{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-max-flow-min-cut-theorem",
  "semantic_family_id": "sf-max-flow-min-cut-theorem",
  "type": "theorem",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "最大流最小割定理"
  },
  "summary": {
    "zh-Hant": "有限容量網路的最大feasible flow value等於全部s–t cuts中的最小capacity。"
  },
  "statement": {
    "zh-Hant": "在完整finite flow network上枚舉每個含source、不含sink的vertex subset S；cut capacity只加總S→V\\S的directed edges，反向edges不計。任一feasible flow對任一cut皆滿足value≤capacity。完整cut enumeration決定minimum cut；若一個feasible flow與一個cut同值，weak bound立即同時證flow maximum與cut minimum。單一feasible flow或單一cut不自動optimal；本輪不宣稱integral-flow、augmenting-path算法或以有限instances取代一般定理證明。"
  },
  "formula": {
    "tex": "F(N)=C(N)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mrow><mi>F</mi><mo>(</mo><mi>N</mi><mo>)</mo></mrow><mo>=</mo><mrow><mi>C</mi><mo>(</mo><mi>N</mi><mo>)</mo></mrow></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "F",
          "glyph": "F"
        },
        "arguments": [
          {
            "type": "symbol",
            "name": "N",
            "glyph": "N"
          }
        ]
      },
      "rhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "C",
          "glyph": "C"
        },
        "arguments": [
          {
            "type": "symbol",
            "name": "N",
            "glyph": "N"
          }
        ]
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "c07d6ce4ab62ef2547ff149585d34adef2d2dcb91bc0196c033154663e97aeeb"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "使用完整有限directed network、nonnegative exact rational capacities與flows。"
    },
    {
      "id": "a2",
      "zh-Hant": "頂點數有界以完整枚舉全部s–t cuts。"
    }
  ],
  "symbols": [
    {
      "token": "F",
      "role": "maximum_flow_value",
      "role_zh": "最大流值",
      "scope": "representative_formula"
    },
    {
      "token": "N",
      "role": "finite_flow_network",
      "role_zh": "有限流網路",
      "scope": "representative_formula"
    },
    {
      "token": "C",
      "role": "minimum_cut_capacity",
      "role_zh": "最小割容量",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-finite-flow-network",
      "title_zh": "有限流網路",
      "reason_zh": "feasible flow與value。"
    },
    {
      "id": "mko-optimization-problem",
      "title_zh": "最佳化問題",
      "reason_zh": "maximum/minimum語義。"
    },
    {
      "id": "mko-linear-programming",
      "title_zh": "線性規劃",
      "reason_zh": "一般定理背景。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "完整枚舉全部合法s–t cuts。",
      "flow≤cut以capacity與conservation逐cut核對。",
      "同值pair給雙optimality certificate。"
    ],
    "common_misconception_zh": "把反向edge算入cut capacity，或從任一feasible flow／cut直接推optimal。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "有限枚舉只證指定網路；不提前宣稱整數性或特定演算法正確性。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "加入augmenting paths與Ford–Fulkerson邊界。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-max-flow-min-cut-theorem.md 整理；2026-10-15 脈衝 R05（使用者手動接續 2026-11-15 工作序列；實際日期仍為 2026-10-15），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-15"
  }
}