{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-bipartite-graph-01",
  "mko_id": "mko-bipartite-graph",
  "title": {
    "zh-Hant": "含孤立點的二分證書",
    "en": "A bipartition certificate with an isolated vertex"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-bipartite-graph-01",
    "source": {
      "mko_id": "mko-bipartite-graph",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "c54116e1bcf2bbb83b39260399271673ba3ad43ec2e0b27732fd148d843f7aa1",
      "english_sha256": "472cf9fff25f9a4141e6af6495349bcca9b7d0bc995a018899ee403da6dc3ab1"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "work"
    ],
    "goal": {
      "zh-Hant": "核兩部的全部責任，區分失敗partition與非二分。",
      "en": "Audit every bipartition obligation and distinguish one failed partition from nonbipartiteness."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "V={0,1,2,3}，E={{0,1},{1,2}}；候選A={0,2,3}，B={1}。",
          "en": "V={0,1,2,3}, E={{0,1},{1,2}}; candidate A={0,2,3}, B={1}."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1",
          "work"
        ],
        "before_tex": "V,E",
        "after_tex": "E\\subseteq\\{\\{u,v\\}:u,v\\in V,\\ u\\ne v\\}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "兩邊皆有合法不同端點，3為孤立點。",
          "en": "Both edges have distinct legal endpoints;3 is isolated."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "work"
          ],
          "explanation": {
            "zh-Hant": "圖只有這兩邊，沒有暗藏0至2的邊。",
            "en": "These are the only edges; no0-to 2 edge is implicit."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "S01",
          "work"
        ],
        "before_tex": "A=\\{0,2,3\\},\\quad B=\\{1\\}",
        "after_tex": "A\\cap B=\\varnothing",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "四個頂點沒有跨部重複。",
          "en": "No vertex appears in both parts."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01",
            "work"
          ],
          "explanation": {
            "zh-Hant": "同一頂點不得同屬兩部。",
            "en": "A vertex cannot belong to both parts."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S03",
        "uses": [
          "G1",
          "S02",
          "work"
        ],
        "before_tex": "A\\cup B=\\{0,1,2,3\\}",
        "after_tex": "A\\cup B=V",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "孤立點3也被納入，完整cover。",
          "en": "The isolated vertex 3 is included, giving complete coverage."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S02",
            "work"
          ],
          "explanation": {
            "zh-Hant": "漏孤立點也會使partition失敗。",
            "en": "Omitting an isolate would still invalidate the partition."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S04",
        "uses": [
          "G1",
          "S03",
          "work"
        ],
        "before_tex": "\\{0,1\\}\\in E",
        "after_tex": "0\\in A,\\quad1\\in B",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "第一邊兩端跨部。",
          "en": "The first edge crosses the parts."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S03",
            "work"
          ],
          "explanation": {
            "zh-Hant": "只核頂點數不會證此邊責任。",
            "en": "Vertex counts alone do not prove this edge obligation."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S05",
        "uses": [
          "G1",
          "S04",
          "work"
        ],
        "before_tex": "\\{1,2\\}\\in E",
        "after_tex": "1\\in B,\\quad2\\in A",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "第二邊跨部，全部邊已窮盡。",
          "en": "The second edge crosses, exhausting all edges."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S04",
            "work"
          ],
          "explanation": {
            "zh-Hant": "0與2可同部因它們不相鄰。",
            "en": "0 and 2 may share a part because they are not adjacent."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S06",
        "uses": [
          "G1",
          "S05",
          "work"
        ],
        "before_tex": "A'=\\{0,1\\},\\quad B'=\\{2,3\\}",
        "after_tex": "\\{0,1\\}\\subseteq A'\\implies(A',B')\\text{ invalid}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "另一候選失敗但原圖仍有上面合法證書。",
          "en": "Another candidate fails while the original graph retains the valid certificate above."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S05",
            "work"
          ],
          "explanation": {
            "zh-Hant": "無效候選不能否定存在某個有效候選。",
            "en": "One invalid candidate cannot refute existence of a valid one."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S07",
        "uses": [
          "G1",
          "S06",
          "work"
        ],
        "before_tex": "H:0-1-2-0\\quad\\text{(separate triangle)}",
        "after_tex": "c(0)\\ne c(1)\\ne c(2)\\implies c(2)=c(0),\\ c(2)\\ne c(0)",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "若三角形只有兩部，沿前兩邊迫使0、2同部，末邊矛盾。",
          "en": "With two parts on a triangle, the first two edges force0 and 2 together, contradicting the last edge."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S06",
            "work"
          ],
          "explanation": {
            "zh-Hant": "這個奇圈反證針對另一完整圖，不混入原圖。",
            "en": "This odd-cycle refutation concerns the separate complete triangle, not the original graph."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S08",
        "uses": [
          "G1",
          "S07",
          "mko_definition"
        ],
        "before_tex": "A\\cap B=\\varnothing,\\quad A\\cup B=V,\\quad\\text{all edges cross}",
        "after_tex": "G\\text{ is bipartite}",
        "relation": "implies",
        "rule": "mko_definition",
        "conditions": [],
        "explanation": {
          "zh-Hant": "合併全部資格完成原圖二分證書。",
          "en": "Combine every obligation to certify the original graph bipartite."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S07",
            "mko_definition"
          ],
          "explanation": {
            "zh-Hant": "無邊圖可用(V,empty)但不能漏頂點。",
            "en": "An edgeless graph may use(V,empty), but may not omit vertices."
          }
        },
        "leaf_basis": "mko_definition"
      }
    ],
    "result": {
      "zh-Hant": "A={0,2,3}、B={1}證原圖二分。",
      "en": "A={0,2,3}, B={1} certify the graph bipartite."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "可漏掉孤立3。",
          "en": "Isolated 3 may be omitted."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "兩部聯集就不等V。",
          "en": "The union would no longer equal V."
        }
      },
      {
        "claim": {
          "zh-Hant": "A'失敗故圖非二分。",
          "en": "Failure of A' proves nonbipartiteness."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "A、B已是合法partition。",
          "en": "A,B already form a valid partition."
        }
      },
      {
        "claim": {
          "zh-Hant": "三角形也可兩部。",
          "en": "A triangle admits two parts."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "沿三邊交替會迫使首尾同部又相鄰。",
          "en": "Alternation around three edges forces adjacent endpoints into the same part."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "mko_definition",
        "kind": "cited",
        "statement": {
          "zh-Hant": "核完整頂點cover、兩部互斥及全部邊跨部。失敗partition不證非二分；奇cycle可作否證。孤立頂點可置任一部，空部是否允許須明示。",
          "en": "Check complete vertex coverage, disjointness of the two parts and every edge crossing. A failed proposed partition does not prove nonbipartiteness; an odd cycle can certify it. Isolated vertices may belong to either part, and whether empty parts are allowed must be explicit."
        },
        "source": {
          "mko_id": "mko-bipartite-graph",
          "sha256": "c54116e1bcf2bbb83b39260399271673ba3ad43ec2e0b27732fd148d843f7aa1",
          "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。 二分證書是互斥且聯集為V的兩部，每邊跨部。",
          "en": "A finite simple undirected graph has loop-free duplicate-free unordered edges; graph properties are audited on all declared V and E. A bipartition certificate has disjoint parts with union V and every edge crossing."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "N01",
        "lhs": "3+1",
        "rhs": "4"
      },
      {
        "id": "N02",
        "lhs": "2%2",
        "rhs": "0"
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "有限無向簡單圖；允許空部但仍需cover。",
      "en": "Finite simple undirected graphs; empty parts are allowed but coverage remains required."
    },
    "md_symbols": [
      [
        "V,E",
        "完整頂點與邊集合",
        "complete vertex and edge sets"
      ],
      [
        "A,B",
        "兩個部",
        "the two parts"
      ]
    ],
    "md_error_controls": [
      {
        "title": "全頂點 / All vertices",
        "zh": "孤立3仍必分配到一部。",
        "en": "Assign isolated 3 to a part."
      },
      {
        "title": "存在量詞 / Existential quantifier",
        "zh": "合法證書證存在；失敗一案不證不存在。",
        "en": "A valid witness proves existence; one failure does not prove nonexistence."
      }
    ],
    "md_open_items": []
  },
  "revision": {
    "id": "ocme-new142-reader-adoption-20261021",
    "collection": "coverage-20261021",
    "source_commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
    "adopted_source_sha256": "bd3b9a767549e7cefb60cebc2278fb332f5bac033b84321bd0c2eb7004197c3c",
    "original_source_sha256": "b94470821c9e899e02832a1b63f4ceaf9869bac586a0967c66175e03ad73bf37",
    "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-S027",
      "BR142-S026",
      "BR142-S022",
      "BR142-S023",
      "BR142-S024",
      "BR142-S025"
    ],
    "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."
  }
}