_1, l1_2, l1_3, l1_6], [c0, c1, l1_0, l1_2, l1_3, l1_6], [c0, c1, l0_0, l1_0, l1_2, l1_3, l1_6], [c0, c1, l0_1, l1_0, l1_2, l1_3, l1_6], [c1, l0_0, l0_1, l1_0, l1_2, l1_3, l1_6], [c0, c1, l0_0, l0_1, l1_0, l1_2, l1_3, l1_6], [c0, c1, l1_1, l1_2, l1_3, l1_6], [c0, c1, l0_0, l1_1, l1_2, l1_3, l1_6], [c0, c1, l0_1, l1_1, l1_2, l1_3, l1_6], [c1, l0_0, l0_1, l1_1, l1_2, l1_3, l1_6], [c0, c1, l0_0, l0_1, l1_1, l1_2, l1_3, l1_6], [c0, c1, l1_0, l1_1, l1_2, l1_3, l1_6], [c0, c1, l0_0, l1_0, l1_1, l1_2, l1_3, l1_6], [c0, c1, l0_1, l1_0, l1_1, l1_2, l1_3, l1_6], [c1, l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_6], [c0, c1, l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_6], [c0, c1, l1_4, l1_6], [c0, c1, l0_0, l1_4, l1_6], [c0, c1, l0_1, l1_4, l1_6], [c1, l0_0, l0_1, l1_4, l1_6], [c0, c1, l0_0, l0_1, l1_4, l1_6], [c0, c1, l1_0, l1_4, l1_6], [c0, c1, l0_0, l1_0, l1_4, l1_6], [c0, c1, l0_1, l1_0, l1_4, l1_6], [c1, l0_0, l0_1, l1_0, l1_4, l1_6], [c0, c1, l0_0, l0_1, l1_0, l1_4, l1_6], [c0, c1, l1_1, l1_4, l1_6], [c0, c1, l0_0, l1_1, l1_4, l1_6], [c0, c1, l0_1, l1_1, l1_4, l1_6], [c1, l0_0, l0_1, l1_1, l1_4, l1_6], [c0, c1, l0_0, l0_1, l1_1, l1_4, l1_6], [c0, c1, l1_0, l1_1, l1_4, l1_6], [c0, c1, l0_0, l1_0, l1_1, l1_4, l1_6], [c0, c1, l0_1, l1_0, l1_1, l1_4, l1_6], [c1, l0_0, l0_1, l1_0, l1_1, l1_4, l1_6], [c0, c1, l0_0, l0_1, l1_0, l1_1, l1_4, l1_6], [c0, c1, l1_2, l1_4, l1_6], [c0, c1, l0_0, l1_2, l1_4, l1_6], [c0, c1, l0_1, l1_2, l1_4, l1_6], [c1, l0_0, l0_1, l1_2, l1_4, l1_6], [c0, c1, l0_0, l0_1, l1_2, l1_4, l1_6], [c0, c1, l1_0, l1_2, l1_4, l1_6], [c0, c1, l0_0, l1_0, l1_2, l1_4, l1_6], [c0, c1, l0_1, l1_0, l1_2, l1_4, l1_6], [c1, l0_0, l0_1, l1_0, l1_2, l1_4, l1_6], [c0, c1, l0_0, l0_1, l1_0, l1_2, l1_4, l1_6], [c0, c1, l1_1, l1_2, l1_4, l1_6], [c0, c1, l0_0, l1_1, l1_2, l1_4, l1_6], [c0, c1, l0_1, l1_1, l1_2, l1_4, l1_6], [c1, l0_0, l0_1, l1_1, l1_2, l1_4, l1_6], [c0, c1, l0_0, l0_1, l1_1, l1_2, l1_4, l1_6], [c0, c1, l1_0, l1_1, l1_2, l1_4, l1_6], [c0, c1, l0_0, l1_0, l1_1, l1_2, l1_4, l1_6], [c0, c1, l0_1, l1_0, l1_1, l1_2, l1_4, l1_6], [c1, l0_0, l0_1, l1_0, l1_1, l1_2, l1_4, l1_6], [c0, c1, l0_0, l0_1, l1_0, l1_1, l1_2, l1_4, l1_6], [c0, c1, l1_3, l1_4, l1_6], [c0, c1, l0_0, l1_3, l1_4, l1_6], [c0, c1, l0_1, l1_3, l1_4, l1_6], [c1, l0_0, l0_1, l1_3, l1_4, l1_6], [c0, c1, l0_0, l0_1, l1_3, l1_4, l1_6], [c0, c1, l1_0, l1_3, l1_4, l1_6], [c0, c1, l0_0, l1_0, l1_3, l1_4, l1_6], [c0, c1, l0_1, l1_0, l1_3, l1_4, l1_6], [c1, l0_0, l0_1, l1_0, l1_3, l1_4, l1_6], [c0, c1, l0_0, l0_1, l1_0, l1_3, l1_4, l1_6], [c0, c1, l1_1, l1_3, l1_4, l1_6], [c0, c1, l0_0, l1_1, l1_3, l1_4, l1_6], [c0, c1, l0_1, l1_1, l1_3, l1_4, l1_6], [c1, l0_0, l0_1, l1_1, l1_3, l1_4, l1_6], [c0, c1, l0_0, l0_1, l1_1, l1_3, l1_4, l1_6], [c0, c1, l1_0, l1_1, l1_3, l1_4, l1_6], [c0, c1, l0_0, l1_0, l1_1, l1_3, l1_4, l1_6], [c0, c1, l0_1, l1_0, l1_1, l1_3, l1_4, l1_6], [c1, l0_0, l0_1, l1_0, l1_1, l1_3, l1_4, l1_6], [c0, c1, l0_0, l0_1, l1_0, l1_1, l1_3, l1_4, l1_6], [c0, c1, l1_2, l1_3, l1_4, l1_6], [c0, c1, l0_0, l1_2, l1_3, l1_4, l1_6], [c0, c1, l0_1, l1_2, l1_3, l1_4, l1_6], [c1, l0_0, l0_1, l1_2, l1_3, l1_4, l1_6], [c0, c1, l0_0, l0_1, l1_2, l1_3, l1_4, l1_6], [c0, c1, l1_0, l1_2, l1_3, l1_4, l1_6], [c0, c1, l0_0, l1_0, l1_2, l1_3, l1_4, l1_6], [c0, c1, l0_1, l1_0, l1_2, l1_3, l1_4, l1_6], [c1, l0_0, l0_1, l1_0, l1_2, l1_3, l1_4, l1_6], [c0, c1, l0_0, l0_1, l1_0, l1_2, l1_3, l1_4, l1_6], [c0, c1, l1_1, l1_2, l1_3, l1_4, l1_6], [c0, c1, l0_0, l1_1, l1_2, l1_3, l1_4, l1_6], [c0, c1, l0_1, l1_1, l1_2, l1_3, l1_4, l1_6], [c1, l0_0, l0_1, l1_1, l1_2, l1_3, l1_4, l1_6], [c0, c1, l0_0, l0_1, l1_1, l1_2, l1_3, l1_4, l1_6], [c0, c1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_6], [c0, c1, l0_0, l1_0, l1_1, l1_2, l1_3, l1_4, l1_6], [c0, c1, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_6], [c1, l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_6], [c0, c1, l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_6], [c0, c1, l1_5, l1_6], [c0, c1, l0_0, l1_5, l1_6], [c0, c1, l0_1, l1_5, l1_6], [c1, l0_0, l0_1, l1_5, l1_6], [c0, c1, l0_0, l0_1, l1_5, l1_6], [c0, c1, l1_0, l1_5, l1_6], [c0, c1, l0_0, l1_0, l1_5, l1_6], [c0, c1, l0_1, l1_0, l1_5, l1_6], [c1, l0_0, l0_1, l1_0, l1_5, l1_6], [c0, c1, l0_0, l0_1, l1_0, l1_5, l1_6], [c0, c1, l1_1, l1_5, l1_6], [c0, c1, l0_0, l1_1, l1_5, l1_6], [c0, c1, l0_1, l1_1, l1_5, l1_6], [c1, l0_0, l0_1, l1_1, l1_5, l1_6], [c0, c1, l0_0, l0_1, l1_1, l1_5, l1_6], [c0, c1, l1_0, l1_1, l1_5, l1_6], [c0, c1, l0_0, l1_0, l1_1, l1_5, l1_6], [c0, c1, l0_1, l1_0, l1_1, l1_5, l1_6], [c1, l0_0, l0_1, l1_0, l1_1, l1_5, l1_6], [c0, c1, l0_0, l0_1, l1_0, l1_1, l1_5, l1_6], [c0, c1, l1_2, l1_5, l1_6], [c0, c1, l0_0, l1_2, l1_5, l1_6], [c0, c1, l0_1, l1_2, l1_5, l1_6], [c1, l0_0, l0_1, l1_2, l1_5, l1_6], [c0, c1, l0_0, l0_1, l1_2, l1_5, l1_6], [c0, c1, l1_0, l1_2, l1_5, l1_6], [c0, c1, l0_0, l1_0, l1_2, l1_5, l1_6], [c0, c1, l0_1, l1_0, l1_2, l1_5, l1_6], [c1, l0_0, l0_1, l1_0, l1_2, l1_5, l1_6], [c0, c1, l0_0, l0_1, l1_0, l1_2, l1_5, l1_6], [c0, c1, l1_1, l1_2, l1_5, l1_6], [c0, c1, l0_0, l1_1, l1_2, l1_5, l1_6], [c0, c1, l0_1, l1_1, l1_2, l1_5, l1_6], [c1, l0_0, l0_1, l1_1, l1_2, l1_5, l1_6], [c0, c1, l0_0, l0_1, l1_1, l1_2, l1_5, l1_6], [c0, c1, l1_0, l1_1, l1_2, l1_5, l1_6], [c0, c1, l0_0, l1_0, l1_1, l1_2, l1_5, l1_6], [c0, c1, l0_1, l1_0, l1_1, l1_2, l1_5, l1_6], [c1, l0_0, l0_1, l1_0, l1_1, l1_2, l1_5, l1_6], [c0, c1, l0_0, l0_1, l1_0, l1_1, l1_2, l1_5, l1_6], [c0, c1, l1_3, l1_5, l1_6], [c0, c1, l0_0, l1_3, l1_5, l1_6], [c0, c1, l0_1, l1_3, l1_5, l1_6], [c1, l0_0, l0_1, l1_3, l1_5, l1_6], [c0, c1, l0_0, l0_1, l1_3, l1_5, l1_6], [c0, c1, l1_0, l1_3, l1_5, l1_6], [c0, c1, l0_0, l1_0, l1_3, l1_5, l1_6], [c0, c1, l0_1, l1_0, l1_3, l1_5, l1_6], [c1, l0_0, l0_1, l1_0, l1_3, l1_5, l1_6], [c0, c1, l0_0, l0_1, l1_0, l1_3, l1_5, l1_6], [c0, c1, l1_1, l1_3, l1_5, l1_6], [c0, c1, l0_0, l1_1, l1_3, l1_5, l1_6], [c0, c1, l0_1, l1_1, l1_3, l1_5, l1_6], [c1, l0_0, l0_1, l1_1, l1_3, l1_5, l1_6], [c0, c1, l0_0, l0_1, l1_1, l1_3, l1_5, l1_6], [c0, c1, l1_0, l1_1, l1_3, l1_5, l1_6], [c0, c1, l0_0, l1_0, l1_1, l1_3, l1_5, l1_6], [c0, c1, l0_1, l1_0, l1_1, l1_3, l1_5, l1_6], [c1, l0_0, l0_1, l1_0, l1_1, l1_3, l1_5, l1_6], [c0, c1, l0_0, l0_1, l1_0, l1_1, l1_3, l1_5, l1_6], [c0, c1, l1_2, l1_3, l1_5, l1_6], [c0, c1, l0_0, l1_2, l1_3, l1_5, l1_6], [c0, c1, l0_1, l1_2, l1_3, l1_5, l1_6], [c1, l0_0, l0_1, l1_2, l1_3, l1_5, l1_6], [c0, c1, l0_0, l0_1, l1_2, l1_3, l1_5, l1_6], [c0, c1, l1_0, l1_2, l1_3, l1_5, l1_6], [c0, c1, l0_0, l1_0, l1_2, l1_3, l1_5, l1_6], [c0, c1, l0_1, l1_0, l1_2, l1_3, l1_5, l1_6], [c1, l0_0, l0_1, l1_0, l1_2, l1_3, l1_5, l1_6], [c0, c1, l0_0, l0_1, l1_0, l1_2, l1_3, l1_5, l1_6], [c0, c1, l1_1, l1_2, l1_3, l1_5, l1_6], [c0, c1, l0_0, l1_1, l1_2, l1_3, l1_5, l1_6], [c0, c1, l0_1, l1_1, l1_2, l1_3, l1_5, l1_6], [c1, l0_0, l0_1, l1_1, l1_2, l1_3, l1_5, l1_6], [c0, c1, l0_0, l0_1, l1_1, l1_2, l1_3, l1_5, l1_6], [c0, c1, l1_0, l1_1, l1_2, l1_3, l1_5, l1_6], [c0, c1, l0_0, l1_0, l1_1, l1_2, l1_3, l1_5, l1_6], [c0, c1, l0_1, l1_0, l1_1, l1_2, l1_3, l1_5, l1_6], [c1, l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_5, l1_6], [c0, c1, l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_5, l1_6], [c0, c1, l1_4, l1_5, l1_6], [c0, c1, l0_0, l1_4, l1_5, l1_6], [c0, c1, l0_1, l1_4, l1_5, l1_6], [c1, l0_0, l0_1, l1_4, l1_5, l1_6], [c0, c1, l0_0, l0_1, l1_4, l1_5, l1_6], [c0, c1, l1_0, l1_4, l1_5, l1_6], [c0, c1, l0_0, l1_0, l1_4, l1_5, l1_6], [c0, c1, l0_1, l1_0, l1_4, l1_5, l1_6], [c1, l0_0, l0_1, l1_0, l1_4, l1_5, l1_6], [c0, c1, l0_0, l0_1, l1_0, l1_4, l1_5, l1_6], [c0, c1, l1_1, l1_4, l1_5, l1_6], [c0, c1, l0_0, l1_1, l1_4, l1_5, l1_6], [c0, c1, l0_1, l1_1, l1_4, l1_5, l1_6], [c1, l0_0, l0_1, l1_1, l1_4, l1_5, l1_6], [c0, c1, l0_0, l0_1, l1_1, l1_4, l1_5, l1_6], [c0, c1, l1_0, l1_1, l1_4, l1_5, l1_6], [c0, c1, l0_0, l1_0, l1_1, l1_4, l1_5, l1_6], [c0, c1, l0_1, l1_0, l1_1, l1_4, l1_5, l1_6], [c1, l0_0, l0_1, l1_0, l1_1, l1_4, l1_5, l1_6], [c0, c1, l0_0, l0_1, l1_0, l1_1, l1_4, l1_5, l1_6], [c0, c1, l1_2, l1_4, l1_5, l1_6], [c0, c1, l0_0, l1_2, l1_4, l1_5, l1_6], [c0, c1, l0_1, l1_2, l1_4, l1_5, l1_6], [c1, l0_0, l0_1, l1_2, l1_4, l1_5, l1_6], [c0, c1, l0_0, l0_1, l1_2, l1_4, l1_5, l1_6], [c0, c1, l1_0, l1_2, l1_4, l1_5, l1_6], [c0, c1, l0_0, l1_0, l1_2, l1_4, l1_5, l1_6], [c0, c1, l0_1, l1_0, l1_2, l1_4, l1_5, l1_6], [c1, l0_0, l0_1, l1_0, l1_2, l1_4, l1_5, l1_6], [c0, c1, l0_0, l0_1, l1_0, l1_2, l1_4, l1_5, l1_6], [c0, c1, l1_1, l1_2, l1_4, l1_5, l1_6], [c0, c1, l0_0, l1_1, l1_2, l1_4, l1_5, l1_6], [c0, c1, l0_1, l1_1, l1_2, l1_4, l1_5, l1_6], [c1, l0_0, l0_1, l1_1, l1_2, l1_4, l1_5, l1_6], [c0, c1, l0_0, l0_1, l1_1, l1_2, l1_4, l1_5, l1_6], [c0, c1, l1_0, l1_1, l1_2, l1_4, l1_5, l1_6], [c0, c1, l0_0, l1_0, l1_1, l1_2, l1_4, l1_5, l1_6], [c0, c1, l0_1, l1_0, l1_1, l1_2, l1_4, l1_5, l1_6], [c1, l0_0, l0_1, l1_0, l1_1, l1_2, l1_4, l1_5, l1_6], [c0, c1, l0_0, l0_1, l1_0, l1_1, l1_2, l1_4, l1_5, l1_6], [c0, c1, l1_3, l1_4, l1_5, l1_6], [c0, c1, l0_0, l1_3, l1_4, l1_5, l1_6], [c0, c1, l0_1, l1_3, l1_4, l1_5, l1_6], [c1, l0_0, l0_1, l1_3, l1_4, l1_5, l1_6], [c0, c1, l0_0, l0_1, l1_3, l1_4, l1_5, l1_6], [c0, c1, l1_0, l1_3, l1_4, l1_5, l1_6], [c0, c1, l0_0, l1_0, l1_3, l1_4, l1_5, l1_6], [c0, c1, l0_1, l1_0, l1_3, l1_4, l1_5, l1_6], [c1, l0_0, l0_1, l1_0, l1_3, l1_4, l1_5, l1_6], [c0, c1, l0_0, l0_1, l1_0, l1_3, l1_4, l1_5, l1_6], [c0, c1, l1_1, l1_3, l1_4, l1_5, l1_6], [c0, c1, l0_0, l1_1, l1_3, l1_4, l1_5, l1_6], [c0, c1, l0_1, l1_1, l1_3, l1_4, l1_5, l1_6], [c1, l0_0, l0_1, l1_1, l1_3, l1_4, l1_5, l1_6], [c0, c1, l0_0, l0_1, l1_1, l1_3, l1_4, l1_5, l1_6], [c0, c1, l1_0, l1_1, l1_3, l1_4, l1_5, l1_6], [c0, c1, l0_0, l1_0, l1_1, l1_3, l1_4, l1_5, l1_6], [c0, c1, l0_1, l1_0, l1_1, l1_3, l1_4, l1_5, l1_6], [c1, l0_0, l0_1, l1_0, l1_1, l1_3, l1_4, l1_5, l1_6], [c0, c1, l0_0, l0_1, l1_0, l1_1, l1_3, l1_4, l1_5, l1_6], [c0, c1, l1_2, l1_3, l1_4, l1_5, l1_6], [c0, c1, l0_0, l1_2, l1_3, l1_4, l1_5, l1_6], [c0, c1, l0_1, l1_2, l1_3, l1_4, l1_5, l1_6], [c1, l0_0, l0_1, l1_2, l1_3, l1_4, l1_5, l1_6], [c0, c1, l0_0, l0_1, l1_2, l1_3, l1_4, l1_5, l1_6], [c0, c1, l1_0, l1_2, l1_3, l1_4, l1_5, l1_6], [c0, c1, l0_0, l1_0, l1_2, l1_3, l1_4, l1_5, l1_6], [c0, c1, l0_1, l1_0, l1_2, l1_3, l1_4, l1_5, l1_6], [c1, l0_0, l0_1, l1_0, l1_2, l1_3, l1_4, l1_5, l1_6], [c0, c1, l0_0, l0_1, l1_0, l1_2, l1_3, l1_4, l1_5, l1_6], [c0, c1, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], [c0, c1, l0_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], [c0, c1, l0_1, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], [c1, l0_0, l0_1, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], [c0, c1, l0_0, l0_1, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], [c0, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], [c0, c1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], [c0, l0_0, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], [c0, c1, l0_0, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], [c0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], [c0, c1, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], [c0, l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], [c1, l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], [c0, c1, l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6]], [0, 0, 1, 10, 43, 105, 161, 161, 106, 46, 11, 1], 2, [[c0, c1]]]",
      "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
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "source roles = [[l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], [[c0, l0_0], [c0, l0_1], [c1, l1_0], [c1, l1_1], [c1, l1_2], [c1, l1_3], [c1, l1_4], [c1, l1_5], [c1, l1_6]]]",
      "after": "derived = [[[[c0, l0_0], 是], [[c0, l0_1], 是], [[c1, l1_0], 是], [[c1, l1_1], 是], [[c1, l1_2], 是], [[c1, l1_3], 是], [[c1, l1_4], 是], [[c1, l1_5], 是], [[c1, l1_6], 是]], 9, 2, [c0, c1]]",
      "relation": "implies",
      "rule_id": "Mincover_actual_candidate_complete_cover_and_smaller_witness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先核candidate真cover；較小解取完整source清單的首minimum，不用任意小集合代替。",
        "en": "First qualify an actual candidate cover.The smaller witness is the first minimum from the complete source list,not an arbitrary small set."
      },
      "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": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "source roles = [[l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], [[l0_0, [l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], 否], [l0_1, [l0_0, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], 否], [l1_0, [l0_0, l0_1, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], 否], [l1_1, [l0_0, l0_1, l1_0, l1_2, l1_3, l1_4, l1_5, l1_6], 否], [l1_2, [l0_0, l0_1, l1_0, l1_1, l1_3, l1_4, l1_5, l1_6], 否], [l1_3, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_4, l1_5, l1_6], 否], [l1_4, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_5, l1_6], 否], [l1_5, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_6], 否], [l1_6, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], 否]]]",
      "after": "derived = [是, 2, [c0, c1], 否]",
      "relation": "implies",
      "rule_id": "Mincover_ALL_single_deletions_minimal_BUT_smaller_cover_exists",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐刪都失cover，所以inclusion-minimal；但完整枚舉已有嚴格更小cover，所以不是minimum。cover對superset單調使逐刪可驗全部proper subset責任，不等同最大／最優大小。",
        "en": "Every single deletion loses coverage,so the candidate is inclusion-minimal.But exhaustive search supplies a strictly smaller cover,so it is not minimum.Superset monotonicity makes deletion tests cover all proper-subset obligations,not maximum or optimal size."
      },
      "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": [
      [
        "l0_0",
        "l0_1",
        "l1_0",
        "l1_1",
        "l1_2",
        "l1_3",
        "l1_4",
        "l1_5",
        "l1_6"
      ],
      [
        [
          "l0_0",
          [
            "l0_1",
            "l1_0",
            "l1_1",
            "l1_2",
            "l1_3",
            "l1_4",
            "l1_5",
            "l1_6"
          ],
          false
        ],
        [
          "l0_1",
          [
            "l0_0",
            "l1_0",
            "l1_1",
            "l1_2",
            "l1_3",
            "l1_4",
            "l1_5",
            "l1_6"
          ],
          false
        ],
        [
          "l1_0",
          [
            "l0_0",
            "l0_1",
            "l1_1",
            "l1_2",
            "l1_3",
            "l1_4",
            "l1_5",
            "l1_6"
          ],
          false
        ],
        [
          "l1_1",
          [
            "l0_0",
            "l0_1",
            "l1_0",
            "l1_2",
            "l1_3",
            "l1_4",
            "l1_5",
            "l1_6"
          ],
          false
        ],
        [
          "l1_2",
          [
            "l0_0",
            "l0_1",
            "l1_0",
            "l1_1",
            "l1_3",
            "l1_4",
            "l1_5",
            "l1_6"
          ],
          false
        ],
        [
          "l1_3",
          [
            "l0_0",
            "l0_1",
            "l1_0",
            "l1_1",
            "l1_2",
            "l1_4",
            "l1_5",
            "l1_6"
          ],
          false
        ],
        [
          "l1_4",
          [
            "l0_0",
            "l0_1",
            "l1_0",
            "l1_1",
            "l1_2",
            "l1_3",
            "l1_5",
            "l1_6"
          ],
          false
        ],
        [
          "l1_5",
          [
            "l0_0",
            "l0_1",
            "l1_0",
            "l1_1",
            "l1_2",
            "l1_3",
            "l1_4",
            "l1_6"
          ],
          false
        ],
        [
          "l1_6",
          [
            "l0_0",
            "l0_1",
            "l1_0",
            "l1_1",
            "l1_2",
            "l1_3",
            "l1_4",
            "l1_5"
          ],
          false
        ]
      ],
      true,
      2,
      [
        "c0",
        "c1"
      ],
      false,
      "inclusion_minimal_is_not_minimum"
    ]
  },
  "conclusion": {
    "canonical_value": "([l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], [[l0_0, [l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], 否], [l0_1, [l0_0, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], 否], [l1_0, [l0_0, l0_1, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], 否], [l1_1, [l0_0, l0_1, l1_0, l1_2, l1_3, l1_4, l1_5, l1_6], 否], [l1_2, [l0_0, l0_1, l1_0, l1_1, l1_3, l1_4, l1_5, l1_6], 否], [l1_3, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_4, l1_5, l1_6], 否], [l1_4, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_5, l1_6], 否], [l1_5, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_6], 否], [l1_6, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], 否]], 是, 2, [c0, c1], 否, inclusion_minimal_is_not_minimum)",
    "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": 66,
      "original_line_sha256": "68bb63d11fdf16cc558ddc44140983bc90abe42fc2497609660a9cc6e8210ef7",
      "question_compact_sha256": "d8fec2b81f4a1ca2bb032866cac66d3d66defbdb01cdf2d608407a084215f506",
      "english_row_compact_sha256": "0a28083fe519fe9b0f00a1334646824e7efed75d0838ce5947523c5665442bc4",
      "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": "43065083931c1f4d6ca7e3f14b866b17260320f15892a27c04f563ce6c558eaa",
    "original_candidate_line_sha256": "43065083931c1f4d6ca7e3f14b866b17260320f15892a27c04f563ce6c558eaa"
  },
  "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": 5,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "fd6e14bd97d02b084f5566014b0bf90a62d91e0f836337fc5571612f894f4fcc",
      "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."
  }
}