{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-clique-and-independent-set-01",
  "mko_id": "mko-clique-and-independent-set",
  "title": {
    "zh-Hant": "三頂點圖的全部clique與independent sets",
    "en": "All cliques and independent sets on three vertices"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-clique-and-independent-set-01",
    "source": {
      "mko_id": "mko-clique-and-independent-set",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "9477f7b7e5b3af0c1c270f6f42b508ec6142c458868b507c850d56c9a5b946fe",
      "english_sha256": "7d32b2cc13a3ecb7fe4cb5c14409d97ce2b9decc8176de4b5d780f0c601b22ef"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "work"
    ],
    "goal": {
      "zh-Hant": "完整2³子集分類求omega與alpha。",
      "en": "Classify all2³ subsets to obtain omega and alpha."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "V={0,1,2}，E={01}，2孤立。",
          "en": "V={0,1,2}, E={01};2 is isolated."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1",
          "work"
        ],
        "before_tex": "|V|=3",
        "after_tex": "|\\mathcal P(V)|=8",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "完整子集數為八。",
          "en": "There are eight vertex subsets in total."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "work"
          ],
          "explanation": {
            "zh-Hant": "clique核所有pair相鄰，independent核所有pair不相鄰。",
            "en": "Cliques require every pair adjacent; independent sets require every pair nonadjacent."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "S01",
          "work"
        ],
        "before_tex": "\\varnothing",
        "after_tex": "\\text{both clique and independent, size }0",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "無頂點對，兩個全稱條件皆空真。",
          "en": "With no pairs, both universal conditions are vacuously true."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01",
            "work"
          ],
          "explanation": {
            "zh-Hant": "empty set依此convention保留。",
            "en": "Retain the empty set under this convention."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S03",
        "uses": [
          "G1",
          "S02",
          "work"
        ],
        "before_tex": "\\{0\\},\\{1\\},\\{2\\}",
        "after_tex": "\\text{all both clique and independent, size }1",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "三個singleton都無不同頂點pair。",
          "en": "All three singletons have no distinct-vertex pairs."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S02",
            "work"
          ],
          "explanation": {
            "zh-Hant": "singleton2是maximal clique但未必maximum。",
            "en": "Singleton2 is a maximal clique but need not be maximum."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S04",
        "uses": [
          "G1",
          "S03",
          "work"
        ],
        "before_tex": "\\{0,1\\}",
        "after_tex": "01\\in E\\implies\\text{clique, not independent}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "這一pair提供大小二clique。",
          "en": "This pair supplies a size-two clique."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S03",
            "work"
          ],
          "explanation": {
            "zh-Hant": "一條已存在邊足以否定該pair independent。",
            "en": "Its existing edge refutes independence of the pair."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S05",
        "uses": [
          "G1",
          "S04",
          "work"
        ],
        "before_tex": "\\{0,2\\},\\{1,2\\}",
        "after_tex": "02,12\\notin E\\implies\\text{independent, not clique}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "另外兩pair提供大小二independent sets。",
          "en": "The other two pairs supply size-two independent sets."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S04",
            "work"
          ],
          "explanation": {
            "zh-Hant": "兩pair都逐一核，而非抽樣。",
            "en": "Both pairs are checked, not sampled."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S06",
        "uses": [
          "G1",
          "S05",
          "work"
        ],
        "before_tex": "\\{0,1,2\\}",
        "after_tex": "02\\notin E\\implies\\text{not clique};\\quad01\\in E\\implies\\text{not independent}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "三點集用一個缺邊與一個有邊分別否證兩性質。",
          "en": "A missing edge and an existing edge separately refute the two properties on the triple."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S05",
            "work"
          ],
          "explanation": {
            "zh-Hant": "兩種否證使用不同pair，不能混淆。",
            "en": "The two refutations use different pairs and must not be conflated."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S07",
        "uses": [
          "G1",
          "S06",
          "work"
        ],
        "before_tex": "\\text{all eight subsets classified}",
        "after_tex": "\\omega(G)=2,\\quad\\alpha(G)=2",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "無合法三點集且有合法二點見證，得兩exact extrema。",
          "en": "No triple qualifies while size-two witnesses do, giving both exact extrema."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S06",
            "work"
          ],
          "explanation": {
            "zh-Hant": "所有maximum clique為{0,1}；maximum independent為{0,2}、{1,2}。",
            "en": "The maximum clique is{0,1}; maximum independent sets are{0,2},{1,2}."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S08",
        "uses": [
          "G1",
          "S07",
          "work"
        ],
        "before_tex": "\\{2\\}\\text{ clique, no vertex addable}",
        "after_tex": "\\{2\\}\\text{ maximal but not maximum since }1<2",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "孤立點singleton不能擴張成clique，但被大小二clique超過。",
          "en": "The isolated singleton cannot be extended as a clique but is smaller than the size-two clique."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S07",
            "work"
          ],
          "explanation": {
            "zh-Hant": "clique一般只給chi≥omega，不給chi=omega定理。",
            "en": "A clique generally gives chi≥omega, not a theorem of equality."
          }
        },
        "leaf_basis": "work"
      }
    ],
    "result": {
      "zh-Hant": "omega=alpha=2；{2}示範maximal不等maximum。",
      "en": "Omega=alpha=2;{2} illustrates maximal versus maximum."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "{2} maximal所以maximum。",
          "en": "{2} is maximal, hence maximum."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "{0,1}是更大clique。",
          "en": "{0,1} is a larger clique."
        }
      },
      {
        "claim": {
          "zh-Hant": "{0,1,2} independent。",
          "en": "{0,1,2} is independent."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "它包含原邊01。",
          "en": "It contains edge 01."
        }
      },
      {
        "claim": {
          "zh-Hant": "由omega=2一般推出chi=2。",
          "en": "Omega=2 generally implies chi=2."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "五圈omega2但奇圈不能二色，需三色。",
          "en": "A five-cycle has omega2 but, being odd, needs three colors."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "mko_definition",
        "kind": "cited",
        "statement": {
          "zh-Hant": "候選集合逐對核相鄰或不相鄰。單一大小r候選只給omega或alpha的下界；宣稱maximum須遍歷全部頂點子集合或另有全域上界。maximal只代表不能加入頂點，且clique大小只推出chi(G)>=omega(G)。",
          "en": "Audit all candidate pairs. One valid size-r candidate gives only a lower bound on omega or alpha; maximum claims require the complete vertex-subset search or another global upper certificate. Maximal means no vertex can be added. Clique number bounds chromatic number below, but does not imply equality in general."
        },
        "source": {
          "mko_id": "mko-clique-and-independent-set",
          "sha256": "9477f7b7e5b3af0c1c270f6f42b508ec6142c458868b507c850d56c9a5b946fe",
          "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。 clique的所有不同點對皆相鄰，independent set的所有不同點對皆不相鄰；ω、α為各自最大基數，maximal指不能再加入頂點。",
          "en": "A finite simple undirected graph has loop-free duplicate-free unordered edges; graph properties are audited on all declared V and E. All distinct pairs in a clique are adjacent; all distinct pairs in an independent set are nonadjacent; omega and alpha are their maximum cardinalities, while maximal means no vertex can be added."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "N01",
        "lhs": "1+3+3+1",
        "rhs": "2**3"
      },
      {
        "id": "N02",
        "lhs": "max(0,1,1,1,2)",
        "rhs": "2"
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "只對完整小圖給exact extrema。",
      "en": "Exact extrema only for the fully specified small graph."
    },
    "md_symbols": [
      [
        "V,E",
        "完整頂點與邊集合",
        "complete vertex and edge sets"
      ],
      [
        "ω,α",
        "clique與independence numbers",
        "clique and independence numbers"
      ]
    ],
    "md_error_controls": [
      {
        "title": "pair責任 / Pair obligation",
        "zh": "clique缺一pair就失敗；independent多一edge就失敗。",
        "en": "One missing pair refutes a clique; one edge refutes independence."
      },
      {
        "title": "完整極值 / Complete extrema",
        "zh": "1空+3singleton+3pair+1triple=8。",
        "en": "One empty, three singletons, three pairs and one triple total eight."
      }
    ],
    "md_open_items": []
  },
  "revision": {
    "id": "ocme-new142-reader-adoption-20261021",
    "collection": "coverage-20261021",
    "source_commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
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    "original_source_sha256": "b12b9f7e4aa7773c8a1042f9d5b03d1d5d69b4be8c23c84776ee6696a0aeb755",
    "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": [
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    ],
    "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."
  }
}