{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-maximum-matching-01",
  "mko_id": "mko-maximum-matching",
  "title": {
    "zh-Hant": "三邊圖的全部八個matching候選",
    "en": "All eight edge-subset candidates on a three-edge graph"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-maximum-matching-01",
    "source": {
      "mko_id": "mko-maximum-matching",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "bb97ee67fcffa420b79baaaf56c7ebabe2977fb5d14f5826d1f1c0452f4ed7f9",
      "english_sha256": "62ea84a565cbbc9bfd9f80ccaf5cdd5474c7027eca349c25cded547f5fbe5886"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "work"
    ],
    "goal": {
      "zh-Hant": "完整2³枚舉求最大matching與全部witness。",
      "en": "Enumerate all2³ subsets for the maximum matching and all witnesses."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "V={0,1,2,3}，E由e0={0,1}、e1={1,2}、e2={2,3}構成；bit mask順序為(e0,e1,e2)。",
          "en": "V={0,1,2,3}; E consists of e0={0,1}, e1={1,2}, e2={2,3}; bit masks use order(e0,e1,e2)."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1",
          "work"
        ],
        "before_tex": "|E|=3",
        "after_tex": "|\\mathcal P(E)|=2^3=8",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "每邊選或不選，恰八個候選。",
          "en": "Each edge is selected or not, giving exactly eight candidates."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "work"
          ],
          "explanation": {
            "zh-Hant": "mask皆為不同子集合，不是抽樣。",
            "en": "Every mask is a distinct subset; none are samples."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "S01",
          "work"
        ],
        "before_tex": "000",
        "after_tex": "\\varnothing\\text{ valid, size }0",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "空集合合法matching，基數零。",
          "en": "The empty set is a valid matching of size zero."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01",
            "work"
          ],
          "explanation": {
            "zh-Hant": "不可漏掉empty case。",
            "en": "The empty case must be retained."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S03",
        "uses": [
          "G1",
          "S02",
          "work"
        ],
        "before_tex": "100,010,001",
        "after_tex": "\\{e_0\\},\\{e_1\\},\\{e_2\\}\\text{ valid, size }1",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "三個singleton均合法，已核原圖邊存在。",
          "en": "All three singletons are valid existing edges."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S02",
            "work"
          ],
          "explanation": {
            "zh-Hant": "共有三個而非任選一個作代表。",
            "en": "All three are listed, not one representative sample."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S04",
        "uses": [
          "G1",
          "S03",
          "work"
        ],
        "before_tex": "110",
        "after_tex": "e_0\\cap e_1=\\{1\\}\\implies\\text{invalid}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "第一個二邊候選端點衝突。",
          "en": "The first two-edge candidate has a shared endpoint."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S03",
            "work"
          ],
          "explanation": {
            "zh-Hant": "無效候選不得參與最大值比較。",
            "en": "Invalid candidates cannot enter the maximum comparison."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S05",
        "uses": [
          "G1",
          "S04",
          "work"
        ],
        "before_tex": "101",
        "after_tex": "e_0\\cap e_2=\\varnothing\\implies\\text{valid, size }2",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "唯一合法二邊候選給大小二下界。",
          "en": "The valid two-edge candidate gives lower bound two."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S04",
            "work"
          ],
          "explanation": {
            "zh-Hant": "其兩邊都屬原圖。",
            "en": "Both selected edges belong to the graph."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S06",
        "uses": [
          "G1",
          "S05",
          "work"
        ],
        "before_tex": "011",
        "after_tex": "e_1\\cap e_2=\\{2\\}\\implies\\text{invalid}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "最後一個二邊候選也有衝突。",
          "en": "The last two-edge candidate also conflicts."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S05",
            "work"
          ],
          "explanation": {
            "zh-Hant": "三個二邊子集合全部分類。",
            "en": "All three two-edge subsets are classified."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S07",
        "uses": [
          "G1",
          "S06",
          "work"
        ],
        "before_tex": "111",
        "after_tex": "\\{e_0,e_1,e_2\\}\\text{ invalid since }e_0\\cap e_1\\ne\\varnothing",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "三邊候選包含已知衝突對，因此不合法。",
          "en": "The three-edge candidate contains a known conflicting pair and is invalid."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S06",
            "work"
          ],
          "explanation": {
            "zh-Hant": "分類數1+3+3+1=8與總候選數一致。",
            "en": "The count 1+3+3+1=8 matches the candidate total."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S08",
        "uses": [
          "G1",
          "S07",
          "work"
        ],
        "before_tex": "\\text{valid sizes }0,1,1,1,2",
        "after_tex": "\\nu(G)=2,\\quad\\operatorname{Argmax}=\\{\\{e_0,e_2\\}\\}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "對全部合法候選取最大，並列出此圖全部maximizers。",
          "en": "Maximize over all valid candidates and list all maximizers of this graph."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S07",
            "work"
          ],
          "explanation": {
            "zh-Hant": "唯一性只由這個完整枚舉得出，不當作一般maximum性質。",
            "en": "Uniqueness follows from this enumeration only, not from the notion of maximum."
          }
        },
        "leaf_basis": "work"
      }
    ],
    "result": {
      "zh-Hant": "八子集合全分類，maximum size2，唯一witness為{e0,e2}。",
      "en": "All eight subsets are classified: maximum size 2 with unique witness{e0,e2}."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "中間邊singleton是maximum。",
          "en": "The middle-edge singleton is maximum."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "{e0,e2}合法且更大。",
          "en": "{e0,e2} is valid and larger."
        }
      },
      {
        "claim": {
          "zh-Hant": "111因三邊最多故是maximum。",
          "en": "Mask111 is maximum because it has most edges."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "它不先滿足matching資格。",
          "en": "It fails matching validity."
        }
      },
      {
        "claim": {
          "zh-Hant": "測過一個二邊候選就足夠列全部maximizers。",
          "en": "One two-edge test lists all maximizers."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "全部八候選才提供完整分類。",
          "en": "The full eight candidates supply complete classification."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "mko_definition",
        "kind": "cited",
        "statement": {
          "zh-Hant": "本輪對小型完整有限簡單圖exhaustively枚舉全部edge subsets並篩matching，給exact maximum size與witnesses。每個候選先驗matching；maximal不等於maximum，多個maximum可共存。",
          "en": "For small, fully specified finite simple graphs, exhaustively enumerate all edge subsets, retain only valid matchings and obtain the exact maximum size and witnesses. Validate every candidate as a matching first. Maximal does not mean maximum, and several maximum matchings may coexist."
        },
        "source": {
          "mko_id": "mko-maximum-matching",
          "sha256": "bb97ee67fcffa420b79baaaf56c7ebabe2977fb5d14f5826d1f1c0452f4ed7f9",
          "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f"
        }
      },
      {
        "id": "work",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "有限無向簡單圖的邊為無loop、無重複unordered pairs；圖性質須對完整明示V、E逐一檢查。 matching選邊端點互斥；maximum是在全部合法matching中基數最大，全部達此值的集合構成argmax。",
          "en": "A finite simple undirected graph has loop-free duplicate-free unordered edges; graph properties are audited on all declared V and E. Matchings have disjoint selected-edge endpoint sets; maximum means greatest cardinality over all valid matchings, and argmax contains every attaining set."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "N01",
        "lhs": "2**3",
        "rhs": "8"
      },
      {
        "id": "N02",
        "lhs": "1+3+3+1",
        "rhs": "8"
      },
      {
        "id": "N03",
        "lhs": "max(0,1,1,1,2)",
        "rhs": "2"
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "exact小圖枚舉，非一般高效演算法。",
      "en": "Exact small-graph enumeration, not a general efficient algorithm."
    },
    "md_symbols": [
      [
        "V,E",
        "完整頂點與邊集合",
        "complete vertex and edge sets"
      ],
      [
        "ν(G)",
        "最大matching基數",
        "maximum matching cardinality"
      ]
    ],
    "md_error_controls": [
      {
        "title": "mask覆蓋 / Mask coverage",
        "zh": "保留000到111全部八案。",
        "en": "Retain all eight masks000 through111."
      },
      {
        "title": "先資格 / Validity first",
        "zh": "只在valid matching上取最大，不在所有subsets上取。",
        "en": "Maximize over valid matchings, not all subsets."
      }
    ],
    "md_open_items": []
  },
  "revision": {
    "id": "ocme-new142-reader-adoption-20261021",
    "collection": "coverage-20261021",
    "source_commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
    "adopted_source_sha256": "94fd437d003f1addeeb74ab8d4d231b1284b7766692069ced54c9aceff607234",
    "original_source_sha256": "d891a3efe19c2e43e545e39e1e80e982def72e3a3b37c914e50437febe7b0dae",
    "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": [
      "mko_definition"
    ],
    "independent_AI_concurrence": "SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW",
    "issue_ids": [
      "BR142-S064",
      "BR142-P020",
      "BR142-S063"
    ],
    "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."
  }
}