{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261015-advance-20261116-r06-topological-ordering-dag-0033",
  "pilot_number": "P1000-0783",
  "number": 783,
  "family_id": "qf-mod47-toposort-all-orders",
  "title": {
    "zh-Hant": "第 783 題：有向無環圖的拓撲排序",
    "en": "Question 783: Audit All Vertex Permutations and Retain Every Valid Order"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261015-advance-20261116-r06-topological-ordering-dag-0033",
    "batch_id": "batch-20261015-advance-20261116-r06-topological-ordering-dag-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-topological-ordering-dag"
    ],
    "title_zh": "有向無環圖的拓撲排序",
    "stem_zh": "finite digraph、完整vertex permutation與all-edge order責任。操作=toposort-all-orders；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"],\"edges\":[[\"v0\",\"v4\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
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[v5, v2, v1, v3, v0, v4], [v5, v2, v3, v0, v1, v4], [v5, v2, v3, v0, v4, v1], [v5, v2, v3, v1, v0, v4], [v5, v3, v0, v1, v2, v4], [v5, v3, v0, v1, v4, v2], [v5, v3, v0, v2, v1, v4], [v5, v3, v0, v2, v4, v1], [v5, v3, v0, v4, v1, v2], [v5, v3, v0, v4, v2, v1], [v5, v3, v1, v0, v2, v4], [v5, v3, v1, v0, v4, v2], [v5, v3, v1, v2, v0, v4], [v5, v3, v2, v0, v1, v4], [v5, v3, v2, v0, v4, v1], [v5, v3, v2, v1, v0, v4]], 360, 是, all_vertex_permutations_are_audited)",
      "explanation_zh": "Wave6第十層核對=([v0, v1, v2, v3, v4, v5], [[v0, v4]], [[v0, v1, v2, v3, v4, v5], [v0, v1, v2, v3, v5, v4], [v0, v1, v2, v4, v3, v5], [v0, v1, v2, v4, v5, v3], [v0, v1, v2, v5, v3, v4], [v0, v1, v2, v5, v4, v3], [v0, v1, v3, v2, v4, v5], [v0, v1, v3, v2, v5, v4], [v0, v1, v3, v4, v2, v5], [v0, v1, v3, v4, v5, v2], [v0, v1, v3, v5, v2, v4], [v0, v1, v3, v5, v4, v2], [v0, v1, v4, v2, v3, v5], [v0, v1, v4, v2, v5, v3], [v0, v1, v4, v3, v2, v5], [v0, v1, v4, v3, v5, v2], [v0, v1, v4, v5, v2, v3], [v0, v1, v4, v5, v3, v2], [v0, v1, v5, v2, v3, v4], [v0, v1, v5, v2, v4, v3], [v0, v1, v5, v3, v2, v4], [v0, v1, v5, v3, v4, v2], [v0, v1, v5, v4, v2, v3], [v0, v1, v5, v4, v3, v2], [v0, v2, v1, v3, v4, v5], [v0, v2, v1, v3, v5, v4], [v0, v2, v1, v4, v3, v5], [v0, v2, v1, v4, v5, v3], [v0, v2, v1, v5, v3, v4], [v0, v2, v1, v5, v4, v3], [v0, v2, v3, v1, v4, v5], [v0, v2, v3, v1, v5, v4], [v0, v2, v3, v4, v1, v5], [v0, v2, v3, v4, v5, v1], [v0, v2, v3, v5, v1, v4], [v0, v2, v3, 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[v5, v0, v1, v4, v3, v2], [v5, v0, v2, v1, v3, v4], [v5, v0, v2, v1, v4, v3], [v5, v0, v2, v3, v1, v4], [v5, v0, v2, v3, v4, v1], [v5, v0, v2, v4, v1, v3], [v5, v0, v2, v4, v3, v1], [v5, v0, v3, v1, v2, v4], [v5, v0, v3, v1, v4, v2], [v5, v0, v3, v2, v1, v4], [v5, v0, v3, v2, v4, v1], [v5, v0, v3, v4, v1, v2], [v5, v0, v3, v4, v2, v1], [v5, v0, v4, v1, v2, v3], [v5, v0, v4, v1, v3, v2], [v5, v0, v4, v2, v1, v3], [v5, v0, v4, v2, v3, v1], [v5, v0, v4, v3, v1, v2], [v5, v0, v4, v3, v2, v1], [v5, v1, v0, v2, v3, v4], [v5, v1, v0, v2, v4, v3], [v5, v1, v0, v3, v2, v4], [v5, v1, v0, v3, v4, v2], [v5, v1, v0, v4, v2, v3], [v5, v1, v0, v4, v3, v2], [v5, v1, v2, v0, v3, v4], [v5, v1, v2, v0, v4, v3], [v5, v1, v2, v3, v0, v4], [v5, v1, v3, v0, v2, v4], [v5, v1, v3, v0, v4, v2], [v5, v1, v3, v2, v0, v4], [v5, v2, v0, v1, v3, v4], [v5, v2, v0, v1, v4, v3], [v5, v2, v0, v3, v1, v4], [v5, v2, v0, v3, v4, v1], [v5, v2, v0, v4, v1, v3], [v5, v2, v0, v4, v3, v1], [v5, v2, v1, v0, v3, v4], [v5, v2, v1, v0, v4, v3], [v5, v2, v1, v3, v0, v4], [v5, v2, v3, v0, v1, v4], [v5, v2, v3, v0, v4, v1], [v5, v2, v3, v1, v0, v4], [v5, v3, v0, v1, v2, v4], [v5, v3, v0, v1, v4, v2], [v5, v3, v0, v2, v1, v4], [v5, v3, v0, v2, v4, v1], [v5, v3, v0, v4, v1, v2], [v5, v3, v0, v4, v2, v1], [v5, v3, v1, v0, v2, v4], [v5, v3, v1, v0, v4, v2], [v5, v3, v1, v2, v0, v4], [v5, v3, v2, v0, v1, v4], [v5, v3, v2, v0, v4, v1], [v5, v3, v2, v1, v0, v4]], 360, 是, all_vertex_permutations_are_audited)；依完整明示資料獨立重算；全部orders、Kahn source-removal trace與cycle witness交叉核對；invalid candidate不等於cycle；multiple orders與empty-edge graph允許且不混同topological space。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "有向無環圖的拓撲排序"
      ]
    },
    "variant": {
      "family_id": "qf-mod47-toposort-all-orders",
      "seed": 202611166033,
      "parameters": {
        "vertices": [
          "v0",
          "v1",
          "v2",
          "v3",
          "v4",
          "v5"
        ],
        "edges": [
          [
            "v0",
            "v4"
          ]
        ],
        "domain": "Complete_finite_topological_ordering",
        "operation": "toposort-all-orders",
        "semantic_template": "qf-mod47-toposort-all-orders/{\"edges\":[[\"v0\",\"v4\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"]}"
      }
    },
    "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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r06-topological-ordering-dag-0033",
    "batch_id": "batch-20261015-advance-20261116-r06-topological-ordering-dag-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "5a91f0799336ad31b18660eb881f7cd447f4b4de6ade778ec3ecceefb082c11b",
      "family_id": "qf-mod47-toposort-all-orders",
      "seed": 202611166033,
      "semantic_template": "qf-mod47-toposort-all-orders/{\"edges\":[[\"v0\",\"v4\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"]}"
    },
    "fields": {
      "title": "Audit All Vertex Permutations and Retain Every Valid Order",
      "stem": "Use unique vertices=[v0, v1, v2, v3, v4, v5] and loop-free duplicate-free directed edges=[[v0, v4]]. Every edge u→v needs position(u)<position(v) in a complete vertex permutation; labels use source lexical order. Give (vertices, edges, all_valid_orders, order_count, order_exists, certificate).",
      "answer": {
        "kind": "expression",
        "value": "([v0, v1, v2, v3, v4, v5], [[v0, v4]], [[v0, v1, v2, v3, v4, v5], [v0, v1, v2, v3, v5, v4], [v0, v1, v2, v4, v3, v5], [v0, v1, v2, v4, v5, v3], [v0, v1, v2, v5, v3, v4], [v0, v1, v2, v5, v4, v3], [v0, v1, v3, v2, v4, v5], [v0, v1, v3, v2, v5, v4], [v0, v1, v3, v4, v2, v5], [v0, v1, v3, v4, v5, v2], [v0, v1, v3, v5, v2, v4], [v0, v1, v3, v5, v4, v2], [v0, v1, v4, v2, v3, v5], [v0, v1, v4, v2, v5, v3], [v0, v1, v4, v3, v2, v5], [v0, v1, v4, v3, v5, v2], [v0, v1, v4, v5, v2, v3], [v0, v1, v4, v5, v3, v2], [v0, v1, v5, v2, v3, v4], [v0, v1, v5, v2, v4, v3], [v0, v1, v5, v3, v2, v4], [v0, v1, v5, v3, v4, v2], [v0, v1, v5, v4, v2, v3], [v0, v1, v5, v4, v3, v2], [v0, v2, v1, v3, v4, v5], [v0, v2, v1, v3, v5, v4], [v0, v2, v1, v4, v3, v5], [v0, v2, v1, v4, v5, v3], [v0, v2, v1, v5, v3, v4], [v0, v2, v1, v5, v4, v3], [v0, v2, v3, v1, v4, v5], [v0, v2, v3, v1, v5, v4], [v0, v2, v3, v4, v1, v5], [v0, v2, v3, v4, v5, v1], [v0, v2, v3, v5, v1, v4], [v0, v2, v3, v5, v4, v1], [v0, v2, v4, v1, v3, v5], [v0, v2, v4, v1, v5, v3], [v0, v2, v4, v3, v1, v5], [v0, v2, v4, v3, v5, v1], [v0, v2, v4, v5, v1, v3], [v0, v2, v4, v5, v3, v1], [v0, v2, v5, v1, v3, v4], [v0, v2, v5, v1, v4, v3], [v0, v2, v5, v3, v1, v4], [v0, v2, v5, v3, v4, v1], [v0, v2, v5, v4, v1, v3], [v0, v2, v5, v4, v3, v1], [v0, v3, v1, v2, v4, v5], [v0, v3, v1, v2, v5, v4], [v0, v3, v1, v4, v2, v5], [v0, v3, v1, v4, v5, v2], [v0, v3, v1, v5, v2, v4], [v0, v3, v1, v5, v4, v2], [v0, v3, v2, v1, v4, v5], [v0, v3, v2, v1, v5, v4], [v0, v3, v2, v4, v1, v5], [v0, v3, v2, v4, v5, v1], [v0, v3, v2, v5, v1, v4], [v0, v3, v2, v5, v4, v1], [v0, v3, v4, v1, v2, v5], [v0, v3, v4, v1, v5, v2], [v0, v3, v4, v2, v1, v5], [v0, v3, v4, v2, v5, v1], [v0, v3, v4, v5, v1, v2], [v0, v3, v4, v5, v2, v1], [v0, v3, v5, v1, v2, v4], [v0, v3, v5, v1, v4, v2], [v0, v3, v5, v2, v1, v4], [v0, v3, v5, v2, v4, v1], [v0, v3, v5, v4, v1, v2], [v0, v3, v5, v4, v2, v1], [v0, v4, v1, v2, v3, v5], [v0, v4, v1, v2, v5, v3], [v0, v4, v1, v3, v2, v5], [v0, v4, v1, v3, v5, v2], [v0, v4, v1, v5, v2, v3], [v0, v4, v1, v5, v3, v2], [v0, v4, v2, v1, v3, v5], [v0, v4, v2, v1, v5, v3], [v0, v4, v2, v3, v1, v5], [v0, v4, v2, v3, v5, v1], [v0, v4, v2, v5, v1, v3], [v0, v4, v2, v5, v3, v1], [v0, v4, v3, v1, v2, v5], [v0, v4, v3, v1, v5, v2], [v0, v4, v3, v2, v1, v5], [v0, v4, v3, v2, v5, v1], [v0, v4, v3, v5, v1, v2], [v0, v4, v3, v5, v2, v1], [v0, v4, v5, v1, v2, v3], [v0, v4, v5, v1, v3, v2], [v0, v4, v5, v2, v1, v3], [v0, v4, v5, v2, v3, v1], [v0, v4, v5, v3, v1, v2], [v0, v4, v5, v3, v2, v1], [v0, v5, v1, v2, v3, v4], [v0, v5, v1, v2, v4, v3], [v0, v5, v1, v3, v2, v4], [v0, v5, v1, v3, v4, v2], [v0, v5, v1, v4, v2, v3], [v0, v5, v1, v4, v3, v2], [v0, v5, v2, v1, v3, v4], [v0, v5, v2, v1, v4, v3], [v0, v5, v2, v3, v1, v4], [v0, v5, v2, v3, v4, v1], [v0, v5, v2, v4, v1, v3], [v0, v5, v2, v4, v3, v1], [v0, v5, v3, v1, v2, v4], [v0, v5, v3, v1, v4, v2], [v0, v5, v3, v2, v1, v4], [v0, v5, v3, v2, v4, v1], [v0, v5, v3, v4, v1, v2], [v0, v5, v3, v4, v2, v1], [v0, v5, v4, v1, v2, v3], [v0, v5, v4, v1, v3, v2], [v0, v5, v4, v2, v1, v3], [v0, v5, v4, v2, v3, v1], [v0, v5, v4, v3, v1, v2], [v0, v5, v4, v3, v2, v1], [v1, v0, v2, v3, v4, v5], [v1, v0, v2, v3, v5, v4], [v1, v0, v2, v4, v3, v5], [v1, v0, v2, v4, v5, v3], [v1, v0, v2, v5, v3, v4], [v1, v0, v2, v5, v4, v3], [v1, v0, v3, v2, v4, v5], [v1, v0, v3, v2, v5, v4], [v1, v0, v3, v4, v2, v5], [v1, v0, v3, v4, v5, v2], [v1, v0, v3, v5, v2, v4], [v1, v0, v3, v5, v4, v2], [v1, v0, v4, v2, v3, v5], [v1, v0, v4, v2, v5, v3], [v1, v0, v4, v3, v2, v5], [v1, v0, v4, v3, v5, v2], [v1, v0, v4, v5, v2, v3], [v1, v0, v4, v5, v3, v2], [v1, v0, v5, v2, v3, v4], [v1, v0, v5, v2, v4, v3], [v1, v0, v5, v3, v2, v4], [v1, v0, v5, v3, v4, v2], [v1, v0, v5, v4, v2, v3], [v1, v0, v5, v4, v3, v2], [v1, v2, v0, v3, v4, v5], [v1, v2, v0, v3, v5, v4], [v1, v2, v0, v4, v3, v5], [v1, v2, v0, v4, v5, v3], [v1, v2, v0, v5, v3, v4], [v1, v2, v0, v5, v4, v3], [v1, v2, v3, v0, v4, v5], [v1, v2, v3, v0, v5, v4], [v1, v2, v3, v5, v0, v4], [v1, v2, v5, v0, v3, v4], [v1, v2, v5, v0, v4, v3], [v1, v2, v5, v3, v0, v4], [v1, v3, v0, v2, v4, v5], [v1, v3, v0, v2, v5, v4], [v1, v3, v0, v4, v2, v5], [v1, v3, v0, v4, v5, v2], [v1, v3, v0, v5, v2, v4], [v1, v3, v0, v5, v4, v2], [v1, v3, v2, v0, v4, v5], [v1, v3, v2, v0, v5, v4], [v1, v3, v2, v5, v0, v4], [v1, v3, v5, v0, v2, v4], [v1, v3, v5, v0, v4, v2], [v1, v3, v5, v2, v0, v4], [v1, v5, v0, v2, v3, v4], [v1, v5, v0, v2, v4, v3], [v1, v5, v0, v3, v2, v4], [v1, v5, v0, v3, v4, v2], [v1, v5, v0, v4, v2, v3], [v1, v5, v0, v4, v3, v2], [v1, v5, v2, v0, v3, v4], [v1, v5, v2, v0, v4, v3], [v1, v5, v2, v3, v0, v4], [v1, v5, v3, v0, v2, v4], [v1, v5, v3, v0, v4, v2], [v1, v5, v3, v2, v0, v4], [v2, v0, v1, v3, v4, v5], [v2, v0, v1, v3, v5, v4], [v2, v0, v1, v4, v3, v5], [v2, v0, v1, v4, v5, v3], [v2, v0, v1, v5, v3, v4], [v2, v0, v1, v5, v4, v3], [v2, v0, v3, v1, v4, v5], [v2, v0, v3, v1, v5, v4], [v2, v0, v3, v4, v1, v5], [v2, v0, v3, v4, v5, v1], [v2, v0, v3, v5, v1, v4], [v2, v0, v3, v5, v4, v1], [v2, v0, v4, v1, v3, v5], [v2, v0, v4, v1, v5, v3], [v2, v0, v4, v3, v1, v5], [v2, v0, v4, v3, v5, v1], [v2, v0, v4, v5, v1, v3], [v2, v0, v4, v5, v3, v1], [v2, v0, v5, v1, v3, v4], [v2, v0, v5, v1, v4, v3], [v2, v0, v5, v3, v1, v4], [v2, v0, v5, v3, v4, v1], [v2, v0, v5, v4, v1, v3], [v2, v0, v5, v4, v3, v1], [v2, v1, v0, v3, v4, v5], [v2, v1, v0, v3, v5, v4], [v2, v1, v0, v4, v3, v5], [v2, v1, v0, v4, v5, v3], [v2, v1, v0, v5, v3, v4], [v2, v1, v0, v5, v4, v3], [v2, v1, v3, v0, v4, v5], [v2, v1, v3, v0, v5, v4], [v2, v1, v3, v5, v0, v4], [v2, v1, v5, v0, v3, v4], [v2, v1, v5, v0, v4, v3], [v2, v1, v5, v3, v0, v4], [v2, v3, v0, v1, v4, v5], [v2, v3, v0, v1, v5, v4], [v2, v3, v0, v4, v1, v5], [v2, v3, v0, v4, v5, v1], [v2, v3, v0, v5, v1, v4], [v2, v3, v0, v5, v4, v1], [v2, v3, v1, v0, v4, v5], [v2, v3, v1, v0, v5, v4], [v2, v3, v1, v5, v0, v4], [v2, v3, v5, v0, v1, v4], [v2, v3, v5, v0, v4, v1], [v2, v3, v5, v1, v0, v4], [v2, v5, v0, v1, v3, v4], [v2, v5, v0, v1, v4, v3], [v2, v5, v0, v3, v1, v4], [v2, v5, v0, v3, v4, v1], [v2, v5, v0, v4, v1, v3], [v2, v5, v0, v4, v3, v1], [v2, v5, v1, v0, v3, v4], [v2, v5, v1, v0, v4, v3], [v2, v5, v1, v3, v0, v4], [v2, v5, v3, v0, v1, v4], [v2, v5, v3, v0, v4, v1], [v2, v5, v3, v1, v0, v4], [v3, v0, v1, v2, v4, v5], [v3, v0, v1, v2, v5, v4], [v3, v0, v1, v4, v2, v5], [v3, v0, v1, v4, v5, v2], [v3, v0, v1, v5, v2, v4], [v3, v0, v1, v5, v4, v2], [v3, v0, v2, v1, v4, v5], [v3, v0, v2, v1, v5, v4], [v3, v0, v2, v4, v1, v5], [v3, v0, v2, v4, v5, v1], [v3, v0, v2, v5, v1, v4], [v3, v0, v2, v5, v4, v1], [v3, v0, v4, v1, v2, v5], [v3, v0, v4, v1, v5, v2], [v3, v0, v4, v2, v1, v5], [v3, v0, v4, v2, v5, v1], [v3, v0, v4, v5, v1, v2], [v3, v0, v4, v5, v2, v1], [v3, v0, v5, v1, v2, v4], [v3, v0, v5, v1, v4, v2], [v3, v0, v5, v2, v1, v4], [v3, v0, v5, v2, v4, v1], [v3, v0, v5, v4, v1, v2], [v3, v0, v5, v4, v2, v1], [v3, v1, v0, v2, v4, v5], [v3, v1, v0, v2, v5, v4], [v3, v1, v0, v4, v2, v5], [v3, v1, v0, v4, v5, v2], [v3, v1, v0, v5, v2, v4], [v3, v1, v0, v5, v4, v2], [v3, v1, v2, v0, v4, v5], [v3, v1, v2, v0, v5, v4], [v3, v1, v2, v5, v0, v4], [v3, v1, v5, v0, v2, v4], [v3, v1, v5, v0, v4, v2], [v3, v1, v5, v2, v0, v4], [v3, v2, v0, v1, v4, v5], [v3, v2, v0, v1, v5, v4], [v3, v2, v0, v4, v1, v5], [v3, v2, v0, v4, v5, v1], [v3, v2, v0, v5, v1, v4], [v3, v2, v0, v5, v4, v1], [v3, v2, v1, v0, v4, v5], [v3, v2, v1, v0, v5, v4], [v3, v2, v1, v5, v0, v4], [v3, v2, v5, v0, v1, v4], [v3, v2, v5, v0, v4, v1], [v3, v2, v5, v1, v0, v4], [v3, v5, v0, v1, v2, v4], [v3, v5, v0, v1, v4, v2], [v3, v5, v0, v2, v1, v4], [v3, v5, v0, v2, v4, v1], [v3, v5, v0, v4, v1, v2], [v3, v5, v0, v4, v2, v1], [v3, v5, v1, v0, v2, v4], [v3, v5, v1, v0, v4, v2], [v3, v5, v1, v2, v0, v4], [v3, v5, v2, v0, v1, v4], [v3, v5, v2, v0, v4, v1], [v3, v5, v2, v1, v0, v4], [v5, v0, v1, v2, v3, v4], [v5, v0, v1, v2, v4, v3], [v5, v0, v1, v3, v2, v4], [v5, v0, v1, v3, v4, v2], [v5, v0, v1, v4, v2, v3], [v5, v0, v1, v4, v3, v2], [v5, v0, v2, v1, v3, v4], [v5, v0, v2, v1, v4, v3], [v5, v0, v2, v3, v1, v4], [v5, v0, v2, v3, v4, v1], [v5, v0, v2, v4, v1, v3], [v5, v0, v2, v4, v3, v1], [v5, v0, v3, v1, v2, v4], [v5, v0, v3, v1, v4, v2], [v5, v0, v3, v2, v1, v4], [v5, v0, v3, v2, v4, v1], [v5, v0, v3, v4, v1, v2], [v5, v0, v3, v4, v2, v1], [v5, v0, v4, v1, v2, v3], [v5, v0, v4, v1, v3, v2], [v5, v0, v4, v2, v1, v3], [v5, v0, v4, v2, v3, v1], [v5, v0, v4, v3, v1, v2], [v5, v0, v4, v3, v2, v1], [v5, v1, v0, v2, v3, v4], [v5, v1, v0, v2, v4, v3], [v5, v1, v0, v3, v2, v4], [v5, v1, v0, v3, v4, v2], [v5, v1, v0, v4, v2, v3], [v5, v1, v0, v4, v3, v2], [v5, v1, v2, v0, v3, v4], [v5, v1, v2, v0, v4, v3], [v5, v1, v2, v3, v0, v4], [v5, v1, v3, v0, v2, v4], [v5, v1, v3, v0, v4, v2], [v5, v1, v3, v2, v0, v4], [v5, v2, v0, v1, v3, v4], [v5, v2, v0, v1, v4, v3], [v5, v2, v0, v3, v1, v4], [v5, v2, v0, v3, v4, v1], [v5, v2, v0, v4, v1, v3], [v5, v2, v0, v4, v3, v1], [v5, v2, v1, v0, v3, v4], [v5, v2, v1, v0, v4, v3], [v5, v2, v1, v3, v0, v4], [v5, v2, v3, v0, v1, v4], [v5, v2, v3, v0, v4, v1], [v5, v2, v3, v1, v0, v4], [v5, v3, v0, v1, v2, v4], [v5, v3, v0, v1, v4, v2], [v5, v3, v0, v2, v1, v4], [v5, v3, v0, v2, v4, v1], [v5, v3, v0, v4, v1, v2], [v5, v3, v0, v4, v2, v1], [v5, v3, v1, v0, v2, v4], [v5, v3, v1, v0, v4, v2], [v5, v3, v1, v2, v0, v4], [v5, v3, v2, v0, v1, v4], [v5, v3, v2, v0, v4, v1], [v5, v3, v2, v1, v0, v4]], 360, Yes, all_vertex_permutations_are_audited)",
        "explanation": "Return every order satisfying all edges, in source canonical permutation order, along with count and existence. Multiple orders are allowed; a zero-vertex graph has one empty permutation. These are graph orders, not topological-space claims. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5], [[v0, v4]], [[v0, v1, v2, v3, v4, v5], [v0, v1, v2, v3, v5, v4], [v0, v1, v2, v4, v3, v5], [v0, v1, v2, v4, v5, v3], [v0, v1, v2, v5, v3, v4], [v0, v1, v2, v5, v4, v3], [v0, v1, v3, v2, v4, v5], [v0, v1, v3, v2, v5, v4], [v0, v1, v3, v4, v2, v5], [v0, v1, v3, v4, v5, v2], [v0, v1, v3, v5, v2, v4], [v0, v1, v3, v5, v4, v2], [v0, v1, v4, v2, v3, v5], [v0, v1, v4, v2, v5, v3], [v0, v1, v4, v3, v2, v5], [v0, v1, v4, v3, v5, v2], [v0, v1, v4, v5, v2, v3], [v0, v1, v4, v5, v3, v2], [v0, v1, v5, v2, v3, v4], [v0, v1, v5, v2, v4, v3], [v0, v1, v5, v3, v2, v4], [v0, v1, v5, v3, v4, v2], [v0, v1, v5, v4, v2, v3], [v0, v1, v5, v4, v3, v2], [v0, v2, v1, v3, v4, v5], [v0, v2, v1, v3, v5, v4], [v0, v2, v1, v4, v3, v5], [v0, v2, v1, v4, v5, v3], [v0, v2, v1, v5, v3, v4], [v0, v2, v1, v5, v4, v3], [v0, v2, v3, v1, v4, v5], [v0, v2, v3, v1, v5, v4], [v0, v2, v3, v4, v1, v5], [v0, v2, v3, v4, v5, v1], [v0, v2, v3, v5, v1, v4], [v0, v2, v3, v5, v4, v1], [v0, v2, v4, v1, v3, v5], [v0, v2, v4, v1, v5, v3], [v0, v2, v4, v3, v1, v5], [v0, v2, v4, v3, v5, v1], [v0, v2, v4, v5, v1, v3], [v0, v2, v4, v5, v3, v1], [v0, v2, v5, v1, v3, v4], [v0, v2, v5, v1, v4, v3], [v0, v2, v5, v3, v1, v4], [v0, v2, v5, v3, v4, v1], [v0, v2, v5, v4, v1, v3], [v0, v2, v5, v4, v3, v1], [v0, v3, v1, v2, v4, v5], [v0, v3, v1, v2, v5, v4], [v0, v3, v1, v4, v2, v5], [v0, v3, v1, v4, v5, v2], [v0, v3, v1, v5, v2, v4], [v0, v3, v1, v5, v4, v2], [v0, v3, v2, v1, v4, v5], [v0, v3, v2, v1, v5, v4], [v0, v3, v2, v4, v1, v5], [v0, v3, v2, v4, v5, v1], [v0, v3, v2, v5, v1, v4], [v0, v3, v2, v5, v4, v1], [v0, v3, v4, v1, v2, v5], [v0, v3, v4, v1, v5, v2], [v0, v3, v4, v2, v1, v5], [v0, v3, v4, v2, v5, v1], [v0, v3, v4, v5, v1, v2], [v0, v3, v4, v5, v2, v1], [v0, v3, v5, v1, v2, v4], [v0, v3, v5, v1, v4, v2], [v0, v3, v5, v2, v1, v4], [v0, v3, v5, v2, v4, v1], [v0, v3, v5, v4, v1, v2], [v0, v3, v5, v4, v2, v1], [v0, v4, v1, v2, v3, v5], [v0, v4, v1, v2, v5, v3], [v0, v4, v1, v3, v2, v5], [v0, v4, v1, v3, v5, v2], [v0, v4, v1, v5, v2, v3], [v0, v4, v1, v5, v3, v2], [v0, v4, v2, v1, v3, v5], [v0, v4, v2, v1, v5, v3], [v0, v4, v2, v3, v1, v5], [v0, v4, v2, v3, v5, v1], [v0, v4, v2, v5, v1, v3], [v0, v4, v2, v5, v3, v1], [v0, v4, v3, v1, v2, v5], [v0, v4, v3, v1, v5, v2], [v0, v4, v3, v2, v1, v5], [v0, v4, v3, v2, v5, v1], [v0, v4, v3, v5, v1, v2], [v0, v4, v3, v5, v2, v1], [v0, v4, v5, v1, v2, v3], [v0, v4, v5, v1, v3, v2], [v0, v4, v5, v2, v1, v3], [v0, v4, v5, v2, v3, v1], [v0, v4, v5, v3, v1, v2], [v0, v4, v5, v3, v2, v1], [v0, v5, v1, v2, v3, v4], [v0, v5, v1, v2, v4, v3], [v0, v5, v1, v3, v2, v4], [v0, v5, v1, v3, v4, v2], [v0, v5, v1, v4, v2, v3], [v0, v5, v1, v4, v3, v2], [v0, v5, v2, v1, v3, v4], [v0, v5, v2, v1, v4, v3], [v0, v5, v2, v3, v1, v4], [v0, v5, v2, v3, v4, v1], [v0, v5, v2, v4, v1, v3], [v0, v5, v2, v4, v3, v1], [v0, v5, v3, v1, v2, v4], [v0, v5, v3, v1, v4, v2], [v0, v5, v3, v2, v1, v4], [v0, v5, v3, v2, v4, v1], [v0, v5, v3, v4, v1, v2], [v0, v5, v3, v4, v2, v1], [v0, v5, v4, v1, v2, v3], [v0, v5, v4, v1, v3, v2], [v0, v5, v4, v2, v1, v3], [v0, v5, v4, v2, v3, v1], [v0, v5, v4, v3, v1, v2], [v0, v5, v4, v3, v2, v1], [v1, v0, v2, v3, v4, v5], [v1, v0, v2, v3, v5, v4], [v1, v0, v2, v4, v3, v5], [v1, v0, v2, v4, v5, v3], [v1, v0, v2, v5, v3, v4], [v1, v0, v2, v5, v4, v3], [v1, v0, v3, v2, v4, v5], [v1, v0, v3, v2, v5, v4], [v1, v0, v3, v4, v2, v5], [v1, v0, v3, v4, v5, v2], [v1, v0, v3, v5, v2, v4], [v1, v0, v3, v5, v4, v2], [v1, v0, v4, v2, v3, v5], [v1, v0, v4, v2, v5, v3], [v1, v0, v4, v3, v2, v5], [v1, v0, v4, v3, v5, v2], [v1, v0, v4, v5, v2, v3], [v1, v0, v4, v5, v3, v2], [v1, v0, v5, v2, v3, v4], [v1, v0, v5, v2, v4, v3], [v1, v0, v5, v3, v2, v4], [v1, v0, v5, v3, v4, v2], [v1, v0, v5, v4, v2, v3], [v1, v0, v5, v4, v3, v2], [v1, v2, v0, v3, v4, v5], [v1, v2, v0, v3, v5, v4], [v1, v2, v0, v4, v3, v5], [v1, v2, v0, v4, v5, v3], [v1, v2, v0, v5, v3, v4], [v1, v2, v0, v5, v4, v3], [v1, v2, v3, v0, v4, v5], [v1, v2, v3, v0, v5, v4], [v1, v2, v3, v5, v0, v4], [v1, v2, v5, v0, v3, v4], [v1, v2, v5, v0, v4, v3], [v1, v2, v5, v3, v0, v4], [v1, v3, v0, v2, v4, v5], [v1, v3, v0, v2, v5, v4], [v1, v3, v0, v4, v2, v5], [v1, v3, v0, v4, v5, v2], [v1, v3, v0, v5, v2, v4], [v1, v3, v0, v5, v4, v2], [v1, v3, v2, v0, v4, v5], [v1, v3, v2, v0, v5, v4], [v1, v3, v2, v5, v0, v4], [v1, v3, v5, v0, v2, v4], [v1, v3, v5, v0, v4, v2], [v1, v3, v5, v2, v0, v4], [v1, v5, v0, v2, v3, v4], [v1, v5, v0, v2, v4, v3], [v1, v5, v0, v3, v2, v4], [v1, v5, v0, v3, v4, v2], [v1, v5, v0, v4, v2, v3], [v1, v5, v0, v4, v3, v2], [v1, v5, v2, v0, v3, v4], [v1, v5, v2, v0, v4, v3], [v1, v5, v2, v3, v0, v4], [v1, v5, v3, v0, v2, v4], [v1, v5, v3, v0, v4, v2], [v1, v5, v3, v2, v0, v4], [v2, v0, v1, v3, v4, v5], [v2, v0, v1, v3, v5, v4], [v2, v0, v1, v4, v3, v5], [v2, v0, v1, v4, v5, v3], [v2, v0, v1, v5, v3, v4], [v2, v0, v1, v5, v4, v3], [v2, v0, v3, v1, v4, v5], [v2, v0, v3, v1, v5, v4], [v2, v0, v3, v4, v1, v5], [v2, v0, v3, v4, v5, v1], [v2, v0, v3, v5, v1, v4], [v2, v0, v3, v5, v4, v1], [v2, v0, v4, v1, v3, v5], [v2, v0, v4, v1, v5, v3], [v2, v0, v4, v3, v1, v5], [v2, v0, v4, v3, v5, v1], [v2, v0, v4, v5, v1, v3], [v2, v0, v4, v5, v3, v1], [v2, v0, v5, v1, v3, v4], [v2, v0, v5, v1, v4, v3], [v2, v0, v5, v3, v1, v4], [v2, v0, v5, v3, v4, v1], [v2, v0, v5, v4, v1, v3], [v2, v0, v5, v4, v3, v1], [v2, v1, v0, v3, v4, v5], [v2, v1, v0, v3, v5, v4], [v2, v1, v0, v4, v3, v5], [v2, v1, v0, v4, v5, v3], [v2, v1, v0, v5, v3, v4], [v2, v1, v0, v5, v4, v3], [v2, v1, v3, v0, v4, v5], [v2, v1, v3, v0, v5, v4], [v2, v1, v3, v5, v0, v4], [v2, v1, v5, v0, v3, v4], [v2, v1, v5, v0, v4, v3], [v2, v1, v5, v3, v0, v4], [v2, v3, v0, v1, v4, v5], [v2, v3, v0, v1, v5, v4], [v2, v3, v0, v4, v1, v5], [v2, v3, v0, v4, v5, v1], [v2, v3, v0, v5, v1, v4], [v2, v3, v0, v5, v4, v1], [v2, v3, v1, v0, v4, v5], [v2, v3, v1, v0, v5, v4], [v2, v3, v1, v5, v0, v4], [v2, v3, v5, v0, v1, v4], [v2, v3, v5, v0, v4, v1], [v2, v3, v5, v1, v0, v4], [v2, v5, v0, v1, v3, v4], [v2, v5, v0, v1, v4, v3], [v2, v5, v0, v3, v1, v4], [v2, v5, v0, v3, v4, v1], [v2, v5, v0, v4, v1, v3], [v2, v5, v0, v4, v3, v1], [v2, v5, v1, v0, v3, v4], [v2, v5, v1, v0, v4, v3], [v2, v5, v1, v3, v0, v4], [v2, v5, v3, v0, v1, v4], [v2, v5, v3, v0, v4, v1], [v2, v5, v3, v1, v0, v4], [v3, v0, v1, v2, v4, v5], [v3, v0, v1, v2, v5, v4], [v3, v0, v1, v4, v2, v5], [v3, v0, v1, v4, v5, v2], [v3, v0, v1, v5, v2, v4], [v3, v0, v1, v5, v4, v2], [v3, v0, v2, v1, v4, v5], [v3, v0, v2, v1, v5, v4], [v3, v0, v2, v4, v1, v5], [v3, v0, v2, v4, v5, v1], [v3, v0, v2, v5, v1, v4], [v3, v0, v2, v5, v4, v1], [v3, v0, v4, v1, v2, v5], [v3, v0, v4, v1, v5, v2], [v3, v0, v4, v2, v1, v5], [v3, v0, v4, v2, v5, v1], [v3, v0, v4, v5, v1, v2], [v3, v0, v4, v5, v2, v1], [v3, v0, v5, v1, v2, v4], [v3, v0, v5, v1, v4, v2], [v3, v0, v5, v2, v1, v4], [v3, v0, v5, v2, v4, v1], [v3, v0, v5, v4, v1, v2], [v3, v0, v5, v4, v2, v1], [v3, v1, v0, v2, v4, v5], [v3, v1, v0, v2, v5, v4], [v3, v1, v0, v4, v2, v5], [v3, v1, v0, v4, v5, v2], [v3, v1, v0, v5, v2, v4], [v3, v1, v0, v5, v4, v2], [v3, v1, v2, v0, v4, v5], [v3, v1, v2, v0, v5, v4], [v3, v1, v2, v5, v0, v4], [v3, v1, v5, v0, v2, v4], [v3, v1, v5, v0, v4, v2], [v3, v1, v5, v2, v0, v4], [v3, v2, v0, v1, v4, v5], [v3, v2, v0, v1, v5, v4], [v3, v2, v0, v4, v1, v5], [v3, v2, v0, v4, v5, v1], [v3, v2, v0, v5, v1, v4], [v3, v2, v0, v5, v4, v1], [v3, v2, v1, v0, v4, v5], [v3, v2, v1, v0, v5, v4], [v3, v2, v1, v5, v0, v4], [v3, v2, v5, v0, v1, v4], [v3, v2, v5, v0, v4, v1], [v3, v2, v5, v1, v0, v4], [v3, v5, v0, v1, v2, v4], [v3, v5, v0, v1, v4, v2], [v3, v5, v0, v2, v1, v4], [v3, v5, v0, v2, v4, v1], [v3, v5, v0, v4, v1, v2], [v3, v5, v0, v4, v2, v1], [v3, v5, v1, v0, v2, v4], [v3, v5, v1, v0, v4, v2], [v3, v5, v1, v2, v0, v4], [v3, v5, v2, v0, v1, v4], [v3, v5, v2, v0, v4, v1], [v3, v5, v2, v1, v0, v4], [v5, v0, v1, v2, v3, v4], [v5, v0, v1, v2, v4, v3], [v5, v0, v1, v3, v2, v4], [v5, v0, v1, v3, v4, v2], [v5, v0, v1, v4, v2, v3], [v5, v0, v1, v4, v3, v2], [v5, v0, v2, v1, v3, v4], [v5, v0, v2, v1, v4, v3], [v5, v0, v2, v3, v1, v4], [v5, v0, v2, v3, v4, v1], [v5, v0, v2, v4, v1, v3], [v5, v0, v2, v4, v3, v1], [v5, v0, v3, v1, v2, v4], [v5, v0, v3, v1, v4, v2], [v5, v0, v3, v2, v1, v4], [v5, v0, v3, v2, v4, v1], [v5, v0, v3, v4, v1, v2], [v5, v0, v3, v4, v2, v1], [v5, v0, v4, v1, v2, v3], [v5, v0, v4, v1, v3, v2], [v5, v0, v4, v2, v1, v3], [v5, v0, v4, v2, v3, v1], [v5, v0, v4, v3, v1, v2], [v5, v0, v4, v3, v2, v1], [v5, v1, v0, v2, v3, v4], [v5, v1, v0, v2, v4, v3], [v5, v1, v0, v3, v2, v4], [v5, v1, v0, v3, v4, v2], [v5, v1, v0, v4, v2, v3], [v5, v1, v0, v4, v3, v2], [v5, v1, v2, v0, v3, v4], [v5, v1, v2, v0, v4, v3], [v5, v1, v2, v3, v0, v4], [v5, v1, v3, v0, v2, v4], [v5, v1, v3, v0, v4, v2], [v5, v1, v3, v2, v0, v4], [v5, v2, v0, v1, v3, v4], [v5, v2, v0, v1, v4, v3], [v5, v2, v0, v3, v1, v4], [v5, v2, v0, v3, v4, v1], [v5, v2, v0, v4, v1, v3], [v5, v2, v0, v4, v3, v1], [v5, v2, v1, v0, v3, v4], [v5, v2, v1, v0, v4, v3], [v5, v2, v1, v3, v0, v4], [v5, v2, v3, v0, v1, v4], [v5, v2, v3, v0, v4, v1], [v5, v2, v3, v1, v0, v4], [v5, v3, v0, v1, v2, v4], [v5, v3, v0, v1, v4, v2], [v5, v3, v0, v2, v1, v4], [v5, v3, v0, v2, v4, v1], [v5, v3, v0, v4, v1, v2], [v5, v3, v0, v4, v2, v1], [v5, v3, v1, v0, v2, v4], [v5, v3, v1, v0, v4, v2], [v5, v3, v1, v2, v0, v4], [v5, v3, v2, v0, v1, v4], [v5, v3, v2, v0, v4, v1], [v5, v3, v2, v1, v0, v4]], 360, Yes, all_vertex_permutations_are_audited)."
      },
      "skills": [
        "Audit All Vertex Permutations and Retain Every Valid Order"
      ],
      "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": "finite digraph、完整vertex permutation與all-edge order責任。操作=toposort-all-orders；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"],\"edges\":[[\"v0\",\"v4\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use unique vertices=[v0, v1, v2, v3, v4, v5] and loop-free duplicate-free directed edges=[[v0, v4]]. Every edge u→v needs position(u)<position(v) in a complete vertex permutation; labels use source lexical order. Give (vertices, edges, all_valid_orders, order_count, order_exists, certificate)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "V=[v0, v1, v2, v3, v4, v5]",
      "after": "all directed edges=[[v0, v4]]",
      "relation": "implies",
      "rule_id": "full_directed_graph",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這是完整指定的有限有向圖，不是圖論術語的完全圖。",
        "en": "This is a fully specified finite digraph, not necessarily a complete digraph."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "核對完整頂點與邊；一個失敗次序不等於有環，全部合法分支不能漏列。",
          "en": "Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "no vertices remain",
      "after": "one empty completion; C(∅)=1",
      "relation": "implies",
      "rule_id": "empty_order_base",
      "conditions": [],
      "explanation": {
        "zh-Hant": "空剩餘問題有一種完成方式：不再放頂點。",
        "en": "The empty remaining problem has one completion: place no further vertex."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "核對完整頂點與邊；一個失敗次序不等於有環，全部合法分支不能漏列。",
          "en": "Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "C(v5) := sum(1)",
      "after": "C(v5) = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個合法首點形成互斥分支，分支完成數相加。",
        "en": "Each legal first vertex defines a disjoint branch; add the branch completion counts."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 C(v5)，不以小數近似替代。",
          "en": "Recompute C(v5) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1"
        ],
        "output": "1",
        "label": "C(v5)"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "remaining=[v5]; eligible first vertices=[v5]",
      "after": "complete continuations=1",
      "relation": "implies",
      "rule_id": "all_source_branches",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任何合法次序的首點必為零入度；反之先取任一零入度點，再接合法尾端即可。故分支沒有漏列。",
        "en": "The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "核對完整頂點與邊；一個失敗次序不等於有環，全部合法分支不能漏列。",
          "en": "Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "C(v4) := sum(1)",
      "after": "C(v4) = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個合法首點形成互斥分支，分支完成數相加。",
        "en": "Each legal first vertex defines a disjoint branch; add the branch completion counts."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 C(v4)，不以小數近似替代。",
          "en": "Recompute C(v4) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1"
        ],
        "output": "1",
        "label": "C(v4)"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "remaining=[v4]; eligible first vertices=[v4]",
      "after": "complete continuations=1",
      "relation": "implies",
      "rule_id": "all_source_branches",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任何合法次序的首點必為零入度；反之先取任一零入度點，再接合法尾端即可。故分支沒有漏列。",
        "en": "The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "核對完整頂點與邊；一個失敗次序不等於有環，全部合法分支不能漏列。",
          "en": "Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "C(v4,v5) := sum(1, 1)",
      "after": "C(v4,v5) = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個合法首點形成互斥分支，分支完成數相加。",
        "en": "Each legal first vertex defines a disjoint branch; add the branch completion counts."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 C(v4,v5)，不以小數近似替代。",
          "en": "Recompute C(v4,v5) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "1"
        ],
        "output": "2",
        "label": "C(v4,v5)"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "remaining=[v4, v5]; eligible first vertices=[v4, v5]",
      "after": "complete continuations=2",
      "relation": "implies",
      "rule_id": "all_source_branches",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任何合法次序的首點必為零入度；反之先取任一零入度點，再接合法尾端即可。故分支沒有漏列。",
        "en": "The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "核對完整頂點與邊；一個失敗次序不等於有環，全部合法分支不能漏列。",
          "en": "Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "C(v3) := sum(1)",
      "after": "C(v3) = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個合法首點形成互斥分支，分支完成數相加。",
        "en": "Each legal first vertex defines a disjoint branch; add the branch completion counts."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 C(v3)，不以小數近似替代。",
          "en": "Recompute C(v3) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1"
        ],
        "output": "1",
        "label": "C(v3)"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "remaining=[v3]; eligible first vertices=[v3]",
      "after": "complete continuations=1",
      "relation": "implies",
      "rule_id": "all_source_branches",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任何合法次序的首點必為零入度；反之先取任一零入度點，再接合法尾端即可。故分支沒有漏列。",
        "en": "The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "核對完整頂點與邊；一個失敗次序不等於有環，全部合法分支不能漏列。",
          "en": "Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "C(v3,v5) := sum(1, 1)",
      "after": "C(v3,v5) = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個合法首點形成互斥分支，分支完成數相加。",
        "en": "Each legal first vertex defines a disjoint branch; add the branch completion counts."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 C(v3,v5)，不以小數近似替代。",
          "en": "Recompute C(v3,v5) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "1"
        ],
        "output": "2",
        "label": "C(v3,v5)"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "remaining=[v3, v5]; eligible first vertices=[v3, v5]",
      "after": "complete continuations=2",
      "relation": "implies",
      "rule_id": "all_source_branches",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任何合法次序的首點必為零入度；反之先取任一零入度點，再接合法尾端即可。故分支沒有漏列。",
        "en": "The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "核對完整頂點與邊；一個失敗次序不等於有環，全部合法分支不能漏列。",
          "en": "Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "C(v3,v4) := sum(1, 1)",
      "after": "C(v3,v4) = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個合法首點形成互斥分支，分支完成數相加。",
        "en": "Each legal first vertex defines a disjoint branch; add the branch completion counts."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 C(v3,v4)，不以小數近似替代。",
          "en": "Recompute C(v3,v4) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "1"
        ],
        "output": "2",
        "label": "C(v3,v4)"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "remaining=[v3, v4]; eligible first vertices=[v3, v4]",
      "after": "complete continuations=2",
      "relation": "implies",
      "rule_id": "all_source_branches",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任何合法次序的首點必為零入度；反之先取任一零入度點，再接合法尾端即可。故分支沒有漏列。",
        "en": "The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "核對完整頂點與邊；一個失敗次序不等於有環，全部合法分支不能漏列。",
          "en": "Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "C(v3,v4,v5) := sum(2, 2, 2)",
      "after": "C(v3,v4,v5) = 6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個合法首點形成互斥分支，分支完成數相加。",
        "en": "Each legal first vertex defines a disjoint branch; add the branch completion counts."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 C(v3,v4,v5)，不以小數近似替代。",
          "en": "Recompute C(v3,v4,v5) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "2",
          "2",
          "2"
        ],
        "output": "6",
        "label": "C(v3,v4,v5)"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "remaining=[v3, v4, v5]; eligible first vertices=[v3, v4, v5]",
      "after": "complete continuations=6",
      "relation": "implies",
      "rule_id": "all_source_branches",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任何合法次序的首點必為零入度；反之先取任一零入度點，再接合法尾端即可。故分支沒有漏列。",
        "en": "The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "核對完整頂點與邊；一個失敗次序不等於有環，全部合法分支不能漏列。",
          "en": "Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "C(v2) := sum(1)",
      "after": "C(v2) = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個合法首點形成互斥分支，分支完成數相加。",
        "en": "Each legal first vertex defines a disjoint branch; add the branch completion counts."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 C(v2)，不以小數近似替代。",
          "en": "Recompute C(v2) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1"
        ],
        "output": "1",
        "label": "C(v2)"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "remaining=[v2]; eligible first vertices=[v2]",
      "after": "complete continuations=1",
      "relation": "implies",
      "rule_id": "all_source_branches",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任何合法次序的首點必為零入度；反之先取任一零入度點，再接合法尾端即可。故分支沒有漏列。",
        "en": "The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "核對完整頂點與邊；一個失敗次序不等於有環，全部合法分支不能漏列。",
          "en": "Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "C(v2,v5) := sum(1, 1)",
      "after": "C(v2,v5) = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個合法首點形成互斥分支，分支完成數相加。",
        "en": "Each legal first vertex defines a disjoint branch; add the branch completion counts."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 C(v2,v5)，不以小數近似替代。",
          "en": "Recompute C(v2,v5) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "1"
        ],
        "output": "2",
        "label": "C(v2,v5)"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "remaining=[v2, v5]; eligible first vertices=[v2, v5]",
      "after": "complete continuations=2",
      "relation": "implies",
      "rule_id": "all_source_branches",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任何合法次序的首點必為零入度；反之先取任一零入度點，再接合法尾端即可。故分支沒有漏列。",
        "en": "The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "核對完整頂點與邊；一個失敗次序不等於有環，全部合法分支不能漏列。",
          "en": "Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "C(v2,v4) := sum(1, 1)",
      "after": "C(v2,v4) = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個合法首點形成互斥分支，分支完成數相加。",
        "en": "Each legal first vertex defines a disjoint branch; add the branch completion counts."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 C(v2,v4)，不以小數近似替代。",
          "en": "Recompute C(v2,v4) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "1"
        ],
        "output": "2",
        "label": "C(v2,v4)"
      }
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "remaining=[v2, v4]; eligible first vertices=[v2, v4]",
      "after": "complete continuations=2",
      "relation": "implies",
      "rule_id": "all_source_branches",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任何合法次序的首點必為零入度；反之先取任一零入度點，再接合法尾端即可。故分支沒有漏列。",
        "en": "The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "核對完整頂點與邊；一個失敗次序不等於有環，全部合法分支不能漏列。",
          "en": "Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "C(v2,v4,v5) := sum(2, 2, 2)",
      "after": "C(v2,v4,v5) = 6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個合法首點形成互斥分支，分支完成數相加。",
        "en": "Each legal first vertex defines a disjoint branch; add the branch completion counts."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 C(v2,v4,v5)，不以小數近似替代。",
          "en": "Recompute C(v2,v4,v5) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "2",
          "2",
          "2"
        ],
        "output": "6",
        "label": "C(v2,v4,v5)"
      }
    },
    {
      "id": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "remaining=[v2, v4, v5]; eligible first vertices=[v2, v4, v5]",
      "after": "complete continuations=6",
      "relation": "implies",
      "rule_id": "all_source_branches",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任何合法次序的首點必為零入度；反之先取任一零入度點，再接合法尾端即可。故分支沒有漏列。",
        "en": "The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "核對完整頂點與邊；一個失敗次序不等於有環，全部合法分支不能漏列。",
          "en": "Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "C(v2,v3) := sum(1, 1)",
      "after": "C(v2,v3) = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個合法首點形成互斥分支，分支完成數相加。",
        "en": "Each legal first vertex defines a disjoint branch; add the branch completion counts."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 C(v2,v3)，不以小數近似替代。",
          "en": "Recompute C(v2,v3) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "1"
        ],
        "output": "2",
        "label": "C(v2,v3)"
      }
    },
    {
      "id": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "remaining=[v2, v3]; eligible first vertices=[v2, v3]",
      "after": "complete continuations=2",
      "relation": "implies",
      "rule_id": "all_source_branches",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任何合法次序的首點必為零入度；反之先取任一零入度點，再接合法尾端即可。故分支沒有漏列。",
        "en": "The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "核對完整頂點與邊；一個失敗次序不等於有環，全部合法分支不能漏列。",
          "en": "Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "C(v2,v3,v5) := sum(2, 2, 2)",
      "after": "C(v2,v3,v5) = 6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個合法首點形成互斥分支，分支完成數相加。",
        "en": "Each legal first vertex defines a disjoint branch; add the branch completion counts."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 C(v2,v3,v5)，不以小數近似替代。",
          "en": "Recompute C(v2,v3,v5) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "2",
          "2",
          "2"
        ],
        "output": "6",
        "label": "C(v2,v3,v5)"
      }
    },
    {
      "id": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "remaining=[v2, v3, v5]; eligible first vertices=[v2, v3, v5]",
      "after": "complete continuations=6",
      "relation": "implies",
      "rule_id": "all_source_branches",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任何合法次序的首點必為零入度；反之先取任一零入度點，再接合法尾端即可。故分支沒有漏列。",
        "en": "The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "核對完整頂點與邊；一個失敗次序不等於有環，全部合法分支不能漏列。",
          "en": "Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "C(v2,v3,v4) := sum(2, 2, 2)",
      "after": "C(v2,v3,v4) = 6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個合法首點形成互斥分支，分支完成數相加。",
        "en": "Each legal first vertex defines a disjoint branch; add the branch completion counts."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 C(v2,v3,v4)，不以小數近似替代。",
          "en": "Recompute C(v2,v3,v4) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "2",
          "2",
          "2"
        ],
        "output": "6",
        "label": "C(v2,v3,v4)"
      }
    },
    {
      "id": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "remaining=[v2, v3, v4]; eligible first vertices=[v2, v3, v4]",
      "after": "complete continuations=6",
      "relation": "implies",
      "rule_id": "all_source_branches",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任何合法次序的首點必為零入度；反之先取任一零入度點，再接合法尾端即可。故分支沒有漏列。",
        "en": "The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "核對完整頂點與邊；一個失敗次序不等於有環，全部合法分支不能漏列。",
          "en": "Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "C(v2,v3,v4,v5) := sum(6, 6, 6, 6)",
      "after": "C(v2,v3,v4,v5) = 24",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個合法首點形成互斥分支，分支完成數相加。",
        "en": "Each legal first vertex defines a disjoint branch; add the branch completion counts."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 C(v2,v3,v4,v5)，不以小數近似替代。",
          "en": "Recompute C(v2,v3,v4,v5) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "6",
          "6",
          "6",
          "6"
        ],
        "output": "24",
        "label": "C(v2,v3,v4,v5)"
      }
    },
    {
      "id": "S032",
      "uses": [
        "G001",
        "S031"
      ],
      "before": "remaining=[v2, v3, v4, v5]; eligible first vertices=[v2, v3, v4, v5]",
      "after": "complete continuations=24",
      "relation": "implies",
      "rule_id": "all_source_branches",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任何合法次序的首點必為零入度；反之先取任一零入度點，再接合法尾端即可。故分支沒有漏列。",
        "en": "The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "核對完整頂點與邊；一個失敗次序不等於有環，全部合法分支不能漏列。",
          "en": "Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S033",
      "uses": [
        "G001",
        "S032"
      ],
      "before": "C(v1) := sum(1)",
      "after": "C(v1) = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個合法首點形成互斥分支，分支完成數相加。",
        "en": "Each legal first vertex defines a disjoint branch; add the branch completion counts."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 C(v1)，不以小數近似替代。",
          "en": "Recompute C(v1) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1"
        ],
        "output": "1",
        "label": "C(v1)"
      }
    },
    {
      "id": "S034",
      "uses": [
        "G001",
        "S033"
      ],
      "before": "remaining=[v1]; eligible first vertices=[v1]",
      "after": "complete continuations=1",
      "relation": "implies",
      "rule_id": "all_source_branches",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任何合法次序的首點必為零入度；反之先取任一零入度點，再接合法尾端即可。故分支沒有漏列。",
        "en": "The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "核對完整頂點與邊；一個失敗次序不等於有環，全部合法分支不能漏列。",
          "en": "Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S035",
      "uses": [
        "G001",
        "S034"
      ],
      "before": "C(v1,v5) := sum(1, 1)",
      "after": "C(v1,v5) = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個合法首點形成互斥分支，分支完成數相加。",
        "en": "Each legal first vertex defines a disjoint branch; add the branch completion counts."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 C(v1,v5)，不以小數近似替代。",
          "en": "Recompute C(v1,v5) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "1"
        ],
        "output": "2",
        "label": "C(v1,v5)"
      }
    },
    {
      "id": "S036",
      "uses": [
        "G001",
        "S035"
      ],
      "before": "remaining=[v1, v5]; eligible first vertices=[v1, v5]",
      "after": "complete continuations=2",
      "relation": "implies",
      "rule_id": "all_source_branches",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任何合法次序的首點必為零入度；反之先取任一零入度點，再接合法尾端即可。故分支沒有漏列。",
        "en": "The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "核對完整頂點與邊；一個失敗次序不等於有環，全部合法分支不能漏列。",
          "en": "Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S037",
      "uses": [
        "G001",
        "S036"
      ],
      "before": "C(v1,v4) := sum(1, 1)",
      "after": "C(v1,v4) = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個合法首點形成互斥分支，分支完成數相加。",
        "en": "Each legal first vertex defines a disjoint branch; add the branch completion counts."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 C(v1,v4)，不以小數近似替代。",
          "en": "Recompute C(v1,v4) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "1"
        ],
        "output": "2",
        "label": "C(v1,v4)"
      }
    },
    {
      "id": "S038",
      "uses": [
        "G001",
        "S037"
      ],
      "before": "remaining=[v1, v4]; eligible first vertices=[v1, v4]",
      "after": "complete continuations=2",
      "relation": "implies",
      "rule_id": "all_source_branches",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任何合法次序的首點必為零入度；反之先取任一零入度點，再接合法尾端即可。故分支沒有漏列。",
        "en": "The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "核對完整頂點與邊；一個失敗次序不等於有環，全部合法分支不能漏列。",
          "en": "Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S039",
      "uses": [
        "G001",
        "S038"
      ],
      "before": "C(v1,v4,v5) := sum(2, 2, 2)",
      "after": "C(v1,v4,v5) = 6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個合法首點形成互斥分支，分支完成數相加。",
        "en": "Each legal first vertex defines a disjoint branch; add the branch completion counts."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 C(v1,v4,v5)，不以小數近似替代。",
          "en": "Recompute C(v1,v4,v5) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "2",
          "2",
          "2"
        ],
        "output": "6",
        "label": "C(v1,v4,v5)"
      }
    },
    {
      "id": "S040",
      "uses": [
        "G001",
        "S039"
      ],
      "before": "remaining=[v1, v4, v5]; eligible first vertices=[v1, v4, v5]",
      "after": "complete continuations=6",
      "relation": "implies",
      "rule_id": "all_source_branches",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任何合法次序的首點必為零入度；反之先取任一零入度點，再接合法尾端即可。故分支沒有漏列。",
        "en": "The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "核對完整頂點與邊；一個失敗次序不等於有環，全部合法分支不能漏列。",
          "en": "Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S041",
      "uses": [
        "G001",
        "S040"
      ],
      "before": "C(v1,v3) := sum(1, 1)",
      "after": "C(v1,v3) = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個合法首點形成互斥分支，分支完成數相加。",
        "en": "Each legal first vertex defines a disjoint branch; add the branch completion counts."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 C(v1,v3)，不以小數近似替代。",
          "en": "Recompute C(v1,v3) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "1"
        ],
        "output": "2",
        "label": "C(v1,v3)"
      }
    },
    {
      "id": "S042",
      "uses": [
        "G001",
        "S041"
      ],
      "before": "remaining=[v1, v3]; eligible first vertices=[v1, v3]",
      "after": "complete continuations=2",
      "relation": "implies",
      "rule_id": "all_source_branches",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任何合法次序的首點必為零入度；反之先取任一零入度點，再接合法尾端即可。故分支沒有漏列。",
        "en": "The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "核對完整頂點與邊；一個失敗次序不等於有環，全部合法分支不能漏列。",
          "en": "Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S043",
      "uses": [
        "G001",
        "S042"
      ],
      "before": "C(v1,v3,v5) := sum(2, 2, 2)",
      "after": "C(v1,v3,v5) = 6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個合法首點形成互斥分支，分支完成數相加。",
        "en": "Each legal first vertex defines a disjoint branch; add the branch completion counts."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 C(v1,v3,v5)，不以小數近似替代。",
          "en": "Recompute C(v1,v3,v5) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "2",
          "2",
          "2"
        ],
        "output": "6",
        "label": "C(v1,v3,v5)"
      }
    },
    {
      "id": "S044",
      "uses": [
        "G001",
        "S043"
      ],
      "before": "remaining=[v1, v3, v5]; eligible first vertices=[v1, v3, v5]",
      "after": "complete continuations=6",
      "relation": "implies",
      "rule_id": "all_source_branches",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任何合法次序的首點必為零入度；反之先取任一零入度點，再接合法尾端即可。故分支沒有漏列。",
        "en": "The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "核對完整頂點與邊；一個失敗次序不等於有環，全部合法分支不能漏列。",
          "en": "Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S045",
      "uses": [
        "G001",
        "S044"
      ],
      "before": "C(v1,v3,v4) := sum(2, 2, 2)",
      "after": "C(v1,v3,v4) = 6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個合法首點形成互斥分支，分支完成數相加。",
        "en": "Each legal first vertex defines a disjoint branch; add the branch completion counts."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 C(v1,v3,v4)，不以小數近似替代。",
          "en": "Recompute C(v1,v3,v4) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "2",
          "2",
          "2"
        ],
        "output": "6",
        "label": "C(v1,v3,v4)"
      }
    },
    {
      "id": "S046",
      "uses": [
        "G001",
        "S045"
      ],
      "before": "remaining=[v1, v3, v4]; eligible first vertices=[v1, v3, v4]",
      "after": "complete continuations=6",
      "relation": "implies",
      "rule_id": "all_source_branches",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任何合法次序的首點必為零入度；反之先取任一零入度點，再接合法尾端即可。故分支沒有漏列。",
        "en": "The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "核對完整頂點與邊；一個失敗次序不等於有環，全部合法分支不能漏列。",
          "en": "Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S047",
      "uses": [
        "G001",
        "S046"
      ],
      "before": "C(v1,v3,v4,v5) := sum(6, 6, 6, 6)",
      "after": "C(v1,v3,v4,v5) = 24",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個合法首點形成互斥分支，分支完成數相加。",
        "en": "Each legal first vertex defines a disjoint branch; add the branch completion counts."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 C(v1,v3,v4,v5)，不以小數近似替代。",
          "en": "Recompute C(v1,v3,v4,v5) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "6",
          "6",
          "6",
          "6"
        ],
        "output": "24",
        "label": "C(v1,v3,v4,v5)"
      }
    },
    {
      "id": "S048",
      "uses": [
        "G001",
        "S047"
      ],
      "before": "remaining=[v1, v3, v4, v5]; eligible first vertices=[v1, v3, v4, v5]",
      "after": "complete continuations=24",
      "relation": "implies",
      "rule_id": "all_source_branches",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任何合法次序的首點必為零入度；反之先取任一零入度點，再接合法尾端即可。故分支沒有漏列。",
        "en": "The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "核對完整頂點與邊；一個失敗次序不等於有環，全部合法分支不能漏列。",
          "en": "Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S049",
      "uses": [
        "G001",
        "S048"
      ],
      "before": "C(v1,v2) := sum(1, 1)",
      "after": "C(v1,v2) = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個合法首點形成互斥分支，分支完成數相加。",
        "en": "Each legal first vertex defines a disjoint branch; add the branch completion counts."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 C(v1,v2)，不以小數近似替代。",
          "en": "Recompute C(v1,v2) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "1"
        ],
        "output": "2",
        "label": "C(v1,v2)"
      }
    },
    {
      "id": "S050",
      "uses": [
        "G001",
        "S049"
      ],
      "before": "remaining=[v1, v2]; eligible first vertices=[v1, v2]",
      "after": "complete continuations=2",
      "relation": "implies",
      "rule_id": "all_source_branches",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任何合法次序的首點必為零入度；反之先取任一零入度點，再接合法尾端即可。故分支沒有漏列。",
        "en": "The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "核對完整頂點與邊；一個失敗次序不等於有環，全部合法分支不能漏列。",
          "en": "Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S051",
      "uses": [
        "G001",
        "S050"
      ],
      "before": "C(v1,v2,v5) := sum(2, 2, 2)",
      "after": "C(v1,v2,v5) = 6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個合法首點形成互斥分支，分支完成數相加。",
        "en": "Each legal first vertex defines a disjoint branch; add the branch completion counts."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 C(v1,v2,v5)，不以小數近似替代。",
          "en": "Recompute C(v1,v2,v5) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "2",
          "2",
          "2"
        ],
        "output": "6",
        "label": "C(v1,v2,v5)"
      }
    },
    {
      "id": "S052",
      "uses": [
        "G001",
        "S051"
      ],
      "before": "remaining=[v1, v2, v5]; eligible first vertices=[v1, v2, v5]",
      "after": "complete continuations=6",
      "relation": "implies",
      "rule_id": "all_source_branches",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任何合法次序的首點必為零入度；反之先取任一零入度點，再接合法尾端即可。故分支沒有漏列。",
        "en": "The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "核對完整頂點與邊；一個失敗次序不等於有環，全部合法分支不能漏列。",
          "en": "Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S053",
      "uses": [
        "G001",
        "S052"
      ],
      "before": "C(v1,v2,v4) := sum(2, 2, 2)",
      "after": "C(v1,v2,v4) = 6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個合法首點形成互斥分支，分支完成數相加。",
        "en": "Each legal first vertex defines a disjoint branch; add the branch completion counts."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 C(v1,v2,v4)，不以小數近似替代。",
          "en": "Recompute C(v1,v2,v4) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "2",
          "2",
          "2"
        ],
        "output": "6",
        "label": "C(v1,v2,v4)"
      }
    },
    {
      "id": "S054",
      "uses": [
        "G001",
        "S053"
      ],
      "before": "remaining=[v1, v2, v4]; eligible first vertices=[v1, v2, v4]",
      "after": "complete continuations=6",
      "relation": "implies",
      "rule_id": "all_source_branches",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任何合法次序的首點必為零入度；反之先取任一零入度點，再接合法尾端即可。故分支沒有漏列。",
        "en": "The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "核對完整頂點與邊；一個失敗次序不等於有環，全部合法分支不能漏列。",
          "en": "Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S055",
      "uses": [
        "G001",
        "S054"
      ],
      "before": "C(v1,v2,v4,v5) := sum(6, 6, 6, 6)",
      "after": "C(v1,v2,v4,v5) = 24",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個合法首點形成互斥分支，分支完成數相加。",
        "en": "Each legal first vertex defines a disjoint branch; add the branch completion counts."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 C(v1,v2,v4,v5)，不以小數近似替代。",
          "en": "Recompute C(v1,v2,v4,v5) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "6",
          "6",
          "6",
          "6"
        ],
        "output": "24",
        "label": "C(v1,v2,v4,v5)"
      }
    },
    {
      "id": "S056",
      "uses": [
        "G001",
        "S055"
      ],
      "before": "remaining=[v1, v2, v4, v5]; eligible first vertices=[v1, v2, v4, v5]",
      "after": "complete continuations=24",
      "relation": "implies",
      "rule_id": "all_source_branches",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任何合法次序的首點必為零入度；反之先取任一零入度點，再接合法尾端即可。故分支沒有漏列。",
        "en": "The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "核對完整頂點與邊；一個失敗次序不等於有環，全部合法分支不能漏列。",
          "en": "Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S057",
      "uses": [
        "G001",
        "S056"
      ],
      "before": "C(v1,v2,v3) := sum(2, 2, 2)",
      "after": "C(v1,v2,v3) = 6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個合法首點形成互斥分支，分支完成數相加。",
        "en": "Each legal first vertex defines a disjoint branch; add the branch completion counts."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 C(v1,v2,v3)，不以小數近似替代。",
          "en": "Recompute C(v1,v2,v3) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "2",
          "2",
          "2"
        ],
        "output": "6",
        "label": "C(v1,v2,v3)"
      }
    },
    {
      "id": "S058",
      "uses": [
        "G001",
        "S057"
      ],
      "before": "remaining=[v1, v2, v3]; eligible first vertices=[v1, v2, v3]",
      "after": "complete continuations=6",
      "relation": "implies",
      "rule_id": "all_source_branches",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任何合法次序的首點必為零入度；反之先取任一零入度點，再接合法尾端即可。故分支沒有漏列。",
        "en": "The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "核對完整頂點與邊；一個失敗次序不等於有環，全部合法分支不能漏列。",
          "en": "Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S059",
      "uses": [
        "G001",
        "S058"
      ],
      "before": "C(v1,v2,v3,v5) := sum(6, 6, 6, 6)",
      "after": "C(v1,v2,v3,v5) = 24",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個合法首點形成互斥分支，分支完成數相加。",
        "en": "Each legal first vertex defines a disjoint branch; add the branch completion counts."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 C(v1,v2,v3,v5)，不以小數近似替代。",
          "en": "Recompute C(v1,v2,v3,v5) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "6",
          "6",
          "6",
          "6"
        ],
        "output": "24",
        "label": "C(v1,v2,v3,v5)"
      }
    },
    {
      "id": "S060",
      "uses": [
        "G001",
        "S059"
      ],
      "before": "remaining=[v1, v2, v3, v5]; eligible first vertices=[v1, v2, v3, v5]",
      "after": "complete continuations=24",
      "relation": "implies",
      "rule_id": "all_source_branches",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任何合法次序的首點必為零入度；反之先取任一零入度點，再接合法尾端即可。故分支沒有漏列。",
        "en": "The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "核對完整頂點與邊；一個失敗次序不等於有環，全部合法分支不能漏列。",
          "en": "Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S061",
      "uses": [
        "G001",
        "S060"
      ],
      "before": "C(v1,v2,v3,v4) := sum(6, 6, 6, 6)",
      "after": "C(v1,v2,v3,v4) = 24",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個合法首點形成互斥分支，分支完成數相加。",
        "en": "Each legal first vertex defines a disjoint branch; add the branch completion counts."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 C(v1,v2,v3,v4)，不以小數近似替代。",
          "en": "Recompute C(v1,v2,v3,v4) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "6",
          "6",
          "6",
          "6"
        ],
        "output": "24",
        "label": "C(v1,v2,v3,v4)"
      }
    },
    {
      "id": "S062",
      "uses": [
        "G001",
        "S061"
      ],
      "before": "remaining=[v1, v2, v3, v4]; eligible first vertices=[v1, v2, v3, v4]",
      "after": "complete continuations=24",
      "relation": "implies",
      "rule_id": "all_source_branches",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任何合法次序的首點必為零入度；反之先取任一零入度點，再接合法尾端即可。故分支沒有漏列。",
        "en": "The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "核對完整頂點與邊；一個失敗次序不等於有環，全部合法分支不能漏列。",
          "en": "Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S063",
      "uses": [
        "G001",
        "S062"
      ],
      "before": "C(v1,v2,v3,v4,v5) := sum(24, 24, 24, 24, 24)",
      "after": "C(v1,v2,v3,v4,v5) = 120",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個合法首點形成互斥分支，分支完成數相加。",
        "en": "Each legal first vertex defines a disjoint branch; add the branch completion counts."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 C(v1,v2,v3,v4,v5)，不以小數近似替代。",
          "en": "Recompute C(v1,v2,v3,v4,v5) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "24",
          "24",
          "24",
          "24",
          "24"
        ],
        "output": "120",
        "label": "C(v1,v2,v3,v4,v5)"
      }
    },
    {
      "id": "S064",
      "uses": [
        "G001",
        "S063"
      ],
      "before": "remaining=[v1, v2, v3, v4, v5]; eligible first vertices=[v1, v2, v3, v4, v5]",
      "after": "complete continuations=120",
      "relation": "implies",
      "rule_id": "all_source_branches",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任何合法次序的首點必為零入度；反之先取任一零入度點，再接合法尾端即可。故分支沒有漏列。",
        "en": "The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "核對完整頂點與邊；一個失敗次序不等於有環，全部合法分支不能漏列。",
          "en": "Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S065",
      "uses": [
        "G001",
        "S064"
      ],
      "before": "C(v0,v4) := sum(1)",
      "after": "C(v0,v4) = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個合法首點形成互斥分支，分支完成數相加。",
        "en": "Each legal first vertex defines a disjoint branch; add the branch completion counts."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 C(v0,v4)，不以小數近似替代。",
          "en": "Recompute C(v0,v4) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1"
        ],
        "output": "1",
        "label": "C(v0,v4)"
      }
    },
    {
      "id": "S066",
      "uses": [
        "G001",
        "S065"
      ],
      "before": "remaining=[v0, v4]; eligible first vertices=[v0]",
      "after": "complete continuations=1",
      "relation": "implies",
      "rule_id": "all_source_branches",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任何合法次序的首點必為零入度；反之先取任一零入度點，再接合法尾端即可。故分支沒有漏列。",
        "en": "The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "核對完整頂點與邊；一個失敗次序不等於有環，全部合法分支不能漏列。",
          "en": "Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S067",
      "uses": [
        "G001",
        "S066"
      ],
      "before": "C(v0,v4,v5) := sum(2, 1)",
      "after": "C(v0,v4,v5) = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個合法首點形成互斥分支，分支完成數相加。",
        "en": "Each legal first vertex defines a disjoint branch; add the branch completion counts."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 C(v0,v4,v5)，不以小數近似替代。",
          "en": "Recompute C(v0,v4,v5) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "2",
          "1"
        ],
        "output": "3",
        "label": "C(v0,v4,v5)"
      }
    },
    {
      "id": "S068",
      "uses": [
        "G001",
        "S067"
      ],
      "before": "remaining=[v0, v4, v5]; eligible first vertices=[v0, v5]",
      "after": "complete continuations=3",
      "relation": "implies",
      "rule_id": "all_source_branches",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任何合法次序的首點必為零入度；反之先取任一零入度點，再接合法尾端即可。故分支沒有漏列。",
        "en": "The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "核對完整頂點與邊；一個失敗次序不等於有環，全部合法分支不能漏列。",
          "en": "Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S069",
      "uses": [
        "G001",
        "S068"
      ],
      "before": "C(v0,v3,v4) := sum(2, 1)",
      "after": "C(v0,v3,v4) = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個合法首點形成互斥分支，分支完成數相加。",
        "en": "Each legal first vertex defines a disjoint branch; add the branch completion counts."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 C(v0,v3,v4)，不以小數近似替代。",
          "en": "Recompute C(v0,v3,v4) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "2",
          "1"
        ],
        "output": "3",
        "label": "C(v0,v3,v4)"
      }
    },
    {
      "id": "S070",
      "uses": [
        "G001",
        "S069"
      ],
      "before": "remaining=[v0, v3, v4]; eligible first vertices=[v0, v3]",
      "after": "complete continuations=3",
      "relation": "implies",
      "rule_id": "all_source_branches",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任何合法次序的首點必為零入度；反之先取任一零入度點，再接合法尾端即可。故分支沒有漏列。",
        "en": "The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "核對完整頂點與邊；一個失敗次序不等於有環，全部合法分支不能漏列。",
          "en": "Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S071",
      "uses": [
        "G001",
        "S070"
      ],
      "before": "C(v0,v3,v4,v5) := sum(6, 3, 3)",
      "after": "C(v0,v3,v4,v5) = 12",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個合法首點形成互斥分支，分支完成數相加。",
        "en": "Each legal first vertex defines a disjoint branch; add the branch completion counts."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 C(v0,v3,v4,v5)，不以小數近似替代。",
          "en": "Recompute C(v0,v3,v4,v5) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "6",
          "3",
          "3"
        ],
        "output": "12",
        "label": "C(v0,v3,v4,v5)"
      }
    },
    {
      "id": "S072",
      "uses": [
        "G001",
        "S071"
      ],
      "before": "remaining=[v0, v3, v4, v5]; eligible first vertices=[v0, v3, v5]",
      "after": "complete continuations=12",
      "relation": "implies",
      "rule_id": "all_source_branches",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任何合法次序的首點必為零入度；反之先取任一零入度點，再接合法尾端即可。故分支沒有漏列。",
        "en": "The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "核對完整頂點與邊；一個失敗次序不等於有環，全部合法分支不能漏列。",
          "en": "Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S073",
      "uses": [
        "G001",
        "S072"
      ],
      "before": "C(v0,v2,v4) := sum(2, 1)",
      "after": "C(v0,v2,v4) = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個合法首點形成互斥分支，分支完成數相加。",
        "en": "Each legal first vertex defines a disjoint branch; add the branch completion counts."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 C(v0,v2,v4)，不以小數近似替代。",
          "en": "Recompute C(v0,v2,v4) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "2",
          "1"
        ],
        "output": "3",
        "label": "C(v0,v2,v4)"
      }
    },
    {
      "id": "S074",
      "uses": [
        "G001",
        "S073"
      ],
      "before": "remaining=[v0, v2, v4]; eligible first vertices=[v0, v2]",
      "after": "complete continuations=3",
      "relation": "implies",
      "rule_id": "all_source_branches",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任何合法次序的首點必為零入度；反之先取任一零入度點，再接合法尾端即可。故分支沒有漏列。",
        "en": "The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "核對完整頂點與邊；一個失敗次序不等於有環，全部合法分支不能漏列。",
          "en": "Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S075",
      "uses": [
        "G001",
        "S074"
      ],
      "before": "C(v0,v2,v4,v5) := sum(6, 3, 3)",
      "after": "C(v0,v2,v4,v5) = 12",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個合法首點形成互斥分支，分支完成數相加。",
        "en": "Each legal first vertex defines a disjoint branch; add the branch completion counts."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 C(v0,v2,v4,v5)，不以小數近似替代。",
          "en": "Recompute C(v0,v2,v4,v5) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "6",
          "3",
          "3"
        ],
        "output": "12",
        "label": "C(v0,v2,v4,v5)"
      }
    },
    {
      "id": "S076",
      "uses": [
        "G001",
        "S075"
      ],
      "before": "remaining=[v0, v2, v4, v5]; eligible first vertices=[v0, v2, v5]",
      "after": "complete continuations=12",
      "relation": "implies",
      "rule_id": "all_source_branches",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任何合法次序的首點必為零入度；反之先取任一零入度點，再接合法尾端即可。故分支沒有漏列。",
        "en": "The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "核對完整頂點與邊；一個失敗次序不等於有環，全部合法分支不能漏列。",
          "en": "Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S077",
      "uses": [
        "G001",
        "S076"
      ],
      "before": "C(v0,v2,v3,v4) := sum(6, 3, 3)",
      "after": "C(v0,v2,v3,v4) = 12",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個合法首點形成互斥分支，分支完成數相加。",
        "en": "Each legal first vertex defines a disjoint branch; add the branch completion counts."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 C(v0,v2,v3,v4)，不以小數近似替代。",
          "en": "Recompute C(v0,v2,v3,v4) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "6",
          "3",
          "3"
        ],
        "output": "12",
        "label": "C(v0,v2,v3,v4)"
      }
    },
    {
      "id": "S078",
      "uses": [
        "G001",
        "S077"
      ],
      "before": "remaining=[v0, v2, v3, v4]; eligible first vertices=[v0, v2, v3]",
      "after": "complete continuations=12",
      "relation": "implies",
      "rule_id": "all_source_branches",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任何合法次序的首點必為零入度；反之先取任一零入度點，再接合法尾端即可。故分支沒有漏列。",
        "en": "The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "核對完整頂點與邊；一個失敗次序不等於有環，全部合法分支不能漏列。",
          "en": "Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S079",
      "uses": [
        "G001",
        "S078"
      ],
      "before": "C(v0,v2,v3,v4,v5) := sum(24, 12, 12, 12)",
      "after": "C(v0,v2,v3,v4,v5) = 60",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個合法首點形成互斥分支，分支完成數相加。",
        "en": "Each legal first vertex defines a disjoint branch; add the branch completion counts."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 C(v0,v2,v3,v4,v5)，不以小數近似替代。",
          "en": "Recompute C(v0,v2,v3,v4,v5) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "24",
          "12",
          "12",
          "12"
        ],
        "output": "60",
        "label": "C(v0,v2,v3,v4,v5)"
      }
    },
    {
      "id": "S080",
      "uses": [
        "G001",
        "S079"
      ],
      "before": "remaining=[v0, v2, v3, v4, v5]; eligible first vertices=[v0, v2, v3, v5]",
      "after": "complete continuations=60",
      "relation": "implies",
      "rule_id": "all_source_branches",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任何合法次序的首點必為零入度；反之先取任一零入度點，再接合法尾端即可。故分支沒有漏列。",
        "en": "The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "核對完整頂點與邊；一個失敗次序不等於有環，全部合法分支不能漏列。",
          "en": "Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S081",
      "uses": [
        "G001",
        "S080"
      ],
      "before": "C(v0,v1,v4) := sum(2, 1)",
      "after": "C(v0,v1,v4) = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個合法首點形成互斥分支，分支完成數相加。",
        "en": "Each legal first vertex defines a disjoint branch; add the branch completion counts."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 C(v0,v1,v4)，不以小數近似替代。",
          "en": "Recompute C(v0,v1,v4) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "2",
          "1"
        ],
        "output": "3",
        "label": "C(v0,v1,v4)"
      }
    },
    {
      "id": "S082",
      "uses": [
        "G001",
        "S081"
      ],
      "before": "remaining=[v0, v1, v4]; eligible first vertices=[v0, v1]",
      "after": "complete continuations=3",
      "relation": "implies",
      "rule_id": "all_source_branches",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任何合法次序的首點必為零入度；反之先取任一零入度點，再接合法尾端即可。故分支沒有漏列。",
        "en": "The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "核對完整頂點與邊；一個失敗次序不等於有環，全部合法分支不能漏列。",
          "en": "Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S083",
      "uses": [
        "G001",
        "S082"
      ],
      "before": "C(v0,v1,v4,v5) := sum(6, 3, 3)",
      "after": "C(v0,v1,v4,v5) = 12",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個合法首點形成互斥分支，分支完成數相加。",
        "en": "Each legal first vertex defines a disjoint branch; add the branch completion counts."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 C(v0,v1,v4,v5)，不以小數近似替代。",
          "en": "Recompute C(v0,v1,v4,v5) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "6",
          "3",
          "3"
        ],
        "output": "12",
        "label": "C(v0,v1,v4,v5)"
      }
    },
    {
      "id": "S084",
      "uses": [
        "G001",
        "S083"
      ],
      "before": "remaining=[v0, v1, v4, v5]; eligible first vertices=[v0, v1, v5]",
      "after": "complete continuations=12",
      "relation": "implies",
      "rule_id": "all_source_branches",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任何合法次序的首點必為零入度；反之先取任一零入度點，再接合法尾端即可。故分支沒有漏列。",
        "en": "The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "核對完整頂點與邊；一個失敗次序不等於有環，全部合法分支不能漏列。",
          "en": "Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S085",
      "uses": [
        "G001",
        "S084"
      ],
      "before": "C(v0,v1,v3,v4) := sum(6, 3, 3)",
      "after": "C(v0,v1,v3,v4) = 12",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個合法首點形成互斥分支，分支完成數相加。",
        "en": "Each legal first vertex defines a disjoint branch; add the branch completion counts."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 C(v0,v1,v3,v4)，不以小數近似替代。",
          "en": "Recompute C(v0,v1,v3,v4) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "6",
          "3",
          "3"
        ],
        "output": "12",
        "label": "C(v0,v1,v3,v4)"
      }
    },
    {
      "id": "S086",
      "uses": [
        "G001",
        "S085"
      ],
      "before": "remaining=[v0, v1, v3, v4]; eligible first vertices=[v0, v1, v3]",
      "after": "complete continuations=12",
      "relation": "implies",
      "rule_id": "all_source_branches",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任何合法次序的首點必為零入度；反之先取任一零入度點，再接合法尾端即可。故分支沒有漏列。",
        "en": "The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "核對完整頂點與邊；一個失敗次序不等於有環，全部合法分支不能漏列。",
          "en": "Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S087",
      "uses": [
        "G001",
        "S086"
      ],
      "before": "C(v0,v1,v3,v4,v5) := sum(24, 12, 12, 12)",
      "after": "C(v0,v1,v3,v4,v5) = 60",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個合法首點形成互斥分支，分支完成數相加。",
        "en": "Each legal first vertex defines a disjoint branch; add the branch completion counts."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 C(v0,v1,v3,v4,v5)，不以小數近似替代。",
          "en": "Recompute C(v0,v1,v3,v4,v5) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "24",
          "12",
          "12",
          "12"
        ],
        "output": "60",
        "label": "C(v0,v1,v3,v4,v5)"
      }
    },
    {
      "id": "S088",
      "uses": [
        "G001",
        "S087"
      ],
      "before": "remaining=[v0, v1, v3, v4, v5]; eligible first vertices=[v0, v1, v3, v5]",
      "after": "complete continuations=60",
      "relation": "implies",
      "rule_id": "all_source_branches",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任何合法次序的首點必為零入度；反之先取任一零入度點，再接合法尾端即可。故分支沒有漏列。",
        "en": "The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "核對完整頂點與邊；一個失敗次序不等於有環，全部合法分支不能漏列。",
          "en": "Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S089",
      "uses": [
        "G001",
        "S088"
      ],
      "before": "C(v0,v1,v2,v4) := sum(6, 3, 3)",
      "after": "C(v0,v1,v2,v4) = 12",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個合法首點形成互斥分支，分支完成數相加。",
        "en": "Each legal first vertex defines a disjoint branch; add the branch completion counts."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 C(v0,v1,v2,v4)，不以小數近似替代。",
          "en": "Recompute C(v0,v1,v2,v4) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "6",
          "3",
          "3"
        ],
        "output": "12",
        "label": "C(v0,v1,v2,v4)"
      }
    },
    {
      "id": "S090",
      "uses": [
        "G001",
        "S089"
      ],
      "before": "remaining=[v0, v1, v2, v4]; eligible first vertices=[v0, v1, v2]",
      "after": "complete continuations=12",
      "relation": "implies",
      "rule_id": "all_source_branches",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任何合法次序的首點必為零入度；反之先取任一零入度點，再接合法尾端即可。故分支沒有漏列。",
        "en": "The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "核對完整頂點與邊；一個失敗次序不等於有環，全部合法分支不能漏列。",
          "en": "Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S091",
      "uses": [
        "G001",
        "S090"
      ],
      "before": "C(v0,v1,v2,v4,v5) := sum(24, 12, 12, 12)",
      "after": "C(v0,v1,v2,v4,v5) = 60",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個合法首點形成互斥分支，分支完成數相加。",
        "en": "Each legal first vertex defines a disjoint branch; add the branch completion counts."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 C(v0,v1,v2,v4,v5)，不以小數近似替代。",
          "en": "Recompute C(v0,v1,v2,v4,v5) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "24",
          "12",
          "12",
          "12"
        ],
        "output": "60",
        "label": "C(v0,v1,v2,v4,v5)"
      }
    },
    {
      "id": "S092",
      "uses": [
        "G001",
        "S091"
      ],
      "before": "remaining=[v0, v1, v2, v4, v5]; eligible first vertices=[v0, v1, v2, v5]",
      "after": "complete continuations=60",
      "relation": "implies",
      "rule_id": "all_source_branches",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任何合法次序的首點必為零入度；反之先取任一零入度點，再接合法尾端即可。故分支沒有漏列。",
        "en": "The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "核對完整頂點與邊；一個失敗次序不等於有環，全部合法分支不能漏列。",
          "en": "Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S093",
      "uses": [
        "G001",
        "S092"
      ],
      "before": "C(v0,v1,v2,v3,v4) := sum(24, 12, 12, 12)",
      "after": "C(v0,v1,v2,v3,v4) = 60",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個合法首點形成互斥分支，分支完成數相加。",
        "en": "Each legal first vertex defines a disjoint branch; add the branch completion counts."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 C(v0,v1,v2,v3,v4)，不以小數近似替代。",
          "en": "Recompute C(v0,v1,v2,v3,v4) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "24",
          "12",
          "12",
          "12"
        ],
        "output": "60",
        "label": "C(v0,v1,v2,v3,v4)"
      }
    },
    {
      "id": "S094",
      "uses": [
        "G001",
        "S093"
      ],
      "before": "remaining=[v0, v1, v2, v3, v4]; eligible first vertices=[v0, v1, v2, v3]",
      "after": "complete continuations=60",
      "relation": "implies",
      "rule_id": "all_source_branches",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任何合法次序的首點必為零入度；反之先取任一零入度點，再接合法尾端即可。故分支沒有漏列。",
        "en": "The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "核對完整頂點與邊；一個失敗次序不等於有環，全部合法分支不能漏列。",
          "en": "Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S095",
      "uses": [
        "G001",
        "S094"
      ],
      "before": "C(v0,v1,v2,v3,v4,v5) := sum(120, 60, 60, 60, 60)",
      "after": "C(v0,v1,v2,v3,v4,v5) = 360",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個合法首點形成互斥分支，分支完成數相加。",
        "en": "Each legal first vertex defines a disjoint branch; add the branch completion counts."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 C(v0,v1,v2,v3,v4,v5)，不以小數近似替代。",
          "en": "Recompute C(v0,v1,v2,v3,v4,v5) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "120",
          "60",
          "60",
          "60",
          "60"
        ],
        "output": "360",
        "label": "C(v0,v1,v2,v3,v4,v5)"
      }
    },
    {
      "id": "S096",
      "uses": [
        "G001",
        "S095"
      ],
      "before": "remaining=[v0, v1, v2, v3, v4, v5]; eligible first vertices=[v0, v1, v2, v3, v5]",
      "after": "complete continuations=360",
      "relation": "implies",
      "rule_id": "all_source_branches",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任何合法次序的首點必為零入度；反之先取任一零入度點，再接合法尾端即可。故分支沒有漏列。",
        "en": "The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "核對完整頂點與邊；一個失敗次序不等於有環，全部合法分支不能漏列。",
          "en": "Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S097",
      "uses": [
        "G001",
        "S096"
      ],
      "before": "all root branches; count=360",
      "after": "enumeration contains each legal permutation exactly once",
      "relation": "implies",
      "rule_id": "enumeration_completeness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "不同首點的分支不重疊，尾端遞迴亦不重複，所以得到全部且僅有合法次序。完整清單保留於答案。",
        "en": "Distinct first-vertex branches do not overlap and their recursive tails do not duplicate, so the answer retains every and only valid order."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "核對完整頂點與邊；一個失敗次序不等於有環，全部合法分支不能漏列。",
          "en": "Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S098",
      "uses": [
        "G001",
        "S097"
      ],
      "before": "valid order count=360",
      "after": "at least one exists: 是",
      "relation": "implies",
      "rule_id": "order_existence",
      "conditions": [],
      "explanation": {
        "zh-Hant": "存在合法次序才證這個有限有向圖無環；多個合法次序不代表矛盾。",
        "en": "An existing valid order certifies acyclicity of this finite digraph; multiple valid orders are compatible."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "核對完整頂點與邊；一個失敗次序不等於有環，全部合法分支不能漏列。",
          "en": "Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
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      ],
      360,
      true,
      "all_vertex_permutations_are_audited"
    ]
  },
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[v5, v2, v1, v3, v0, v4], [v5, v2, v3, v0, v1, v4], [v5, v2, v3, v0, v4, v1], [v5, v2, v3, v1, v0, v4], [v5, v3, v0, v1, v2, v4], [v5, v3, v0, v1, v4, v2], [v5, v3, v0, v2, v1, v4], [v5, v3, v0, v2, v4, v1], [v5, v3, v0, v4, v1, v2], [v5, v3, v0, v4, v2, v1], [v5, v3, v1, v0, v2, v4], [v5, v3, v1, v0, v4, v2], [v5, v3, v1, v2, v0, v4], [v5, v3, v2, v0, v1, v4], [v5, v3, v2, v0, v4, v1], [v5, v3, v2, v1, v0, v4]], 360, 是, all_vertex_permutations_are_audited)",
    "explanation": {
      "zh-Hant": "上述完整代值與條件核對得到原題答案；欄位次序及真假標記保持原始合約。",
      "en": "The full substitutions and premise audits recover the original answer, retaining its field order and verdict meanings."
    }
  },
  "definition_note": {
    "zh-Hant": "有限有向圖；合法次序的首點必為零入度點，依首點分支遞迴可不漏不重地列出全部次序。",
    "en": "For a finite digraph every valid order starts at a zero-indegree vertex; branching on that first vertex lists every order once."
  },
  "source": {
    "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db",
    "question_binding": {
      "batch_id": "batch-20261015-advance-20261116-r06-topological-ordering-dag-foundations",
      "batch_path": "public/data/questions/batches/batch-20261015-advance-20261116-r06-topological-ordering-dag-foundations/questions.jsonl",
      "batch_sha256": "d93ad638d602b169ce66fdacb2c40107b89c1577baa28a2b3fc1b46f2b105460",
      "original_line": 33,
      "original_line_sha256": "916c563f7273a545266ac09aa607fd2e776b6f59d63558289c1f61ef9d6b6d4b",
      "question_compact_sha256": "5a91f0799336ad31b18660eb881f7cd447f4b4de6ade778ec3ecceefb082c11b",
      "english_manifest_path": "public/data/i18n/editions/batch-20261015-advance-20261116-r06-topological-ordering-dag-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261015-advance-20261116-r06-topological-ordering-dag-foundations.en.jsonl",
      "english_row_compact_sha256": "ac882fdb8839c279f47fcfe0aa669503b9ec3384ebf557924eafaa183f343da1"
    },
    "mko_bindings": {
      "mko-topological-ordering-dag": {
        "original_path": "public/data/mko/source13-20261018-topological-ordering-dag.json",
        "english_path": "public/data/i18n/editions/mko-topological-ordering-dag.en.json",
        "sha256": "826499909e9a38a23996ab91062cdf86e73a350078b0b11c2a5fb8828d12e5f9",
        "english_sha256": "3063e85d7b49e8d03329c2cbf2fe9c9b69db6122f1ad297fcebacdb7df784a96"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "afa7c0e9f7084cc5929124dcffe9929d5db7def32e40047056d19ea25535873e",
    "original_candidate_sha256": "d646ce3f67e47e5176685a021a482fdc2f3142d70866bb1e0843d2840a1e84d4",
    "original_candidate_line_sha256": "d646ce3f67e47e5176685a021a482fdc2f3142d70866bb1e0843d2840a1e84d4"
  },
  "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": "來源綁定的教學推導候選；精確算術及有限列舉不是獨立AI審查或一般形式化證明。原題、答案、MKO、公式、AST及歷史保持不變。",
      "en": "Source-bound pedagogical derivation candidate: exact arithmetic and finite enumeration are not independent AI 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-20261026-003-03",
    "review_date": "2026-10-26",
    "owner_authorized_reader_publication": true,
    "scope": "Thirty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct sentence shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full answer tuples, model results 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": "6c2c6d46aded7582a11ac41a8c0b3bdde2669a53bdb6e796eea2039994a02772",
      "question_first_witness_sha256": "77e43898df9efdbf8094ad81fe4ab91cf4b4bd82f481eb9b4699b73cb4da612b"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 98,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "b48810031213ba9423f4bb120e87dcad9550c2fe9b8e44d14c54e7cc098e5027",
      "arithmetic_steps": 47
    },
    "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."
  }
}