{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-max-flow-min-cut-theorem-01",
  "mko_id": "mko-max-flow-min-cut-theorem",
  "title": {
    "zh-Hant": "完整四個 cut 與等值雙證書",
    "en": "All four cuts and an equal-value two-sided certificate"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-max-flow-min-cut-theorem-01",
    "source": {
      "mko_id": "mko-max-flow-min-cut-theorem",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "3757bdd0ce86cc1644a36f998a60ce8dbf6971de426f7cf31249a035f64db37c",
      "english_sha256": "89aabf967cba45b9d682914827a0a0ceb7937f58c3977640ba0b4fb92fb3a30a"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "algebra",
      "sets",
      "flow_def",
      "cut_def"
    ],
    "goal": {
      "zh-Hant": "枚舉四點網路全部 s-t cuts，給值 5 的可行流與同值 cut。",
      "en": "Enumerate every s-t cut of a four-vertex network and give a feasible flow of value 5 together with a cut of capacity 5."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "V={s,a,b,t}；邊依序 sa,sb,ab,at,bt,ba，容量 (3,2,1,2,3,1)，流 (3,2,1,2,3,0)。",
          "en": "V={s,a,b,t}; edges sa,sb,ab,at,bt,ba in that order have capacities (3,2,1,2,3,1) and flows (3,2,1,2,3,0)."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1"
        ],
        "before_tex": "E=\\{sa,sb,ab,at,bt,ba\\},\\quad c=(3,2,1,2,3,1)",
        "after_tex": "\\text{all edges loop-free and unique},\\quad0\\le f\\le c",
        "relation": "implies",
        "rule": "flow_def",
        "conditions": [],
        "explanation": {
          "zh-Hant": "完整六邊無重複有向對，反平行合法，逐邊核流容量界。",
          "en": "All six directed pairs are unique and loop-free, antiparallel edges are legal, and every flow obeys its capacity."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "網路資格及每邊可行性先行。",
            "en": "Network qualification and per-edge feasibility come first."
          }
        },
        "leaf_basis": "flow_def"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "S01"
        ],
        "before_tex": "\\operatorname{in}(a)=3+0,\\quad\\operatorname{in}(b)=2+1",
        "after_tex": "\\operatorname{out}(a)=1+2=3,\\quad\\operatorname{out}(b)=3+0=3",
        "relation": "implies",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "完整兩內點守恆，流確可行。",
          "en": "Both internal vertices conserve flow, qualifying the flow as feasible."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01"
          ],
          "explanation": {
            "zh-Hant": "ba 必在 a 入流及 b 出流中保留。",
            "en": "ba is retained in a's inflow and b's outflow."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S03",
        "uses": [
          "G1",
          "S02"
        ],
        "before_tex": "|f|=f_{sa}+f_{sb}=3+2",
        "after_tex": "|f|=5=f_{at}+f_{bt}=2+3",
        "relation": "implies",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "來源與終點淨值一致。",
          "en": "Source and sink net values agree."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S02"
          ],
          "explanation": {
            "zh-Hant": "所有內點守恆已核，非只算端點。",
            "en": "Every internal conservation check precedes the endpoint value computation."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S04",
        "uses": [
          "G1"
        ],
        "before_tex": "s\\in S,\\quad t\\notin S",
        "after_tex": "S\\in\\{\\{s\\},\\{s,a\\},\\{s,b\\},\\{s,a,b\\}\\}",
        "relation": "definition",
        "rule": "cut_def",
        "conditions": [],
        "explanation": {
          "zh-Hant": "a,b 各可在或不在 S，完整列四種而不混補集同一 cut。",
          "en": "a,b may independently belong to S, giving exactly four subsets without confusing complementary sides."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "s,t 歸屬固定，故 2^2 種已完整。",
            "en": "The fixed membership of s,t leaves exactly 2^2 choices."
          }
        },
        "leaf_basis": "cut_def"
      },
      {
        "id": "S05",
        "uses": [
          "S01",
          "S02",
          "S04"
        ],
        "before_tex": "|h|=\\sum_{e:S\\to S^c}h_e-\\sum_{e:S^c\\to S}h_e\\quad(h\\text{ feasible})",
        "after_tex": "|h|\\le\\sum_{e:S\\to S^c}h_e\\le\\sum_{e:S\\to S^c}c_e=c(S)",
        "relation": "implies",
        "rule": "algebra",
        "conditions": [
          "h\\text{ feasible},\\ s\\in S,\\ t\\notin S"
        ],
        "explanation": {
          "zh-Hant": "加總 S 內點的淨流，內部邊抵銷，只留 cut 淨流；入流非負和容量界給全可行流弱界。",
          "en": "Sum divergences over S: internal edges cancel, leaving cut net flow. Nonnegative incoming flow and capacities give the bound for every feasible flow."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S01",
            "S02",
            "S04"
          ],
          "explanation": {
            "zh-Hant": "弱界不假定 h 為主候選 f，才可用來證最優。",
            "en": "The weak bound applies to arbitrary h, not only the candidate f, enabling optimality."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S06",
        "uses": [
          "G1",
          "S04"
        ],
        "before_tex": "c(\\{s\\}),c(\\{s,a\\}),c(\\{s,b\\}),c(\\{s,a,b\\})",
        "after_tex": "3+2=5,\\quad2+1+2=5,\\quad3+3+1=7,\\quad2+3=5",
        "relation": "implies",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "只數向外邊。對 {s,a}，ba 是流入不計；對 {s,b}，ba 才是向外並計容量 1。",
          "en": "Count only outgoing edges. For {s,a}, ba is incoming and excluded; for {s,b}, ba is outgoing and contributes one."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S04"
          ],
          "explanation": {
            "zh-Hant": "四個合法 cut 全算完且方向逐一明示。",
            "en": "All four cuts are computed with directions explicitly checked."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S07",
        "uses": [
          "S04",
          "S06"
        ],
        "before_tex": "\\{c(S):s\\in S,t\\notin S\\}=\\{5,5,7,5\\}",
        "after_tex": "\\min_S c(S)=5\\text{ attained at }S=\\{s\\}",
        "relation": "implies",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "完整清單最小值為 5，可選來源單點 cut 當證書。",
          "en": "The complete list has minimum five, attained by the source-singleton cut."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S04",
            "S06"
          ],
          "explanation": {
            "zh-Hant": "不是因單獨找到一個容量 5 cut 就稱最小；完整表或弱界才支持。",
            "en": "A lone cut of capacity five would not prove minimality; the complete table or weak bound is needed."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S08",
        "uses": [
          "S03",
          "S05",
          "S07"
        ],
        "before_tex": "|f|=5=c(\\{s\\})",
        "after_tex": "f\\text{ maximum},\\quad\\{s\\}\\text{ minimum cut},\\quad\\max|h|=\\min c(S)=5",
        "relation": "implies",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "任意可行 h≤此 cut 5，候選 f 達到 5；任意 cut 又≥f 值，兩邊同時最優。",
          "en": "Every feasible h is bounded by the cut's five and f attains it; every cut is at least f's value, certifying both sides optimal."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S03",
            "S05",
            "S07"
          ],
          "explanation": {
            "zh-Hant": "這是給定有限網路的等值雙證書，不宣稱算法或一般整數流性。",
            "en": "This is an equal-value certificate for the supplied finite network, not an algorithm or general integral-flow claim."
          }
        },
        "leaf_basis": "algebra"
      }
    ],
    "result": {
      "zh-Hant": "最大流與最小 cut 值皆 5；全部四個 cut 為 5、5、7、5。",
      "en": "Maximum flow and minimum cut both have value five; all four cut capacities are 5,5,7,5."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "cut 兩個方向容量都加。",
          "en": "Add capacities in both cut directions."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "ba 在 {s,a} cut 是入邊，不能算入。",
          "en": "ba is incoming for the {s,a} cut and must be excluded."
        }
      },
      {
        "claim": {
          "zh-Hant": "任何可行流已最大。",
          "en": "Any feasible flow is maximum."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "只有與 cut 弱界達同值的候選才由此證最優。",
          "en": "This proof certifies a candidate only when it attains the cut weak bound."
        }
      },
      {
        "claim": {
          "zh-Hant": "此實例證所有 augmenting-path 演算法正確。",
          "en": "This instance proves every augmenting-path algorithm correct."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "本稿沒有算法或一般終止證明。",
          "en": "No algorithm or general termination proof is provided."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "mko_definition",
        "kind": "cited",
        "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。",
          "en": "Enumerate each vertex subset S containing the source and excluding the sink. Its capacity sums only directed edges from S to its complement; incoming edges are not counted. Every feasible flow value is at most every cut capacity. Complete cut enumeration determines the minimum cut. If a feasible flow and a cut have equal values, this bound immediately certifies both a maximum flow and a minimum cut. An arbitrary feasible flow or single cut is not automatically optimal. This slice does not claim integral flow, an augmenting-path algorithm proof or a general theorem proof from finitely many instances."
        },
        "source": {
          "mko_id": "mko-max-flow-min-cut-theorem",
          "sha256": "3757bdd0ce86cc1644a36f998a60ce8dbf6971de426f7cf31249a035f64db37c",
          "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f"
        }
      },
      {
        "id": "algebra",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "實數域的等式運算；除法前分母必非零，不等式乘正數保持方向。",
          "en": "Use real-field identities; division requires a nonzero denominator and multiplication by a positive number preserves inequalities."
        }
      },
      {
        "id": "sets",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "集合的聯、交、補集、包含與函數逆像以逐元素定義計算。",
          "en": "Compute unions, intersections, complements, inclusion and preimages element by element."
        }
      },
      {
        "id": "flow_def",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "有限簡單有向網路無 self-loop、無重複 ordered pair，但容許反平行邊；可行流滿足每邊 0≤f≤c 及每內點流入等於流出，值為來源淨流出及終點淨流入。",
          "en": "A finite simple directed network has no self-loop or duplicate ordered pair, but permits antiparallel edges; feasible flows satisfy every capacity bound and internal conservation, with value equal to source net outflow and sink net inflow."
        }
      },
      {
        "id": "cut_def",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "合法 s-t cut 是含 s 不含 t 的頂點子集 S；容量只加 S 到其補集的有向容量。",
          "en": "A legal s-t cut is a vertex subset S containing s and excluding t; its capacity sums only directed capacities from S to its complement."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "cut-sa",
        "lhs": "2+1+2",
        "rhs": "5",
        "step": "S06"
      },
      {
        "id": "cut-sb",
        "lhs": "3+3+1",
        "rhs": "7",
        "step": "S06"
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "有限完整證書；不證 augmenting-path、整數性或用實例取代一般定理。",
      "en": "A complete finite certificate, without augmenting-path correctness, integrality or a general theorem from instances."
    },
    "md_symbols": [
      [
        "S^c",
        "相對 V 的 cut 補集",
        "cut complement relative to V"
      ],
      [
        "c(S)",
        "只計向外邊的容量",
        "outgoing-only cut capacity"
      ]
    ],
    "md_error_controls": [
      {
        "title": "cut 方向 / Cut direction",
        "zh": "只算 S→V\\S，不算反向。",
        "en": "Count only S to its complement, not incoming edges."
      },
      {
        "title": "任意比較流 / Arbitrary competing flow",
        "zh": "弱界量化全部可行 h，才可證 maximum。",
        "en": "The weak bound must cover every feasible h to prove a maximum."
      }
    ],
    "md_open_items": []
  },
  "revision": {
    "id": "ocme-new142-reader-adoption-20261021",
    "collection": "coverage-20261021",
    "source_commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
    "adopted_source_sha256": "a3f6ca7038c79f9eea7d99baeadfe28a147465af5256279afd33c427ad4be560",
    "original_source_sha256": "05cf7399183d2ca5c249a6bc36b388edd4cd3634a6674052196c12852140714a",
    "owner_adopted": true,
    "bilingual_sync": "complete_source_bound",
    "formal_Lean": "NOT_RUN",
    "reader_review": "142_of_142_source_bound",
    "wording_review": "original_Lead_decisions_resolved",
    "glossary_reader_paraphrases": [],
    "independent_AI_concurrence": "SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW",
    "issue_ids": [
      "BR142-P073"
    ],
    "parent_manifest_sha256": "d29f4cae8229ebd24f60d496cb1d3cd4ad8c0a4dca9d44daab0c3245c544283a",
    "scope": "Owner-adopted reader revision. Lead and bilingual review concern the recorded text and premises; no new universal or Lean proof, canonical promotion or historical-label rewrite."
  }
}