{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261015-advance-20261113-r07-finite-flow-network-0031",
  "pilot_number": "R0047-24306",
  "number": 24306,
  "family_id": "qf-mod44-flow-capacity-audit",
  "title": {
    "zh-Hant": "第 24306 題：有限流網路",
    "en": "Question 24306: Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261015-advance-20261113-r07-finite-flow-network-0031",
    "batch_id": "batch-20261015-advance-20261113-r07-finite-flow-network-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-finite-flow-network"
    ],
    "title_zh": "有限流網路",
    "stem_zh": "finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-capacity-audit；資料={\"vertices\":[\"s5\",\"a5\",\"t5\",\"u5\"],\"edges\":[[\"s5\",\"a5\"],[\"a5\",\"s5\"],[\"a5\",\"t5\"],[\"t5\",\"a5\"]],\"source\":\"s5\",\"sink\":\"t5\",\"capacities\":[30,30,30,30],\"flows\":[24,12,18,6]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(s5, t5, [[[a5, s5], 30, 12], [[a5, t5], 30, 18], [[s5, a5], 30, 24], [[t5, a5], 30, 6]], [[[a5, s5], 12, 30, 是], [[a5, t5], 18, 30, 是], [[s5, a5], 24, 30, 是], [[t5, a5], 6, 30, 是]], 是, null, every_edge_requires_zero_less_equal_flow_less_equal_capacity)",
      "explanation_zh": "Wave6第七層核對=(s5, t5, [[[a5, s5], 30, 12], [[a5, t5], 30, 18], [[s5, a5], 30, 24], [[t5, a5], 30, 6]], [[[a5, s5], 12, 30, 是], [[a5, t5], 18, 30, 是], [[s5, a5], 24, 30, 是], [[t5, a5], 6, 30, 是]], 是, null, every_edge_requires_zero_less_equal_flow_less_equal_capacity)；依完整明示資料獨立重算；antiparallel edges明示，zero capacity與disconnected vertices允許；flow value由source與sink交叉計算；feasible不等於maximum，不提前宣稱min-cut equality或augmenting-path optimality。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "有限流網路"
      ]
    },
    "variant": {
      "family_id": "qf-mod44-flow-capacity-audit",
      "seed": 202611137031,
      "parameters": {
        "vertices": [
          "s5",
          "a5",
          "t5",
          "u5"
        ],
        "edges": [
          [
            "s5",
            "a5"
          ],
          [
            "a5",
            "s5"
          ],
          [
            "a5",
            "t5"
          ],
          [
            "t5",
            "a5"
          ]
        ],
        "source": "s5",
        "sink": "t5",
        "capacities": [
          30,
          30,
          30,
          30
        ],
        "flows": [
          24,
          12,
          18,
          6
        ],
        "domain": "Exact_finite_flow_network",
        "operation": "flow-capacity-audit",
        "semantic_template": "qf-mod44-flow-capacity-audit/{\"capacities\":[[30,1],[30,1],[30,1],[30,1]],\"edges\":[[\"a5\",\"s5\"],[\"a5\",\"t5\"],[\"s5\",\"a5\"],[\"t5\",\"a5\"]],\"flows\":[[12,1],[18,1],[24,1],[6,1]],\"sink\":\"t5\",\"source\":\"s5\",\"vertices\":[\"a5\",\"s5\",\"t5\",\"u5\"]}"
      }
    },
    "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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r07-finite-flow-network-0031",
    "batch_id": "batch-20261015-advance-20261113-r07-finite-flow-network-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "9441d7507b1c6eb2597a16800ae5c7a1080a5072d8faadac85e58a3a7423db1a",
      "family_id": "qf-mod44-flow-capacity-audit",
      "seed": 202611137031,
      "semantic_template": "qf-mod44-flow-capacity-audit/{\"capacities\":[[30,1],[30,1],[30,1],[30,1]],\"edges\":[[\"a5\",\"s5\"],[\"a5\",\"t5\"],[\"s5\",\"a5\"],[\"t5\",\"a5\"]],\"flows\":[[12,1],[18,1],[24,1],[6,1]],\"sink\":\"t5\",\"source\":\"s5\",\"vertices\":[\"a5\",\"s5\",\"t5\",\"u5\"]}"
    },
    "fields": {
      "title": "Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity",
      "stem": "Use vertices=[s5, a5, t5, u5], ordered directed edges=[[s5, a5], [a5, s5], [a5, t5], [t5, a5]], distinct source=s5, sink=t5, aligned capacities=[30, 30, 30, 30] and candidate flows=[24, 12, 18, 6]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, capacity_rows, capacity_feasible, first_failed_edge_or_null, certificate).",
      "answer": {
        "kind": "expression",
        "value": "(s5, t5, [[[a5, s5], 30, 12], [[a5, t5], 30, 18], [[s5, a5], 30, 24], [[t5, a5], 30, 6]], [[[a5, s5], 12, 30, Yes], [[a5, t5], 18, 30, Yes], [[s5, a5], 24, 30, Yes], [[t5, a5], 6, 30, Yes]], Yes, null, every_edge_requires_zero_less_equal_flow_less_equal_capacity)",
        "explanation": "Keep network rows [edge,capacity,flow] distinct from audit rows [edge,flow,capacity,pass]. Report the first failing edge in canonical order, or null. Negative/over-capacity candidates are rejected, not rewritten; local capacity success alone does not establish conservation. The exact result, retaining the requested order, is (s5, t5, [[[a5, s5], 30, 12], [[a5, t5], 30, 18], [[s5, a5], 30, 24], [[t5, a5], 30, 6]], [[[a5, s5], 12, 30, Yes], [[a5, t5], 18, 30, Yes], [[s5, a5], 24, 30, Yes], [[t5, a5], 6, 30, Yes]], Yes, null, every_edge_requires_zero_less_equal_flow_less_equal_capacity)."
      },
      "skills": [
        "Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity"
      ],
      "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."
    }
  },
  "givens": [
    {
      "id": "G001",
      "text": {
        "zh-Hant": "finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-capacity-audit；資料={\"vertices\":[\"s5\",\"a5\",\"t5\",\"u5\"],\"edges\":[[\"s5\",\"a5\"],[\"a5\",\"s5\"],[\"a5\",\"t5\"],[\"t5\",\"a5\"]],\"source\":\"s5\",\"sink\":\"t5\",\"capacities\":[30,30,30,30],\"flows\":[24,12,18,6]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use vertices=[s5, a5, t5, u5], ordered directed edges=[[s5, a5], [a5, s5], [a5, t5], [t5, a5]], distinct source=s5, sink=t5, aligned capacities=[30, 30, 30, 30] and candidate flows=[24, 12, 18, 6]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, capacity_rows, capacity_feasible, first_failed_edge_or_null, certificate)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [flow-capacity-audit]",
      "after": "derived = [[source, sink, edge_capacity_flow_rows, capacity_rows, capacity_feasible, first_failed_edge_or_null, certificate]]",
      "relation": "implies",
      "rule_id": "source_ordered_fields_without_JS",
      "conditions": [],
      "explanation": {
        "zh-Hant": "本頁列完整來源欄序，不需JavaScript；gradient boundary依原mode選6／8／7欄，不補欄也不由答案猜mode。",
        "en": "List the source field order without JavaScript.Gradient boundary uses its original mode to choose6,8 or7 fields;no padding or answer-derived mode."
      },
      "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 = [[s5, a5, t5, u5], [[s5, a5], [a5, s5], [a5, t5], [t5, a5]], [30, 30, 30, 30], [24, 12, 18, 6], s5, t5]",
      "after": "derived = [[a5, s5, t5, u5], [[[a5, s5], 30, 12], [[a5, t5], 30, 18], [[s5, a5], 30, 24], [[t5, a5], 30, 6]], 2, 是]",
      "relation": "implies",
      "rule_id": "Flow_zip_source_edges_capacities_flows_BEFORE_order",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先依原index配好edge／capacity／flow，再按來源English canonical(edge)序排；vertices用UTF16。反向邊是不同edge，zero容量與孤立點合法，負flow保留供容量判失敗，不修改。",
        "en": "Zip edge,capacity and flow by original index before source English canonical-edge sorting;vertices use UTF16.Opposite directions are distinct,zero capacity and isolated vertices are legal,and negative flow is retained for capacity failure rather than rewritten."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "source roles = [0, [a5, s5], 12, 30]",
      "after": "derived = [是, 是, 是]",
      "relation": "implies",
      "rule_id": "Flow_EACH_edge_two_sided_capacity_obligation",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每條有向邊都要0<=flow<=capacity；網路signature合格並不保證候選flow可行。",
        "en": "Every directed edge needs0<=flow<=capacity.A qualified network signature does not imply candidate-flow feasibility."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "source roles = [1, [a5, t5], 18, 30]",
      "after": "derived = [是, 是, 是]",
      "relation": "implies",
      "rule_id": "Flow_EACH_edge_two_sided_capacity_obligation",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每條有向邊都要0<=flow<=capacity；網路signature合格並不保證候選flow可行。",
        "en": "Every directed edge needs0<=flow<=capacity.A qualified network signature does not imply candidate-flow feasibility."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "source roles = [2, [s5, a5], 24, 30]",
      "after": "derived = [是, 是, 是]",
      "relation": "implies",
      "rule_id": "Flow_EACH_edge_two_sided_capacity_obligation",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每條有向邊都要0<=flow<=capacity；網路signature合格並不保證候選flow可行。",
        "en": "Every directed edge needs0<=flow<=capacity.A qualified network signature does not imply candidate-flow feasibility."
      },
      "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": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "source roles = [3, [t5, a5], 6, 30]",
      "after": "derived = [是, 是, 是]",
      "relation": "implies",
      "rule_id": "Flow_EACH_edge_two_sided_capacity_obligation",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每條有向邊都要0<=flow<=capacity；網路signature合格並不保證候選flow可行。",
        "en": "Every directed edge needs0<=flow<=capacity.A qualified network signature does not imply candidate-flow feasibility."
      },
      "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": "Flow ALL incoming at vertex a5 := sum(24, 6)",
      "after": "Flow ALL incoming at vertex a5 = 30",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Flow ALL incoming at vertex a5，不以小數近似替代。",
          "en": "Recompute Flow ALL incoming at vertex a5 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "24",
          "6"
        ],
        "output": "30",
        "label": "Flow ALL incoming at vertex a5"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "Flow ALL outgoing at vertex a5 := sum(12, 18)",
      "after": "Flow ALL outgoing at vertex a5 = 30",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Flow ALL outgoing at vertex a5，不以小數近似替代。",
          "en": "Recompute Flow ALL outgoing at vertex a5 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "12",
          "18"
        ],
        "output": "30",
        "label": "Flow ALL outgoing at vertex a5"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "source roles = [a5, [2, 3], [0, 1]]",
      "after": "derived = [30, 30, 是, 是]",
      "relation": "implies",
      "rule_id": "Flow_EACH_vertex_complete_directional_incidence",
      "conditions": [],
      "explanation": {
        "zh-Hant": "入邊／出邊各依完整清單加總；孤立點是0=0。守恆只要求internal vertices，源匯仍計完整balance以求net值。",
        "en": "Sum all incoming and outgoing edges separately;an isolated vertex has0=0.Conservation is required only internally,but compute source/sink balances too for net values."
      },
      "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": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "Flow ALL incoming at vertex s5 := sum(12)",
      "after": "Flow ALL incoming at vertex s5 = 12",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Flow ALL incoming at vertex s5，不以小數近似替代。",
          "en": "Recompute Flow ALL incoming at vertex s5 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "12"
        ],
        "output": "12",
        "label": "Flow ALL incoming at vertex s5"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "Flow ALL outgoing at vertex s5 := sum(24)",
      "after": "Flow ALL outgoing at vertex s5 = 24",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Flow ALL outgoing at vertex s5，不以小數近似替代。",
          "en": "Recompute Flow ALL outgoing at vertex s5 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "24"
        ],
        "output": "24",
        "label": "Flow ALL outgoing at vertex s5"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "source roles = [s5, [0], [2]]",
      "after": "derived = [12, 24, 否, 否]",
      "relation": "implies",
      "rule_id": "Flow_EACH_vertex_complete_directional_incidence",
      "conditions": [],
      "explanation": {
        "zh-Hant": "入邊／出邊各依完整清單加總；孤立點是0=0。守恆只要求internal vertices，源匯仍計完整balance以求net值。",
        "en": "Sum all incoming and outgoing edges separately;an isolated vertex has0=0.Conservation is required only internally,but compute source/sink balances too for net values."
      },
      "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": "Flow ALL incoming at vertex t5 := sum(18)",
      "after": "Flow ALL incoming at vertex t5 = 18",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Flow ALL incoming at vertex t5，不以小數近似替代。",
          "en": "Recompute Flow ALL incoming at vertex t5 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "18"
        ],
        "output": "18",
        "label": "Flow ALL incoming at vertex t5"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "Flow ALL outgoing at vertex t5 := sum(6)",
      "after": "Flow ALL outgoing at vertex t5 = 6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Flow ALL outgoing at vertex t5，不以小數近似替代。",
          "en": "Recompute Flow ALL outgoing at vertex t5 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "6"
        ],
        "output": "6",
        "label": "Flow ALL outgoing at vertex t5"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "source roles = [t5, [1], [3]]",
      "after": "derived = [18, 6, 否, 否]",
      "relation": "implies",
      "rule_id": "Flow_EACH_vertex_complete_directional_incidence",
      "conditions": [],
      "explanation": {
        "zh-Hant": "入邊／出邊各依完整清單加總；孤立點是0=0。守恆只要求internal vertices，源匯仍計完整balance以求net值。",
        "en": "Sum all incoming and outgoing edges separately;an isolated vertex has0=0.Conservation is required only internally,but compute source/sink balances too for net values."
      },
      "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": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "Flow ALL incoming at vertex u5 := sum()",
      "after": "Flow ALL incoming at vertex u5 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Flow ALL incoming at vertex u5，不以小數近似替代。",
          "en": "Recompute Flow ALL incoming at vertex u5 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [],
        "output": "0",
        "label": "Flow ALL incoming at vertex u5"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "Flow ALL outgoing at vertex u5 := sum()",
      "after": "Flow ALL outgoing at vertex u5 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Flow ALL outgoing at vertex u5，不以小數近似替代。",
          "en": "Recompute Flow ALL outgoing at vertex u5 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [],
        "output": "0",
        "label": "Flow ALL outgoing at vertex u5"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "source roles = [u5, [], []]",
      "after": "derived = [0, 0, 是, 是]",
      "relation": "implies",
      "rule_id": "Flow_EACH_vertex_complete_directional_incidence",
      "conditions": [],
      "explanation": {
        "zh-Hant": "入邊／出邊各依完整清單加總；孤立點是0=0。守恆只要求internal vertices，源匯仍計完整balance以求net值。",
        "en": "Sum all incoming and outgoing edges separately;an isolated vertex has0=0.Conservation is required only internally,but compute source/sink balances too for net values."
      },
      "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": "Flow source NET outflow not gross outgoing := sub(24, 12)",
      "after": "Flow source NET outflow not gross outgoing = 12",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Flow source NET outflow not gross outgoing，不以小數近似替代。",
          "en": "Recompute Flow source NET outflow not gross outgoing with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "24",
          "12"
        ],
        "output": "12",
        "label": "Flow source NET outflow not gross outgoing"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "Flow sink NET inflow not gross incoming := sub(18, 6)",
      "after": "Flow sink NET inflow not gross incoming = 12",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Flow sink NET inflow not gross incoming，不以小數近似替代。",
          "en": "Recompute Flow sink NET inflow not gross incoming with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "18",
          "6"
        ],
        "output": "12",
        "label": "Flow sink NET inflow not gross incoming"
      }
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "source roles = [s5, t5, [[[a5, s5], 12, 30, 是], [[a5, t5], 18, 30, 是], [[s5, a5], 24, 30, 是], [[t5, a5], 6, 30, 是]], [[a5, 30, 30, 是], [u5, 0, 0, 是]], 12, 12]",
      "after": "derived = [是, 是, 是, 是, 否]",
      "relation": "implies",
      "rule_id": "Flow_value_capacity_conservation_and_maximum_separate",
      "conditions": [],
      "explanation": {
        "zh-Hant": "全vertex總in−out中每edge相消，故全部internal守恆必得兩net相等；反向不成立，internal imbalance可互抵。容量、逐點守恆、兩net值分別核，feasible不推出maximum/min-cut。最後No是不提供最優證書，不否定特例恰好最優。",
        "en": "Every edge cancels in the sum of all vertex balances,so internal conservation implies equal source/sink net values.The converse fails when internal imbalances cancel.Capacity,each balance and net equality remain separate.Feasibility does not certify maximum/min-cut;No does not deny special optimal examples."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      "s5",
      "t5",
      [
        [
          [
            "a5",
            "s5"
          ],
          "30",
          "12"
        ],
        [
          [
            "a5",
            "t5"
          ],
          "30",
          "18"
        ],
        [
          [
            "s5",
            "a5"
          ],
          "30",
          "24"
        ],
        [
          [
            "t5",
            "a5"
          ],
          "30",
          "6"
        ]
      ],
      [
        [
          [
            "a5",
            "s5"
          ],
          "12",
          "30",
          true
        ],
        [
          [
            "a5",
            "t5"
          ],
          "18",
          "30",
          true
        ],
        [
          [
            "s5",
            "a5"
          ],
          "24",
          "30",
          true
        ],
        [
          [
            "t5",
            "a5"
          ],
          "6",
          "30",
          true
        ]
      ],
      true,
      null,
      "every_edge_requires_zero_less_equal_flow_less_equal_capacity"
    ]
  },
  "conclusion": {
    "canonical_value": "(s5, t5, [[[a5, s5], 30, 12], [[a5, t5], 30, 18], [[s5, a5], 30, 24], [[t5, a5], 30, 6]], [[[a5, s5], 12, 30, 是], [[a5, t5], 18, 30, 是], [[s5, a5], 24, 30, 是], [[t5, a5], 6, 30, 是]], 是, null, every_edge_requires_zero_less_equal_flow_less_equal_capacity)",
    "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-20261113-r07-finite-flow-network-foundations",
      "batch_path": "public/data/questions/batches/batch-20261015-advance-20261113-r07-finite-flow-network-foundations/questions.jsonl",
      "batch_sha256": "41cbc89643a2ed064435cd6f3f92fc72d62249d452101a67f47d6a4ac53a089b",
      "original_line": 31,
      "original_line_sha256": "d59e679b160fd25d98eea25775097dd7127cf5ce4e1cfbbdb0192bb0850b4cbe",
      "question_compact_sha256": "9441d7507b1c6eb2597a16800ae5c7a1080a5072d8faadac85e58a3a7423db1a",
      "english_row_compact_sha256": "a31e94cd3cd259a37669320e129abc6c7355f0e739573902666c8a483a40b02b",
      "english_manifest_path": "public/data/i18n/editions/batch-20261015-advance-20261113-r07-finite-flow-network-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261015-advance-20261113-r07-finite-flow-network-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-finite-flow-network": {
        "sha256": "c6cc774f53f5114e2b238ff77e24ed61a2adc5f2d6d7ffa5a969b25f287ad5d0",
        "english_sha256": "eaeb9873e61c9b144bb18d9037f62d7125afdee46da133467f5a106dbddc04b7",
        "original_path": "public/data/mko/finite-flow-network.json",
        "english_path": "public/data/i18n/editions/mko-finite-flow-network.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "b3aad195dad743751e4c8a4a6ce44f091c3824da1e52c21e9266d4ccd1dc1b68",
    "original_candidate_sha256": "d15fc68cd3130ff7f6546af1c5fb08a4c73c8bc1715325311cf7ef50686f61f5",
    "original_candidate_line_sha256": "d15fc68cd3130ff7f6546af1c5fb08a4c73c8bc1715325311cf7ef50686f61f5"
  },
  "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-20261026-009-p48",
    "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": "9b43b9823e9bd2478d6caa5b57ffed161c2c8956af86974e901d5662ad994386",
      "question_first_witness_sha256": "eb97596068e2d5648033bab2b8f6ca9b653bbe8cb77c0f8e1d16d0c40e46f50b"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 21,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "d518e47f65833e23b087373fad8ab28dac9807bee6165169b55b7659859134f9",
      "arithmetic_steps": 10
    },
    "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."
  }
}