{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0008",
  "pilot_number": "R0050-25908",
  "number": 25908,
  "family_id": "qf-mod46-mfmc-complete-cut-enumeration",
  "title": {
    "zh-Hant": "第 25908 題：最大流最小割定理",
    "en": "Question 25908: Every Source-Side Vertex Subset Defines One Directed Cut Audit"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0008",
    "batch_id": "batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-max-flow-min-cut-theorem"
    ],
    "title_zh": "最大流最小割定理",
    "stem_zh": "完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-complete-cut-enumeration；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"a3\",\"a4\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"],[\"s\",\"a3\"],[\"a3\",\"t\"],[\"s\",\"a4\"],[\"a4\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[9,9,10,10,11,11,12,12,13,13]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([a0, a1, a2, a3, a4, s, t], s, t, [[[s], [[[s, a0], 9], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, s], [[[a0, t], 9], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a1, s], [[[a1, t], 10], [[s, a0], 9], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, a1, s], [[[a0, t], 9], [[a1, t], 10], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a2, s], [[[a2, t], 11], [[s, a0], 9], [[s, a1], 10], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, a2, s], [[[a0, t], 9], [[a2, t], 11], [[s, a1], 10], [[s, a3], 12], [[s, a4], 13]], 55], [[a1, a2, s], [[[a1, t], 10], [[a2, t], 11], [[s, a0], 9], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, a1, a2, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a3, s], [[[a3, t], 12], [[s, a0], 9], [[s, a1], 10], [[s, a2], 11], [[s, a4], 13]], 55], [[a0, a3, s], [[[a0, t], 9], [[a3, t], 12], [[s, a1], 10], [[s, a2], 11], [[s, a4], 13]], 55], [[a1, a3, s], [[[a1, t], 10], [[a3, t], 12], [[s, a0], 9], [[s, a2], 11], [[s, a4], 13]], 55], [[a0, a1, a3, s], [[[a0, t], 9], [[a1, t], 10], [[a3, t], 12], [[s, a2], 11], [[s, a4], 13]], 55], [[a2, a3, s], [[[a2, t], 11], [[a3, t], 12], [[s, a0], 9], [[s, a1], 10], [[s, a4], 13]], 55], [[a0, a2, a3, s], [[[a0, t], 9], [[a2, t], 11], [[a3, t], 12], [[s, a1], 10], [[s, a4], 13]], 55], [[a1, a2, a3, s], [[[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[s, a0], 9], [[s, a4], 13]], 55], [[a0, a1, a2, a3, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[s, a4], 13]], 55], [[a4, s], [[[a4, t], 13], [[s, a0], 9], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12]], 55], [[a0, a4, s], [[[a0, t], 9], [[a4, t], 13], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12]], 55], [[a1, a4, s], [[[a1, t], 10], [[a4, t], 13], [[s, a0], 9], [[s, a2], 11], [[s, a3], 12]], 55], [[a0, a1, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a4, t], 13], [[s, a2], 11], [[s, a3], 12]], 55], [[a2, a4, s], [[[a2, t], 11], [[a4, t], 13], [[s, a0], 9], [[s, a1], 10], [[s, a3], 12]], 55], [[a0, a2, a4, s], [[[a0, t], 9], [[a2, t], 11], [[a4, t], 13], [[s, a1], 10], [[s, a3], 12]], 55], [[a1, a2, a4, s], [[[a1, t], 10], [[a2, t], 11], [[a4, t], 13], [[s, a0], 9], [[s, a3], 12]], 55], [[a0, a1, a2, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[a4, t], 13], [[s, a3], 12]], 55], [[a3, a4, s], [[[a3, t], 12], [[a4, t], 13], [[s, a0], 9], [[s, a1], 10], [[s, a2], 11]], 55], [[a0, a3, a4, s], [[[a0, t], 9], [[a3, t], 12], [[a4, t], 13], [[s, a1], 10], [[s, a2], 11]], 55], [[a1, a3, a4, s], [[[a1, t], 10], [[a3, t], 12], [[a4, t], 13], [[s, a0], 9], [[s, a2], 11]], 55], [[a0, a1, a3, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a3, t], 12], [[a4, t], 13], [[s, a2], 11]], 55], [[a2, a3, a4, s], [[[a2, t], 11], [[a3, t], 12], [[a4, t], 13], [[s, a0], 9], [[s, a1], 10]], 55], [[a0, a2, a3, a4, s], [[[a0, t], 9], [[a2, t], 11], [[a3, t], 12], [[a4, t], 13], [[s, a1], 10]], 55], [[a1, a2, a3, a4, s], [[[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[a4, t], 13], [[s, a0], 9]], 55], [[a0, a1, a2, a3, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[a4, t], 13]], 55]], 32, 32, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)",
      "explanation_zh": "Wave6第九層核對=([a0, a1, a2, a3, a4, s, t], s, t, [[[s], [[[s, a0], 9], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, s], [[[a0, t], 9], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a1, s], [[[a1, t], 10], [[s, a0], 9], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, a1, s], [[[a0, t], 9], [[a1, t], 10], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a2, s], [[[a2, t], 11], [[s, a0], 9], [[s, a1], 10], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, a2, s], [[[a0, t], 9], [[a2, t], 11], [[s, a1], 10], [[s, a3], 12], [[s, a4], 13]], 55], [[a1, a2, s], [[[a1, t], 10], [[a2, t], 11], [[s, a0], 9], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, a1, a2, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a3, s], [[[a3, t], 12], [[s, a0], 9], [[s, a1], 10], [[s, a2], 11], [[s, a4], 13]], 55], [[a0, a3, s], [[[a0, t], 9], [[a3, t], 12], [[s, a1], 10], [[s, a2], 11], [[s, a4], 13]], 55], [[a1, a3, s], [[[a1, t], 10], [[a3, t], 12], [[s, a0], 9], [[s, a2], 11], [[s, a4], 13]], 55], [[a0, a1, a3, s], [[[a0, t], 9], [[a1, t], 10], [[a3, t], 12], [[s, a2], 11], [[s, a4], 13]], 55], [[a2, a3, s], [[[a2, t], 11], [[a3, t], 12], [[s, a0], 9], [[s, a1], 10], [[s, a4], 13]], 55], [[a0, a2, a3, s], [[[a0, t], 9], [[a2, t], 11], [[a3, t], 12], [[s, a1], 10], [[s, a4], 13]], 55], [[a1, a2, a3, s], [[[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[s, a0], 9], [[s, a4], 13]], 55], [[a0, a1, a2, a3, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[s, a4], 13]], 55], [[a4, s], [[[a4, t], 13], [[s, a0], 9], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12]], 55], [[a0, a4, s], [[[a0, t], 9], [[a4, t], 13], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12]], 55], [[a1, a4, s], [[[a1, t], 10], [[a4, t], 13], [[s, a0], 9], [[s, a2], 11], [[s, a3], 12]], 55], [[a0, a1, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a4, t], 13], [[s, a2], 11], [[s, a3], 12]], 55], [[a2, a4, s], [[[a2, t], 11], [[a4, t], 13], [[s, a0], 9], [[s, a1], 10], [[s, a3], 12]], 55], [[a0, a2, a4, s], [[[a0, t], 9], [[a2, t], 11], [[a4, t], 13], [[s, a1], 10], [[s, a3], 12]], 55], [[a1, a2, a4, s], [[[a1, t], 10], [[a2, t], 11], [[a4, t], 13], [[s, a0], 9], [[s, a3], 12]], 55], [[a0, a1, a2, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[a4, t], 13], [[s, a3], 12]], 55], [[a3, a4, s], [[[a3, t], 12], [[a4, t], 13], [[s, a0], 9], [[s, a1], 10], [[s, a2], 11]], 55], [[a0, a3, a4, s], [[[a0, t], 9], [[a3, t], 12], [[a4, t], 13], [[s, a1], 10], [[s, a2], 11]], 55], [[a1, a3, a4, s], [[[a1, t], 10], [[a3, t], 12], [[a4, t], 13], [[s, a0], 9], [[s, a2], 11]], 55], [[a0, a1, a3, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a3, t], 12], [[a4, t], 13], [[s, a2], 11]], 55], [[a2, a3, a4, s], [[[a2, t], 11], [[a3, t], 12], [[a4, t], 13], [[s, a0], 9], [[s, a1], 10]], 55], [[a0, a2, a3, a4, s], [[[a0, t], 9], [[a2, t], 11], [[a3, t], 12], [[a4, t], 13], [[s, a1], 10]], 55], [[a1, a2, a3, a4, s], [[[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[a4, t], 13], [[s, a0], 9]], 55], [[a0, a1, a2, a3, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[a4, t], 13]], 55]], 32, 32, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "最大流最小割定理"
      ]
    },
    "variant": {
      "family_id": "qf-mod46-mfmc-complete-cut-enumeration",
      "seed": 202611155008,
      "parameters": {
        "vertices": [
          "s",
          "a0",
          "a1",
          "a2",
          "a3",
          "a4",
          "t"
        ],
        "edges": [
          [
            "s",
            "a0"
          ],
          [
            "a0",
            "t"
          ],
          [
            "s",
            "a1"
          ],
          [
            "a1",
            "t"
          ],
          [
            "s",
            "a2"
          ],
          [
            "a2",
            "t"
          ],
          [
            "s",
            "a3"
          ],
          [
            "a3",
            "t"
          ],
          [
            "s",
            "a4"
          ],
          [
            "a4",
            "t"
          ]
        ],
        "source": "s",
        "sink": "t",
        "capacities": [
          9,
          9,
          10,
          10,
          11,
          11,
          12,
          12,
          13,
          13
        ],
        "domain": "Complete_finite_max_flow_min_cut",
        "operation": "mfmc-complete-cut-enumeration",
        "semantic_template": "qf-mod46-mfmc-complete-cut-enumeration/{\"capacities\":[[9,1],[10,1],[11,1],[12,1],[13,1],[9,1],[10,1],[11,1],[12,1],[13,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"a3\",\"t\"],[\"a4\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"],[\"s\",\"a3\"],[\"s\",\"a4\"]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"a3\",\"a4\",\"s\",\"t\"]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_or_symbolic_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-15",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0008",
    "batch_id": "batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "4de1a0766af89515f3e3139cc4bb286b93c3bb78e17ccbbd974f613807abd4da",
      "family_id": "qf-mod46-mfmc-complete-cut-enumeration",
      "seed": 202611155008,
      "semantic_template": "qf-mod46-mfmc-complete-cut-enumeration/{\"capacities\":[[9,1],[10,1],[11,1],[12,1],[13,1],[9,1],[10,1],[11,1],[12,1],[13,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"a3\",\"t\"],[\"a4\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"],[\"s\",\"a3\"],[\"s\",\"a4\"]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"a3\",\"a4\",\"s\",\"t\"]}"
    },
    "fields": {
      "title": "Every Source-Side Vertex Subset Defines One Directed Cut Audit",
      "stem": "Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [s, a4], [a4, t]], source=s, sink=t, aligned nonnegative exact capacities=[9, 9, 10, 10, 11, 11, 12, 12, 13, 13]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate).",
      "answer": {
        "kind": "expression",
        "value": "([a0, a1, a2, a3, a4, s, t], s, t, [[[s], [[[s, a0], 9], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, s], [[[a0, t], 9], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a1, s], [[[a1, t], 10], [[s, a0], 9], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, a1, s], [[[a0, t], 9], [[a1, t], 10], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a2, s], [[[a2, t], 11], [[s, a0], 9], [[s, a1], 10], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, a2, s], [[[a0, t], 9], [[a2, t], 11], [[s, a1], 10], [[s, a3], 12], [[s, a4], 13]], 55], [[a1, a2, s], [[[a1, t], 10], [[a2, t], 11], [[s, a0], 9], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, a1, a2, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a3, s], [[[a3, t], 12], [[s, a0], 9], [[s, a1], 10], [[s, a2], 11], [[s, a4], 13]], 55], [[a0, a3, s], [[[a0, t], 9], [[a3, t], 12], [[s, a1], 10], [[s, a2], 11], [[s, a4], 13]], 55], [[a1, a3, s], [[[a1, t], 10], [[a3, t], 12], [[s, a0], 9], [[s, a2], 11], [[s, a4], 13]], 55], [[a0, a1, a3, s], [[[a0, t], 9], [[a1, t], 10], [[a3, t], 12], [[s, a2], 11], [[s, a4], 13]], 55], [[a2, a3, s], [[[a2, t], 11], [[a3, t], 12], [[s, a0], 9], [[s, a1], 10], [[s, a4], 13]], 55], [[a0, a2, a3, s], [[[a0, t], 9], [[a2, t], 11], [[a3, t], 12], [[s, a1], 10], [[s, a4], 13]], 55], [[a1, a2, a3, s], [[[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[s, a0], 9], [[s, a4], 13]], 55], [[a0, a1, a2, a3, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[s, a4], 13]], 55], [[a4, s], [[[a4, t], 13], [[s, a0], 9], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12]], 55], [[a0, a4, s], [[[a0, t], 9], [[a4, t], 13], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12]], 55], [[a1, a4, s], [[[a1, t], 10], [[a4, t], 13], [[s, a0], 9], [[s, a2], 11], [[s, a3], 12]], 55], [[a0, a1, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a4, t], 13], [[s, a2], 11], [[s, a3], 12]], 55], [[a2, a4, s], [[[a2, t], 11], [[a4, t], 13], [[s, a0], 9], [[s, a1], 10], [[s, a3], 12]], 55], [[a0, a2, a4, s], [[[a0, t], 9], [[a2, t], 11], [[a4, t], 13], [[s, a1], 10], [[s, a3], 12]], 55], [[a1, a2, a4, s], [[[a1, t], 10], [[a2, t], 11], [[a4, t], 13], [[s, a0], 9], [[s, a3], 12]], 55], [[a0, a1, a2, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[a4, t], 13], [[s, a3], 12]], 55], [[a3, a4, s], [[[a3, t], 12], [[a4, t], 13], [[s, a0], 9], [[s, a1], 10], [[s, a2], 11]], 55], [[a0, a3, a4, s], [[[a0, t], 9], [[a3, t], 12], [[a4, t], 13], [[s, a1], 10], [[s, a2], 11]], 55], [[a1, a3, a4, s], [[[a1, t], 10], [[a3, t], 12], [[a4, t], 13], [[s, a0], 9], [[s, a2], 11]], 55], [[a0, a1, a3, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a3, t], 12], [[a4, t], 13], [[s, a2], 11]], 55], [[a2, a3, a4, s], [[[a2, t], 11], [[a3, t], 12], [[a4, t], 13], [[s, a0], 9], [[s, a1], 10]], 55], [[a0, a2, a3, a4, s], [[[a0, t], 9], [[a2, t], 11], [[a3, t], 12], [[a4, t], 13], [[s, a1], 10]], 55], [[a1, a2, a3, a4, s], [[[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[a4, t], 13], [[s, a0], 9]], 55], [[a0, a1, a2, a3, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[a4, t], 13]], 55]], 32, 32, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)",
        "explanation": "Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([a0, a1, a2, a3, a4, s, t], s, t, [[[s], [[[s, a0], 9], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, s], [[[a0, t], 9], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a1, s], [[[a1, t], 10], [[s, a0], 9], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, a1, s], [[[a0, t], 9], [[a1, t], 10], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a2, s], [[[a2, t], 11], [[s, a0], 9], [[s, a1], 10], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, a2, s], [[[a0, t], 9], [[a2, t], 11], [[s, a1], 10], [[s, a3], 12], [[s, a4], 13]], 55], [[a1, a2, s], [[[a1, t], 10], [[a2, t], 11], [[s, a0], 9], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, a1, a2, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a3, s], [[[a3, t], 12], [[s, a0], 9], [[s, a1], 10], [[s, a2], 11], [[s, a4], 13]], 55], [[a0, a3, s], [[[a0, t], 9], [[a3, t], 12], [[s, a1], 10], [[s, a2], 11], [[s, a4], 13]], 55], [[a1, a3, s], [[[a1, t], 10], [[a3, t], 12], [[s, a0], 9], [[s, a2], 11], [[s, a4], 13]], 55], [[a0, a1, a3, s], [[[a0, t], 9], [[a1, t], 10], [[a3, t], 12], [[s, a2], 11], [[s, a4], 13]], 55], [[a2, a3, s], [[[a2, t], 11], [[a3, t], 12], [[s, a0], 9], [[s, a1], 10], [[s, a4], 13]], 55], [[a0, a2, a3, s], [[[a0, t], 9], [[a2, t], 11], [[a3, t], 12], [[s, a1], 10], [[s, a4], 13]], 55], [[a1, a2, a3, s], [[[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[s, a0], 9], [[s, a4], 13]], 55], [[a0, a1, a2, a3, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[s, a4], 13]], 55], [[a4, s], [[[a4, t], 13], [[s, a0], 9], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12]], 55], [[a0, a4, s], [[[a0, t], 9], [[a4, t], 13], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12]], 55], [[a1, a4, s], [[[a1, t], 10], [[a4, t], 13], [[s, a0], 9], [[s, a2], 11], [[s, a3], 12]], 55], [[a0, a1, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a4, t], 13], [[s, a2], 11], [[s, a3], 12]], 55], [[a2, a4, s], [[[a2, t], 11], [[a4, t], 13], [[s, a0], 9], [[s, a1], 10], [[s, a3], 12]], 55], [[a0, a2, a4, s], [[[a0, t], 9], [[a2, t], 11], [[a4, t], 13], [[s, a1], 10], [[s, a3], 12]], 55], [[a1, a2, a4, s], [[[a1, t], 10], [[a2, t], 11], [[a4, t], 13], [[s, a0], 9], [[s, a3], 12]], 55], [[a0, a1, a2, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[a4, t], 13], [[s, a3], 12]], 55], [[a3, a4, s], [[[a3, t], 12], [[a4, t], 13], [[s, a0], 9], [[s, a1], 10], [[s, a2], 11]], 55], [[a0, a3, a4, s], [[[a0, t], 9], [[a3, t], 12], [[a4, t], 13], [[s, a1], 10], [[s, a2], 11]], 55], [[a1, a3, a4, s], [[[a1, t], 10], [[a3, t], 12], [[a4, t], 13], [[s, a0], 9], [[s, a2], 11]], 55], [[a0, a1, a3, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a3, t], 12], [[a4, t], 13], [[s, a2], 11]], 55], [[a2, a3, a4, s], [[[a2, t], 11], [[a3, t], 12], [[a4, t], 13], [[s, a0], 9], [[s, a1], 10]], 55], [[a0, a2, a3, a4, s], [[[a0, t], 9], [[a2, t], 11], [[a3, t], 12], [[a4, t], 13], [[s, a1], 10]], 55], [[a1, a2, a3, a4, s], [[[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[a4, t], 13], [[s, a0], 9]], 55], [[a0, a1, a2, a3, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[a4, t], 13]], 55]], 32, 32, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)."
      },
      "skills": [
        "Every Source-Side Vertex Subset Defines One Directed Cut Audit"
      ],
      "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": "完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-complete-cut-enumeration；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"a3\",\"a4\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"],[\"s\",\"a3\"],[\"a3\",\"t\"],[\"s\",\"a4\"],[\"a4\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[9,9,10,10,11,11,12,12,13,13]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [s, a4], [a4, t]], source=s, sink=t, aligned nonnegative exact capacities=[9, 9, 10, 10, 11, 11, 12, 12, 13, 13]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [mfmc-complete-cut-enumeration]",
      "after": "derived = [[vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate]]",
      "relation": "implies",
      "rule_id": "source_ordered_fields_without_JS",
      "conditions": [],
      "explanation": {
        "zh-Hant": "直接列完整來源欄序，不需JavaScript。LP weak為13欄、zero與strong為15欄，前提與角色依原operation／mode，不補欄。",
        "en": "List the full source field order without JavaScript.LP weak uses13 fields,zero/strong15;premises and roles follow the original operation/mode without padding."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "source roles = [[s, a0, a1, a2, a3, a4, t], [[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [s, a4], [a4, t]], [9, 9, 10, 10, 11, 11, 12, 12, 13, 13], null, s, t]",
      "after": "derived = [[a0, a1, a2, a3, a4, s, t], [[a0, t], [a1, t], [a2, t], [a3, t], [a4, t], [s, a0], [s, a1], [s, a2], [s, a3], [s, a4]], [9, 10, 11, 12, 13, 9, 10, 11, 12, 13], null]",
      "relation": "implies",
      "rule_id": "MFMC_complete_directed_network_ZIP_before_source_order",
      "conditions": [],
      "explanation": {
        "zh-Hant": "2..12個vertices；ordered edges與capacity/flow先按原index配好再排序，反向邊是獨立arc。source/sink不同、loop/duplicate拒絕，zero容量、空edge與isolates合法。",
        "en": "Use2..12 vertices.Zip directed edges with capacity/flow by original index before sorting;reverse arcs remain distinct.Source and sink differ;reject loops/duplicates but allow zero capacities,no edges and isolated vertices."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "MFMC cut mask0 OUTWARD capacity := sum(9, 10, 11, 12, 13)",
      "after": "MFMC cut mask0 OUTWARD capacity = 55",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。",
        "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 MFMC cut mask0 OUTWARD capacity，不以小數近似替代。",
          "en": "Recompute MFMC cut mask0 OUTWARD capacity with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "9",
          "10",
          "11",
          "12",
          "13"
        ],
        "output": "55",
        "label": "MFMC cut mask0 OUTWARD capacity"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "source roles = [0, [a0, a1, a2, a3, a4], [s]]",
      "after": "derived = [[5, 6, 7, 8, 9], [], [[[s, a0], 9], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55]",
      "relation": "implies",
      "rule_id": "MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only",
      "conditions": [],
      "explanation": {
        "zh-Hant": "S必含source排除sink；mask逐全部other subsets，S內UTF16排序但outer清單保numeric mask次序。capacity只加OUTWARD，不加incoming；ties也不另按English重排。",
        "en": "S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "MFMC cut mask1 OUTWARD capacity := sum(9, 10, 11, 12, 13)",
      "after": "MFMC cut mask1 OUTWARD capacity = 55",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。",
        "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 MFMC cut mask1 OUTWARD capacity，不以小數近似替代。",
          "en": "Recompute MFMC cut mask1 OUTWARD capacity with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "9",
          "10",
          "11",
          "12",
          "13"
        ],
        "output": "55",
        "label": "MFMC cut mask1 OUTWARD capacity"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "source roles = [1, [a0, a1, a2, a3, a4], [a0, s]]",
      "after": "derived = [[0, 6, 7, 8, 9], [], [[[a0, t], 9], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55]",
      "relation": "implies",
      "rule_id": "MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only",
      "conditions": [],
      "explanation": {
        "zh-Hant": "S必含source排除sink；mask逐全部other subsets，S內UTF16排序但outer清單保numeric mask次序。capacity只加OUTWARD，不加incoming；ties也不另按English重排。",
        "en": "S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "MFMC cut mask2 OUTWARD capacity := sum(10, 9, 11, 12, 13)",
      "after": "MFMC cut mask2 OUTWARD capacity = 55",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。",
        "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 MFMC cut mask2 OUTWARD capacity，不以小數近似替代。",
          "en": "Recompute MFMC cut mask2 OUTWARD capacity with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "10",
          "9",
          "11",
          "12",
          "13"
        ],
        "output": "55",
        "label": "MFMC cut mask2 OUTWARD capacity"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "source roles = [2, [a0, a1, a2, a3, a4], [a1, s]]",
      "after": "derived = [[1, 5, 7, 8, 9], [], [[[a1, t], 10], [[s, a0], 9], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55]",
      "relation": "implies",
      "rule_id": "MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only",
      "conditions": [],
      "explanation": {
        "zh-Hant": "S必含source排除sink；mask逐全部other subsets，S內UTF16排序但outer清單保numeric mask次序。capacity只加OUTWARD，不加incoming；ties也不另按English重排。",
        "en": "S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "MFMC cut mask3 OUTWARD capacity := sum(9, 10, 11, 12, 13)",
      "after": "MFMC cut mask3 OUTWARD capacity = 55",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。",
        "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 MFMC cut mask3 OUTWARD capacity，不以小數近似替代。",
          "en": "Recompute MFMC cut mask3 OUTWARD capacity with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "9",
          "10",
          "11",
          "12",
          "13"
        ],
        "output": "55",
        "label": "MFMC cut mask3 OUTWARD capacity"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "source roles = [3, [a0, a1, a2, a3, a4], [a0, a1, s]]",
      "after": "derived = [[0, 1, 7, 8, 9], [], [[[a0, t], 9], [[a1, t], 10], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55]",
      "relation": "implies",
      "rule_id": "MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only",
      "conditions": [],
      "explanation": {
        "zh-Hant": "S必含source排除sink；mask逐全部other subsets，S內UTF16排序但outer清單保numeric mask次序。capacity只加OUTWARD，不加incoming；ties也不另按English重排。",
        "en": "S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "MFMC cut mask4 OUTWARD capacity := sum(11, 9, 10, 12, 13)",
      "after": "MFMC cut mask4 OUTWARD capacity = 55",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。",
        "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 MFMC cut mask4 OUTWARD capacity，不以小數近似替代。",
          "en": "Recompute MFMC cut mask4 OUTWARD capacity with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "11",
          "9",
          "10",
          "12",
          "13"
        ],
        "output": "55",
        "label": "MFMC cut mask4 OUTWARD capacity"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "source roles = [4, [a0, a1, a2, a3, a4], [a2, s]]",
      "after": "derived = [[2, 5, 6, 8, 9], [], [[[a2, t], 11], [[s, a0], 9], [[s, a1], 10], [[s, a3], 12], [[s, a4], 13]], 55]",
      "relation": "implies",
      "rule_id": "MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only",
      "conditions": [],
      "explanation": {
        "zh-Hant": "S必含source排除sink；mask逐全部other subsets，S內UTF16排序但outer清單保numeric mask次序。capacity只加OUTWARD，不加incoming；ties也不另按English重排。",
        "en": "S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "MFMC cut mask5 OUTWARD capacity := sum(9, 11, 10, 12, 13)",
      "after": "MFMC cut mask5 OUTWARD capacity = 55",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。",
        "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 MFMC cut mask5 OUTWARD capacity，不以小數近似替代。",
          "en": "Recompute MFMC cut mask5 OUTWARD capacity with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "9",
          "11",
          "10",
          "12",
          "13"
        ],
        "output": "55",
        "label": "MFMC cut mask5 OUTWARD capacity"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "source roles = [5, [a0, a1, a2, a3, a4], [a0, a2, s]]",
      "after": "derived = [[0, 2, 6, 8, 9], [], [[[a0, t], 9], [[a2, t], 11], [[s, a1], 10], [[s, a3], 12], [[s, a4], 13]], 55]",
      "relation": "implies",
      "rule_id": "MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only",
      "conditions": [],
      "explanation": {
        "zh-Hant": "S必含source排除sink；mask逐全部other subsets，S內UTF16排序但outer清單保numeric mask次序。capacity只加OUTWARD，不加incoming；ties也不另按English重排。",
        "en": "S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "MFMC cut mask6 OUTWARD capacity := sum(10, 11, 9, 12, 13)",
      "after": "MFMC cut mask6 OUTWARD capacity = 55",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。",
        "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 MFMC cut mask6 OUTWARD capacity，不以小數近似替代。",
          "en": "Recompute MFMC cut mask6 OUTWARD capacity with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "10",
          "11",
          "9",
          "12",
          "13"
        ],
        "output": "55",
        "label": "MFMC cut mask6 OUTWARD capacity"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "source roles = [6, [a0, a1, a2, a3, a4], [a1, a2, s]]",
      "after": "derived = [[1, 2, 5, 8, 9], [], [[[a1, t], 10], [[a2, t], 11], [[s, a0], 9], [[s, a3], 12], [[s, a4], 13]], 55]",
      "relation": "implies",
      "rule_id": "MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only",
      "conditions": [],
      "explanation": {
        "zh-Hant": "S必含source排除sink；mask逐全部other subsets，S內UTF16排序但outer清單保numeric mask次序。capacity只加OUTWARD，不加incoming；ties也不另按English重排。",
        "en": "S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "MFMC cut mask7 OUTWARD capacity := sum(9, 10, 11, 12, 13)",
      "after": "MFMC cut mask7 OUTWARD capacity = 55",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。",
        "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 MFMC cut mask7 OUTWARD capacity，不以小數近似替代。",
          "en": "Recompute MFMC cut mask7 OUTWARD capacity with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "9",
          "10",
          "11",
          "12",
          "13"
        ],
        "output": "55",
        "label": "MFMC cut mask7 OUTWARD capacity"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "source roles = [7, [a0, a1, a2, a3, a4], [a0, a1, a2, s]]",
      "after": "derived = [[0, 1, 2, 8, 9], [], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[s, a3], 12], [[s, a4], 13]], 55]",
      "relation": "implies",
      "rule_id": "MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only",
      "conditions": [],
      "explanation": {
        "zh-Hant": "S必含source排除sink；mask逐全部other subsets，S內UTF16排序但outer清單保numeric mask次序。capacity只加OUTWARD，不加incoming；ties也不另按English重排。",
        "en": "S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "MFMC cut mask8 OUTWARD capacity := sum(12, 9, 10, 11, 13)",
      "after": "MFMC cut mask8 OUTWARD capacity = 55",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。",
        "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 MFMC cut mask8 OUTWARD capacity，不以小數近似替代。",
          "en": "Recompute MFMC cut mask8 OUTWARD capacity with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "12",
          "9",
          "10",
          "11",
          "13"
        ],
        "output": "55",
        "label": "MFMC cut mask8 OUTWARD capacity"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "source roles = [8, [a0, a1, a2, a3, a4], [a3, s]]",
      "after": "derived = [[3, 5, 6, 7, 9], [], [[[a3, t], 12], [[s, a0], 9], [[s, a1], 10], [[s, a2], 11], [[s, a4], 13]], 55]",
      "relation": "implies",
      "rule_id": "MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only",
      "conditions": [],
      "explanation": {
        "zh-Hant": "S必含source排除sink；mask逐全部other subsets，S內UTF16排序但outer清單保numeric mask次序。capacity只加OUTWARD，不加incoming；ties也不另按English重排。",
        "en": "S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "MFMC cut mask9 OUTWARD capacity := sum(9, 12, 10, 11, 13)",
      "after": "MFMC cut mask9 OUTWARD capacity = 55",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。",
        "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 MFMC cut mask9 OUTWARD capacity，不以小數近似替代。",
          "en": "Recompute MFMC cut mask9 OUTWARD capacity with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "9",
          "12",
          "10",
          "11",
          "13"
        ],
        "output": "55",
        "label": "MFMC cut mask9 OUTWARD capacity"
      }
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "source roles = [9, [a0, a1, a2, a3, a4], [a0, a3, s]]",
      "after": "derived = [[0, 3, 6, 7, 9], [], [[[a0, t], 9], [[a3, t], 12], [[s, a1], 10], [[s, a2], 11], [[s, a4], 13]], 55]",
      "relation": "implies",
      "rule_id": "MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only",
      "conditions": [],
      "explanation": {
        "zh-Hant": "S必含source排除sink；mask逐全部other subsets，S內UTF16排序但outer清單保numeric mask次序。capacity只加OUTWARD，不加incoming；ties也不另按English重排。",
        "en": "S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "MFMC cut mask10 OUTWARD capacity := sum(10, 12, 9, 11, 13)",
      "after": "MFMC cut mask10 OUTWARD capacity = 55",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。",
        "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 MFMC cut mask10 OUTWARD capacity，不以小數近似替代。",
          "en": "Recompute MFMC cut mask10 OUTWARD capacity with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "10",
          "12",
          "9",
          "11",
          "13"
        ],
        "output": "55",
        "label": "MFMC cut mask10 OUTWARD capacity"
      }
    },
    {
      "id": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "source roles = [10, [a0, a1, a2, a3, a4], [a1, a3, s]]",
      "after": "derived = [[1, 3, 5, 7, 9], [], [[[a1, t], 10], [[a3, t], 12], [[s, a0], 9], [[s, a2], 11], [[s, a4], 13]], 55]",
      "relation": "implies",
      "rule_id": "MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only",
      "conditions": [],
      "explanation": {
        "zh-Hant": "S必含source排除sink；mask逐全部other subsets，S內UTF16排序但outer清單保numeric mask次序。capacity只加OUTWARD，不加incoming；ties也不另按English重排。",
        "en": "S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "MFMC cut mask11 OUTWARD capacity := sum(9, 10, 12, 11, 13)",
      "after": "MFMC cut mask11 OUTWARD capacity = 55",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。",
        "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 MFMC cut mask11 OUTWARD capacity，不以小數近似替代。",
          "en": "Recompute MFMC cut mask11 OUTWARD capacity with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "9",
          "10",
          "12",
          "11",
          "13"
        ],
        "output": "55",
        "label": "MFMC cut mask11 OUTWARD capacity"
      }
    },
    {
      "id": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "source roles = [11, [a0, a1, a2, a3, a4], [a0, a1, a3, s]]",
      "after": "derived = [[0, 1, 3, 7, 9], [], [[[a0, t], 9], [[a1, t], 10], [[a3, t], 12], [[s, a2], 11], [[s, a4], 13]], 55]",
      "relation": "implies",
      "rule_id": "MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only",
      "conditions": [],
      "explanation": {
        "zh-Hant": "S必含source排除sink；mask逐全部other subsets，S內UTF16排序但outer清單保numeric mask次序。capacity只加OUTWARD，不加incoming；ties也不另按English重排。",
        "en": "S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "MFMC cut mask12 OUTWARD capacity := sum(11, 12, 9, 10, 13)",
      "after": "MFMC cut mask12 OUTWARD capacity = 55",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。",
        "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 MFMC cut mask12 OUTWARD capacity，不以小數近似替代。",
          "en": "Recompute MFMC cut mask12 OUTWARD capacity with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "11",
          "12",
          "9",
          "10",
          "13"
        ],
        "output": "55",
        "label": "MFMC cut mask12 OUTWARD capacity"
      }
    },
    {
      "id": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "source roles = [12, [a0, a1, a2, a3, a4], [a2, a3, s]]",
      "after": "derived = [[2, 3, 5, 6, 9], [], [[[a2, t], 11], [[a3, t], 12], [[s, a0], 9], [[s, a1], 10], [[s, a4], 13]], 55]",
      "relation": "implies",
      "rule_id": "MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only",
      "conditions": [],
      "explanation": {
        "zh-Hant": "S必含source排除sink；mask逐全部other subsets，S內UTF16排序但outer清單保numeric mask次序。capacity只加OUTWARD，不加incoming；ties也不另按English重排。",
        "en": "S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "MFMC cut mask13 OUTWARD capacity := sum(9, 11, 12, 10, 13)",
      "after": "MFMC cut mask13 OUTWARD capacity = 55",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。",
        "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 MFMC cut mask13 OUTWARD capacity，不以小數近似替代。",
          "en": "Recompute MFMC cut mask13 OUTWARD capacity with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "9",
          "11",
          "12",
          "10",
          "13"
        ],
        "output": "55",
        "label": "MFMC cut mask13 OUTWARD capacity"
      }
    },
    {
      "id": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "source roles = [13, [a0, a1, a2, a3, a4], [a0, a2, a3, s]]",
      "after": "derived = [[0, 2, 3, 6, 9], [], [[[a0, t], 9], [[a2, t], 11], [[a3, t], 12], [[s, a1], 10], [[s, a4], 13]], 55]",
      "relation": "implies",
      "rule_id": "MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only",
      "conditions": [],
      "explanation": {
        "zh-Hant": "S必含source排除sink；mask逐全部other subsets，S內UTF16排序但outer清單保numeric mask次序。capacity只加OUTWARD，不加incoming；ties也不另按English重排。",
        "en": "S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "MFMC cut mask14 OUTWARD capacity := sum(10, 11, 12, 9, 13)",
      "after": "MFMC cut mask14 OUTWARD capacity = 55",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。",
        "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 MFMC cut mask14 OUTWARD capacity，不以小數近似替代。",
          "en": "Recompute MFMC cut mask14 OUTWARD capacity with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "10",
          "11",
          "12",
          "9",
          "13"
        ],
        "output": "55",
        "label": "MFMC cut mask14 OUTWARD capacity"
      }
    },
    {
      "id": "S032",
      "uses": [
        "G001",
        "S031"
      ],
      "before": "source roles = [14, [a0, a1, a2, a3, a4], [a1, a2, a3, s]]",
      "after": "derived = [[1, 2, 3, 5, 9], [], [[[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[s, a0], 9], [[s, a4], 13]], 55]",
      "relation": "implies",
      "rule_id": "MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only",
      "conditions": [],
      "explanation": {
        "zh-Hant": "S必含source排除sink；mask逐全部other subsets，S內UTF16排序但outer清單保numeric mask次序。capacity只加OUTWARD，不加incoming；ties也不另按English重排。",
        "en": "S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S033",
      "uses": [
        "G001",
        "S032"
      ],
      "before": "MFMC cut mask15 OUTWARD capacity := sum(9, 10, 11, 12, 13)",
      "after": "MFMC cut mask15 OUTWARD capacity = 55",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。",
        "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 MFMC cut mask15 OUTWARD capacity，不以小數近似替代。",
          "en": "Recompute MFMC cut mask15 OUTWARD capacity with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "9",
          "10",
          "11",
          "12",
          "13"
        ],
        "output": "55",
        "label": "MFMC cut mask15 OUTWARD capacity"
      }
    },
    {
      "id": "S034",
      "uses": [
        "G001",
        "S033"
      ],
      "before": "source roles = [15, [a0, a1, a2, a3, a4], [a0, a1, a2, a3, s]]",
      "after": "derived = [[0, 1, 2, 3, 9], [], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[s, a4], 13]], 55]",
      "relation": "implies",
      "rule_id": "MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only",
      "conditions": [],
      "explanation": {
        "zh-Hant": "S必含source排除sink；mask逐全部other subsets，S內UTF16排序但outer清單保numeric mask次序。capacity只加OUTWARD，不加incoming；ties也不另按English重排。",
        "en": "S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S035",
      "uses": [
        "G001",
        "S034"
      ],
      "before": "MFMC cut mask16 OUTWARD capacity := sum(13, 9, 10, 11, 12)",
      "after": "MFMC cut mask16 OUTWARD capacity = 55",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。",
        "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 MFMC cut mask16 OUTWARD capacity，不以小數近似替代。",
          "en": "Recompute MFMC cut mask16 OUTWARD capacity with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "13",
          "9",
          "10",
          "11",
          "12"
        ],
        "output": "55",
        "label": "MFMC cut mask16 OUTWARD capacity"
      }
    },
    {
      "id": "S036",
      "uses": [
        "G001",
        "S035"
      ],
      "before": "source roles = [16, [a0, a1, a2, a3, a4], [a4, s]]",
      "after": "derived = [[4, 5, 6, 7, 8], [], [[[a4, t], 13], [[s, a0], 9], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12]], 55]",
      "relation": "implies",
      "rule_id": "MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only",
      "conditions": [],
      "explanation": {
        "zh-Hant": "S必含source排除sink；mask逐全部other subsets，S內UTF16排序但outer清單保numeric mask次序。capacity只加OUTWARD，不加incoming；ties也不另按English重排。",
        "en": "S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S037",
      "uses": [
        "G001",
        "S036"
      ],
      "before": "MFMC cut mask17 OUTWARD capacity := sum(9, 13, 10, 11, 12)",
      "after": "MFMC cut mask17 OUTWARD capacity = 55",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。",
        "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 MFMC cut mask17 OUTWARD capacity，不以小數近似替代。",
          "en": "Recompute MFMC cut mask17 OUTWARD capacity with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "9",
          "13",
          "10",
          "11",
          "12"
        ],
        "output": "55",
        "label": "MFMC cut mask17 OUTWARD capacity"
      }
    },
    {
      "id": "S038",
      "uses": [
        "G001",
        "S037"
      ],
      "before": "source roles = [17, [a0, a1, a2, a3, a4], [a0, a4, s]]",
      "after": "derived = [[0, 4, 6, 7, 8], [], [[[a0, t], 9], [[a4, t], 13], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12]], 55]",
      "relation": "implies",
      "rule_id": "MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only",
      "conditions": [],
      "explanation": {
        "zh-Hant": "S必含source排除sink；mask逐全部other subsets，S內UTF16排序但outer清單保numeric mask次序。capacity只加OUTWARD，不加incoming；ties也不另按English重排。",
        "en": "S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S039",
      "uses": [
        "G001",
        "S038"
      ],
      "before": "MFMC cut mask18 OUTWARD capacity := sum(10, 13, 9, 11, 12)",
      "after": "MFMC cut mask18 OUTWARD capacity = 55",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。",
        "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 MFMC cut mask18 OUTWARD capacity，不以小數近似替代。",
          "en": "Recompute MFMC cut mask18 OUTWARD capacity with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "10",
          "13",
          "9",
          "11",
          "12"
        ],
        "output": "55",
        "label": "MFMC cut mask18 OUTWARD capacity"
      }
    },
    {
      "id": "S040",
      "uses": [
        "G001",
        "S039"
      ],
      "before": "source roles = [18, [a0, a1, a2, a3, a4], [a1, a4, s]]",
      "after": "derived = [[1, 4, 5, 7, 8], [], [[[a1, t], 10], [[a4, t], 13], [[s, a0], 9], [[s, a2], 11], [[s, a3], 12]], 55]",
      "relation": "implies",
      "rule_id": "MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only",
      "conditions": [],
      "explanation": {
        "zh-Hant": "S必含source排除sink；mask逐全部other subsets，S內UTF16排序但outer清單保numeric mask次序。capacity只加OUTWARD，不加incoming；ties也不另按English重排。",
        "en": "S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S041",
      "uses": [
        "G001",
        "S040"
      ],
      "before": "MFMC cut mask19 OUTWARD capacity := sum(9, 10, 13, 11, 12)",
      "after": "MFMC cut mask19 OUTWARD capacity = 55",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。",
        "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 MFMC cut mask19 OUTWARD capacity，不以小數近似替代。",
          "en": "Recompute MFMC cut mask19 OUTWARD capacity with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "9",
          "10",
          "13",
          "11",
          "12"
        ],
        "output": "55",
        "label": "MFMC cut mask19 OUTWARD capacity"
      }
    },
    {
      "id": "S042",
      "uses": [
        "G001",
        "S041"
      ],
      "before": "source roles = [19, [a0, a1, a2, a3, a4], [a0, a1, a4, s]]",
      "after": "derived = [[0, 1, 4, 7, 8], [], [[[a0, t], 9], [[a1, t], 10], [[a4, t], 13], [[s, a2], 11], [[s, a3], 12]], 55]",
      "relation": "implies",
      "rule_id": "MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only",
      "conditions": [],
      "explanation": {
        "zh-Hant": "S必含source排除sink；mask逐全部other subsets，S內UTF16排序但outer清單保numeric mask次序。capacity只加OUTWARD，不加incoming；ties也不另按English重排。",
        "en": "S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S043",
      "uses": [
        "G001",
        "S042"
      ],
      "before": "MFMC cut mask20 OUTWARD capacity := sum(11, 13, 9, 10, 12)",
      "after": "MFMC cut mask20 OUTWARD capacity = 55",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。",
        "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 MFMC cut mask20 OUTWARD capacity，不以小數近似替代。",
          "en": "Recompute MFMC cut mask20 OUTWARD capacity with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "11",
          "13",
          "9",
          "10",
          "12"
        ],
        "output": "55",
        "label": "MFMC cut mask20 OUTWARD capacity"
      }
    },
    {
      "id": "S044",
      "uses": [
        "G001",
        "S043"
      ],
      "before": "source roles = [20, [a0, a1, a2, a3, a4], [a2, a4, s]]",
      "after": "derived = [[2, 4, 5, 6, 8], [], [[[a2, t], 11], [[a4, t], 13], [[s, a0], 9], [[s, a1], 10], [[s, a3], 12]], 55]",
      "relation": "implies",
      "rule_id": "MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only",
      "conditions": [],
      "explanation": {
        "zh-Hant": "S必含source排除sink；mask逐全部other subsets，S內UTF16排序但outer清單保numeric mask次序。capacity只加OUTWARD，不加incoming；ties也不另按English重排。",
        "en": "S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S045",
      "uses": [
        "G001",
        "S044"
      ],
      "before": "MFMC cut mask21 OUTWARD capacity := sum(9, 11, 13, 10, 12)",
      "after": "MFMC cut mask21 OUTWARD capacity = 55",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。",
        "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 MFMC cut mask21 OUTWARD capacity，不以小數近似替代。",
          "en": "Recompute MFMC cut mask21 OUTWARD capacity with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "9",
          "11",
          "13",
          "10",
          "12"
        ],
        "output": "55",
        "label": "MFMC cut mask21 OUTWARD capacity"
      }
    },
    {
      "id": "S046",
      "uses": [
        "G001",
        "S045"
      ],
      "before": "source roles = [21, [a0, a1, a2, a3, a4], [a0, a2, a4, s]]",
      "after": "derived = [[0, 2, 4, 6, 8], [], [[[a0, t], 9], [[a2, t], 11], [[a4, t], 13], [[s, a1], 10], [[s, a3], 12]], 55]",
      "relation": "implies",
      "rule_id": "MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only",
      "conditions": [],
      "explanation": {
        "zh-Hant": "S必含source排除sink；mask逐全部other subsets，S內UTF16排序但outer清單保numeric mask次序。capacity只加OUTWARD，不加incoming；ties也不另按English重排。",
        "en": "S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S047",
      "uses": [
        "G001",
        "S046"
      ],
      "before": "MFMC cut mask22 OUTWARD capacity := sum(10, 11, 13, 9, 12)",
      "after": "MFMC cut mask22 OUTWARD capacity = 55",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。",
        "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 MFMC cut mask22 OUTWARD capacity，不以小數近似替代。",
          "en": "Recompute MFMC cut mask22 OUTWARD capacity with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "10",
          "11",
          "13",
          "9",
          "12"
        ],
        "output": "55",
        "label": "MFMC cut mask22 OUTWARD capacity"
      }
    },
    {
      "id": "S048",
      "uses": [
        "G001",
        "S047"
      ],
      "before": "source roles = [22, [a0, a1, a2, a3, a4], [a1, a2, a4, s]]",
      "after": "derived = [[1, 2, 4, 5, 8], [], [[[a1, t], 10], [[a2, t], 11], [[a4, t], 13], [[s, a0], 9], [[s, a3], 12]], 55]",
      "relation": "implies",
      "rule_id": "MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only",
      "conditions": [],
      "explanation": {
        "zh-Hant": "S必含source排除sink；mask逐全部other subsets，S內UTF16排序但outer清單保numeric mask次序。capacity只加OUTWARD，不加incoming；ties也不另按English重排。",
        "en": "S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S049",
      "uses": [
        "G001",
        "S048"
      ],
      "before": "MFMC cut mask23 OUTWARD capacity := sum(9, 10, 11, 13, 12)",
      "after": "MFMC cut mask23 OUTWARD capacity = 55",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。",
        "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 MFMC cut mask23 OUTWARD capacity，不以小數近似替代。",
          "en": "Recompute MFMC cut mask23 OUTWARD capacity with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "9",
          "10",
          "11",
          "13",
          "12"
        ],
        "output": "55",
        "label": "MFMC cut mask23 OUTWARD capacity"
      }
    },
    {
      "id": "S050",
      "uses": [
        "G001",
        "S049"
      ],
      "before": "source roles = [23, [a0, a1, a2, a3, a4], [a0, a1, a2, a4, s]]",
      "after": "derived = [[0, 1, 2, 4, 8], [], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[a4, t], 13], [[s, a3], 12]], 55]",
      "relation": "implies",
      "rule_id": "MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only",
      "conditions": [],
      "explanation": {
        "zh-Hant": "S必含source排除sink；mask逐全部other subsets，S內UTF16排序但outer清單保numeric mask次序。capacity只加OUTWARD，不加incoming；ties也不另按English重排。",
        "en": "S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S051",
      "uses": [
        "G001",
        "S050"
      ],
      "before": "MFMC cut mask24 OUTWARD capacity := sum(12, 13, 9, 10, 11)",
      "after": "MFMC cut mask24 OUTWARD capacity = 55",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。",
        "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 MFMC cut mask24 OUTWARD capacity，不以小數近似替代。",
          "en": "Recompute MFMC cut mask24 OUTWARD capacity with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "12",
          "13",
          "9",
          "10",
          "11"
        ],
        "output": "55",
        "label": "MFMC cut mask24 OUTWARD capacity"
      }
    },
    {
      "id": "S052",
      "uses": [
        "G001",
        "S051"
      ],
      "before": "source roles = [24, [a0, a1, a2, a3, a4], [a3, a4, s]]",
      "after": "derived = [[3, 4, 5, 6, 7], [], [[[a3, t], 12], [[a4, t], 13], [[s, a0], 9], [[s, a1], 10], [[s, a2], 11]], 55]",
      "relation": "implies",
      "rule_id": "MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only",
      "conditions": [],
      "explanation": {
        "zh-Hant": "S必含source排除sink；mask逐全部other subsets，S內UTF16排序但outer清單保numeric mask次序。capacity只加OUTWARD，不加incoming；ties也不另按English重排。",
        "en": "S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S053",
      "uses": [
        "G001",
        "S052"
      ],
      "before": "MFMC cut mask25 OUTWARD capacity := sum(9, 12, 13, 10, 11)",
      "after": "MFMC cut mask25 OUTWARD capacity = 55",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。",
        "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 MFMC cut mask25 OUTWARD capacity，不以小數近似替代。",
          "en": "Recompute MFMC cut mask25 OUTWARD capacity with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "9",
          "12",
          "13",
          "10",
          "11"
        ],
        "output": "55",
        "label": "MFMC cut mask25 OUTWARD capacity"
      }
    },
    {
      "id": "S054",
      "uses": [
        "G001",
        "S053"
      ],
      "before": "source roles = [25, [a0, a1, a2, a3, a4], [a0, a3, a4, s]]",
      "after": "derived = [[0, 3, 4, 6, 7], [], [[[a0, t], 9], [[a3, t], 12], [[a4, t], 13], [[s, a1], 10], [[s, a2], 11]], 55]",
      "relation": "implies",
      "rule_id": "MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only",
      "conditions": [],
      "explanation": {
        "zh-Hant": "S必含source排除sink；mask逐全部other subsets，S內UTF16排序但outer清單保numeric mask次序。capacity只加OUTWARD，不加incoming；ties也不另按English重排。",
        "en": "S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S055",
      "uses": [
        "G001",
        "S054"
      ],
      "before": "MFMC cut mask26 OUTWARD capacity := sum(10, 12, 13, 9, 11)",
      "after": "MFMC cut mask26 OUTWARD capacity = 55",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。",
        "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 MFMC cut mask26 OUTWARD capacity，不以小數近似替代。",
          "en": "Recompute MFMC cut mask26 OUTWARD capacity with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "10",
          "12",
          "13",
          "9",
          "11"
        ],
        "output": "55",
        "label": "MFMC cut mask26 OUTWARD capacity"
      }
    },
    {
      "id": "S056",
      "uses": [
        "G001",
        "S055"
      ],
      "before": "source roles = [26, [a0, a1, a2, a3, a4], [a1, a3, a4, s]]",
      "after": "derived = [[1, 3, 4, 5, 7], [], [[[a1, t], 10], [[a3, t], 12], [[a4, t], 13], [[s, a0], 9], [[s, a2], 11]], 55]",
      "relation": "implies",
      "rule_id": "MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only",
      "conditions": [],
      "explanation": {
        "zh-Hant": "S必含source排除sink；mask逐全部other subsets，S內UTF16排序但outer清單保numeric mask次序。capacity只加OUTWARD，不加incoming；ties也不另按English重排。",
        "en": "S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S057",
      "uses": [
        "G001",
        "S056"
      ],
      "before": "MFMC cut mask27 OUTWARD capacity := sum(9, 10, 12, 13, 11)",
      "after": "MFMC cut mask27 OUTWARD capacity = 55",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。",
        "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 MFMC cut mask27 OUTWARD capacity，不以小數近似替代。",
          "en": "Recompute MFMC cut mask27 OUTWARD capacity with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "9",
          "10",
          "12",
          "13",
          "11"
        ],
        "output": "55",
        "label": "MFMC cut mask27 OUTWARD capacity"
      }
    },
    {
      "id": "S058",
      "uses": [
        "G001",
        "S057"
      ],
      "before": "source roles = [27, [a0, a1, a2, a3, a4], [a0, a1, a3, a4, s]]",
      "after": "derived = [[0, 1, 3, 4, 7], [], [[[a0, t], 9], [[a1, t], 10], [[a3, t], 12], [[a4, t], 13], [[s, a2], 11]], 55]",
      "relation": "implies",
      "rule_id": "MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only",
      "conditions": [],
      "explanation": {
        "zh-Hant": "S必含source排除sink；mask逐全部other subsets，S內UTF16排序但outer清單保numeric mask次序。capacity只加OUTWARD，不加incoming；ties也不另按English重排。",
        "en": "S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S059",
      "uses": [
        "G001",
        "S058"
      ],
      "before": "MFMC cut mask28 OUTWARD capacity := sum(11, 12, 13, 9, 10)",
      "after": "MFMC cut mask28 OUTWARD capacity = 55",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。",
        "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 MFMC cut mask28 OUTWARD capacity，不以小數近似替代。",
          "en": "Recompute MFMC cut mask28 OUTWARD capacity with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "11",
          "12",
          "13",
          "9",
          "10"
        ],
        "output": "55",
        "label": "MFMC cut mask28 OUTWARD capacity"
      }
    },
    {
      "id": "S060",
      "uses": [
        "G001",
        "S059"
      ],
      "before": "source roles = [28, [a0, a1, a2, a3, a4], [a2, a3, a4, s]]",
      "after": "derived = [[2, 3, 4, 5, 6], [], [[[a2, t], 11], [[a3, t], 12], [[a4, t], 13], [[s, a0], 9], [[s, a1], 10]], 55]",
      "relation": "implies",
      "rule_id": "MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only",
      "conditions": [],
      "explanation": {
        "zh-Hant": "S必含source排除sink；mask逐全部other subsets，S內UTF16排序但outer清單保numeric mask次序。capacity只加OUTWARD，不加incoming；ties也不另按English重排。",
        "en": "S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S061",
      "uses": [
        "G001",
        "S060"
      ],
      "before": "MFMC cut mask29 OUTWARD capacity := sum(9, 11, 12, 13, 10)",
      "after": "MFMC cut mask29 OUTWARD capacity = 55",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。",
        "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 MFMC cut mask29 OUTWARD capacity，不以小數近似替代。",
          "en": "Recompute MFMC cut mask29 OUTWARD capacity with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "9",
          "11",
          "12",
          "13",
          "10"
        ],
        "output": "55",
        "label": "MFMC cut mask29 OUTWARD capacity"
      }
    },
    {
      "id": "S062",
      "uses": [
        "G001",
        "S061"
      ],
      "before": "source roles = [29, [a0, a1, a2, a3, a4], [a0, a2, a3, a4, s]]",
      "after": "derived = [[0, 2, 3, 4, 6], [], [[[a0, t], 9], [[a2, t], 11], [[a3, t], 12], [[a4, t], 13], [[s, a1], 10]], 55]",
      "relation": "implies",
      "rule_id": "MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only",
      "conditions": [],
      "explanation": {
        "zh-Hant": "S必含source排除sink；mask逐全部other subsets，S內UTF16排序但outer清單保numeric mask次序。capacity只加OUTWARD，不加incoming；ties也不另按English重排。",
        "en": "S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S063",
      "uses": [
        "G001",
        "S062"
      ],
      "before": "MFMC cut mask30 OUTWARD capacity := sum(10, 11, 12, 13, 9)",
      "after": "MFMC cut mask30 OUTWARD capacity = 55",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。",
        "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 MFMC cut mask30 OUTWARD capacity，不以小數近似替代。",
          "en": "Recompute MFMC cut mask30 OUTWARD capacity with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "10",
          "11",
          "12",
          "13",
          "9"
        ],
        "output": "55",
        "label": "MFMC cut mask30 OUTWARD capacity"
      }
    },
    {
      "id": "S064",
      "uses": [
        "G001",
        "S063"
      ],
      "before": "source roles = [30, [a0, a1, a2, a3, a4], [a1, a2, a3, a4, s]]",
      "after": "derived = [[1, 2, 3, 4, 5], [], [[[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[a4, t], 13], [[s, a0], 9]], 55]",
      "relation": "implies",
      "rule_id": "MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only",
      "conditions": [],
      "explanation": {
        "zh-Hant": "S必含source排除sink；mask逐全部other subsets，S內UTF16排序但outer清單保numeric mask次序。capacity只加OUTWARD，不加incoming；ties也不另按English重排。",
        "en": "S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S065",
      "uses": [
        "G001",
        "S064"
      ],
      "before": "MFMC cut mask31 OUTWARD capacity := sum(9, 10, 11, 12, 13)",
      "after": "MFMC cut mask31 OUTWARD capacity = 55",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。",
        "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 MFMC cut mask31 OUTWARD capacity，不以小數近似替代。",
          "en": "Recompute MFMC cut mask31 OUTWARD capacity with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "9",
          "10",
          "11",
          "12",
          "13"
        ],
        "output": "55",
        "label": "MFMC cut mask31 OUTWARD capacity"
      }
    },
    {
      "id": "S066",
      "uses": [
        "G001",
        "S065"
      ],
      "before": "source roles = [31, [a0, a1, a2, a3, a4], [a0, a1, a2, a3, a4, s]]",
      "after": "derived = [[0, 1, 2, 3, 4], [], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[a4, t], 13]], 55]",
      "relation": "implies",
      "rule_id": "MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only",
      "conditions": [],
      "explanation": {
        "zh-Hant": "S必含source排除sink；mask逐全部other subsets，S內UTF16排序但outer清單保numeric mask次序。capacity只加OUTWARD，不加incoming；ties也不另按English重排。",
        "en": "S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S067",
      "uses": [
        "G001",
        "S066"
      ],
      "before": "source roles = [[a0, a1, a2, a3, a4, s, t], [a0, a1, a2, a3, a4]]",
      "after": "derived = [32, 32, 55, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]]]",
      "relation": "implies",
      "rule_id": "MFMC_ALL_cut_family_complete_minimum_and_ALL_ties",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個s–t cut唯一對應other的subset，故已完整比較2^(n−2)個cuts；最小值與全部minimizers由同一完整族取得，保原mask tie次序。",
        "en": "Every s–t cut uniquely corresponds to a subset of other vertices,so all2^(n−2) cuts are compared.Select the minimum and all minimizers from this same complete family,retaining mask tie order."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
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        "a1",
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              "9"
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              "10"
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              "11"
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        ],
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          [
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                "a3",
                "t"
              ],
              "12"
            ],
            [
              [
                "a4",
                "t"
              ],
              "13"
            ],
            [
              [
                "s",
                "a0"
              ],
              "9"
            ],
            [
              [
                "s",
                "a2"
              ],
              "11"
            ]
          ],
          "55"
        ],
        [
          [
            "a0",
            "a1",
            "a3",
            "a4",
            "s"
          ],
          [
            [
              [
                "a0",
                "t"
              ],
              "9"
            ],
            [
              [
                "a1",
                "t"
              ],
              "10"
            ],
            [
              [
                "a3",
                "t"
              ],
              "12"
            ],
            [
              [
                "a4",
                "t"
              ],
              "13"
            ],
            [
              [
                "s",
                "a2"
              ],
              "11"
            ]
          ],
          "55"
        ],
        [
          [
            "a2",
            "a3",
            "a4",
            "s"
          ],
          [
            [
              [
                "a2",
                "t"
              ],
              "11"
            ],
            [
              [
                "a3",
                "t"
              ],
              "12"
            ],
            [
              [
                "a4",
                "t"
              ],
              "13"
            ],
            [
              [
                "s",
                "a0"
              ],
              "9"
            ],
            [
              [
                "s",
                "a1"
              ],
              "10"
            ]
          ],
          "55"
        ],
        [
          [
            "a0",
            "a2",
            "a3",
            "a4",
            "s"
          ],
          [
            [
              [
                "a0",
                "t"
              ],
              "9"
            ],
            [
              [
                "a2",
                "t"
              ],
              "11"
            ],
            [
              [
                "a3",
                "t"
              ],
              "12"
            ],
            [
              [
                "a4",
                "t"
              ],
              "13"
            ],
            [
              [
                "s",
                "a1"
              ],
              "10"
            ]
          ],
          "55"
        ],
        [
          [
            "a1",
            "a2",
            "a3",
            "a4",
            "s"
          ],
          [
            [
              [
                "a1",
                "t"
              ],
              "10"
            ],
            [
              [
                "a2",
                "t"
              ],
              "11"
            ],
            [
              [
                "a3",
                "t"
              ],
              "12"
            ],
            [
              [
                "a4",
                "t"
              ],
              "13"
            ],
            [
              [
                "s",
                "a0"
              ],
              "9"
            ]
          ],
          "55"
        ],
        [
          [
            "a0",
            "a1",
            "a2",
            "a3",
            "a4",
            "s"
          ],
          [
            [
              [
                "a0",
                "t"
              ],
              "9"
            ],
            [
              [
                "a1",
                "t"
              ],
              "10"
            ],
            [
              [
                "a2",
                "t"
              ],
              "11"
            ],
            [
              [
                "a3",
                "t"
              ],
              "12"
            ],
            [
              [
                "a4",
                "t"
              ],
              "13"
            ]
          ],
          "55"
        ]
      ],
      32,
      32,
      "every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted"
    ]
  },
  "conclusion": {
    "canonical_value": "([a0, a1, a2, a3, a4, s, t], s, t, [[[s], [[[s, a0], 9], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, s], [[[a0, t], 9], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a1, s], [[[a1, t], 10], [[s, a0], 9], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, a1, s], [[[a0, t], 9], [[a1, t], 10], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a2, s], [[[a2, t], 11], [[s, a0], 9], [[s, a1], 10], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, a2, s], [[[a0, t], 9], [[a2, t], 11], [[s, a1], 10], [[s, a3], 12], [[s, a4], 13]], 55], [[a1, a2, s], [[[a1, t], 10], [[a2, t], 11], [[s, a0], 9], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, a1, a2, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a3, s], [[[a3, t], 12], [[s, a0], 9], [[s, a1], 10], [[s, a2], 11], [[s, a4], 13]], 55], [[a0, a3, s], [[[a0, t], 9], [[a3, t], 12], [[s, a1], 10], [[s, a2], 11], [[s, a4], 13]], 55], [[a1, a3, s], [[[a1, t], 10], [[a3, t], 12], [[s, a0], 9], [[s, a2], 11], [[s, a4], 13]], 55], [[a0, a1, a3, s], [[[a0, t], 9], [[a1, t], 10], [[a3, t], 12], [[s, a2], 11], [[s, a4], 13]], 55], [[a2, a3, s], [[[a2, t], 11], [[a3, t], 12], [[s, a0], 9], [[s, a1], 10], [[s, a4], 13]], 55], [[a0, a2, a3, s], [[[a0, t], 9], [[a2, t], 11], [[a3, t], 12], [[s, a1], 10], [[s, a4], 13]], 55], [[a1, a2, a3, s], [[[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[s, a0], 9], [[s, a4], 13]], 55], [[a0, a1, a2, a3, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[s, a4], 13]], 55], [[a4, s], [[[a4, t], 13], [[s, a0], 9], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12]], 55], [[a0, a4, s], [[[a0, t], 9], [[a4, t], 13], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12]], 55], [[a1, a4, s], [[[a1, t], 10], [[a4, t], 13], [[s, a0], 9], [[s, a2], 11], [[s, a3], 12]], 55], [[a0, a1, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a4, t], 13], [[s, a2], 11], [[s, a3], 12]], 55], [[a2, a4, s], [[[a2, t], 11], [[a4, t], 13], [[s, a0], 9], [[s, a1], 10], [[s, a3], 12]], 55], [[a0, a2, a4, s], [[[a0, t], 9], [[a2, t], 11], [[a4, t], 13], [[s, a1], 10], [[s, a3], 12]], 55], [[a1, a2, a4, s], [[[a1, t], 10], [[a2, t], 11], [[a4, t], 13], [[s, a0], 9], [[s, a3], 12]], 55], [[a0, a1, a2, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[a4, t], 13], [[s, a3], 12]], 55], [[a3, a4, s], [[[a3, t], 12], [[a4, t], 13], [[s, a0], 9], [[s, a1], 10], [[s, a2], 11]], 55], [[a0, a3, a4, s], [[[a0, t], 9], [[a3, t], 12], [[a4, t], 13], [[s, a1], 10], [[s, a2], 11]], 55], [[a1, a3, a4, s], [[[a1, t], 10], [[a3, t], 12], [[a4, t], 13], [[s, a0], 9], [[s, a2], 11]], 55], [[a0, a1, a3, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a3, t], 12], [[a4, t], 13], [[s, a2], 11]], 55], [[a2, a3, a4, s], [[[a2, t], 11], [[a3, t], 12], [[a4, t], 13], [[s, a0], 9], [[s, a1], 10]], 55], [[a0, a2, a3, a4, s], [[[a0, t], 9], [[a2, t], 11], [[a3, t], 12], [[a4, t], 13], [[s, a1], 10]], 55], [[a1, a2, a3, a4, s], [[[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[a4, t], 13], [[s, a0], 9]], 55], [[a0, a1, a2, a3, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[a4, t], 13]], 55]], 32, 32, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)",
    "explanation": {
      "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。",
      "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged."
    }
  },
  "definition_note": {
    "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。",
    "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate."
  },
  "source": {
    "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db",
    "question_binding": {
      "batch_id": "batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations",
      "batch_path": "public/data/questions/batches/batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations/questions.jsonl",
      "batch_sha256": "9fdcb690fb88ee203e6aef12d1ff12e4b22bd7b93b128b707a8eea9595c11ae5",
      "original_line": 8,
      "original_line_sha256": "54e21c59d1047e0f544e673d000d3576638b03296d4f548ec49c4eacf0966ac9",
      "question_compact_sha256": "4de1a0766af89515f3e3139cc4bb286b93c3bb78e17ccbbd974f613807abd4da",
      "english_row_compact_sha256": "e45adb63a9e014566fd804fea42e5250eefbba7cc28c26efc39d1a99fbc532fb",
      "english_manifest_path": "public/data/i18n/editions/batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-max-flow-min-cut-theorem": {
        "sha256": "3757bdd0ce86cc1644a36f998a60ce8dbf6971de426f7cf31249a035f64db37c",
        "english_sha256": "89aabf967cba45b9d682914827a0a0ceb7937f58c3977640ba0b4fb92fb3a30a",
        "original_path": "public/data/mko/max-flow-min-cut-theorem.json",
        "english_path": "public/data/i18n/editions/mko-max-flow-min-cut-theorem.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "4624c29c8cf22048332daea19f430f922e032c30410aa690ba3f1924d8b9ee45",
    "original_candidate_sha256": "ec4e01a803787526be4f9d32be763fac0a6ec0b0c787538f21b55380e72fc1c2",
    "original_candidate_line_sha256": "ec4e01a803787526be4f9d32be763fac0a6ec0b0c787538f21b55380e72fc1c2"
  },
  "historical_source_state": {
    "review_state": {
      "derivation": "candidate_authored_and_locally_checked",
      "source_question_state": "candidate",
      "independent_AI_or_human_review": "NOT_PERFORMED",
      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
    },
    "scope_note": {
      "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。",
      "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged."
    },
    "label": "Frozen producer metadata; current per-question review is separate below."
  },
  "revision": {
    "id": "review-20261026-010-p51",
    "review_date": "2026-10-26",
    "owner_authorized_reader_publication": true,
    "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.",
    "review_receipts": {
      "semantic_review_sha256": "74c63a2c12285a3b2073f8d9f102bc2ba5fb7ed890c5b7c3b8afa94f032eda82",
      "question_first_witness_sha256": "6d56428610f8e86c45fd00a2ad3b27ad9c052956946f0b91d71a68536f79d95f"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 67,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "5d5608af7eedbb23851c7720943df10269894e2ab7d80855fb46cecf748f70e6",
      "arithmetic_steps": 32
    },
    "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."
  }
}