{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-chromatic-number-01",
  "mko_id": "mko-chromatic-number",
  "title": {
    "zh-Hant": "三角形全部二色配置的排除",
    "en": "Excluding every two-color assignment on a triangle"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-chromatic-number-01",
    "source": {
      "mko_id": "mko-chromatic-number",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "c6461c8fe8f1d88f27aaea185ce8ec6d8d3e4318855946ae3b5ef9daf6f5f109",
      "english_sha256": "c01822b4c5a7c77a39857d06850f1e4afce84eeea3ca3c93ec2fdaab43fec31e"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "work"
    ],
    "goal": {
      "zh-Hant": "以上界與完整低色數搜尋證chi=3。",
      "en": "Prove chi=3 using an upper witness and complete lower-color search."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "V={0,1,2}，E={01,02,12}；二色labels為0、1，配置向量順序(0,1,2)。",
          "en": "V={0,1,2}, E={01,02,12}; two-color labels are 0,1, with assignment vectors ordered by vertices(0,1,2)."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1",
          "work"
        ],
        "before_tex": "c=(0,1,2)",
        "after_tex": "c(0)\\ne c(1),\\quad c(0)\\ne c(2),\\quad c(1)\\ne c(2)",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "三色見證逐邊核，給chi≤3。",
          "en": "Audit every edge of the three-color witness, giving chi≤3."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "work"
          ],
          "explanation": {
            "zh-Hant": "仍需排除較少色數才有等號。",
            "en": "Equality still requires excluding fewer colors."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "S01",
          "work"
        ],
        "before_tex": "c:V\\to\\{0,1\\}",
        "after_tex": "2^3=8\\text{ assignments}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "完整搜尋有八個向量，容許一色作二色子情況。",
          "en": "The complete search has eight vectors and includes one-color assignments as subcases."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01",
            "work"
          ],
          "explanation": {
            "zh-Hant": "使用最多二色不是必須兩色皆出現。",
            "en": "At most two colors does not require both to appear."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S03",
        "uses": [
          "G1",
          "S02",
          "work"
        ],
        "before_tex": "000,001",
        "after_tex": "c(0)=c(1)=0\\implies01\\text{ violated}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "前兩配置均在邊01衝突。",
          "en": "The first two assignments both conflict on edge 01."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S02",
            "work"
          ],
          "explanation": {
            "zh-Hant": "兩案分別列出，不留未核尾端。",
            "en": "Both cases are explicitly listed."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S04",
        "uses": [
          "G1",
          "S03",
          "work"
        ],
        "before_tex": "010,101",
        "after_tex": "c(0)=c(2)\\implies02\\text{ violated}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "兩個交替配置都在02衝突。",
          "en": "Both alternating assignments conflict on02."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S03",
            "work"
          ],
          "explanation": {
            "zh-Hant": "不只檢相鄰編號邊，02也在圖中。",
            "en": "Check02 too; it is an edge despite nonconsecutive labels."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S05",
        "uses": [
          "G1",
          "S04",
          "work"
        ],
        "before_tex": "011,100",
        "after_tex": "c(1)=c(2)\\implies12\\text{ violated}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "兩個配置的後兩頂點同色。",
          "en": "These two assignments give equal colors to the last two vertices."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S04",
            "work"
          ],
          "explanation": {
            "zh-Hant": "0、1色號對換不改違規性。",
            "en": "Swapping color labels does not remove the violation."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S06",
        "uses": [
          "G1",
          "S05",
          "work"
        ],
        "before_tex": "110,111",
        "after_tex": "c(0)=c(1)=1\\implies01\\text{ violated}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "最後兩個配置也違規，八案全部分類。",
          "en": "The final two assignments also fail, exhausting all eight cases."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S05",
            "work"
          ],
          "explanation": {
            "zh-Hant": "配置集合為000、001、010、011、100、101、110、111。",
            "en": "The set is 000,001,010,011,100,101,110,111."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S07",
        "uses": [
          "G1",
          "S06",
          "work"
        ],
        "before_tex": "\\text{no proper map }V\\to\\{0,1\\}",
        "after_tex": "\\chi(G)>2",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "任何0色或1色配置也不能繞過二色搜尋，且V非空。",
          "en": "Zero- or one-color assignments cannot evade the two-color exclusion, with V nonempty."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S06",
            "work"
          ],
          "explanation": {
            "zh-Hant": "兩色排除是全域下界，不是一個失敗候選。",
            "en": "This is a global lower bound, not one failed candidate."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S08",
        "uses": [
          "G1",
          "S07",
          "work"
        ],
        "before_tex": "2<\\chi(G)\\le3,\\quad\\chi(G)\\in\\mathbb N",
        "after_tex": "\\chi(G)=3",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "整數性合併上、下界得exact chromatic number。",
          "en": "Integrality combines upper and lower bounds into the exact chromatic number."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S07",
            "work"
          ],
          "explanation": {
            "zh-Hant": "空圖chi=0、非空無邊chi=1是另明示conventions。",
            "en": "The separate conventions are chi=0 for no vertices and chi=1 for nonempty edgeless graphs."
          }
        },
        "leaf_basis": "work"
      }
    ],
    "result": {
      "zh-Hant": "完整八案排除二色且有三色見證，故chi=3。",
      "en": "All eight two-color cases fail and a three-color witness exists, hence chi=3."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "找到三色就直接證chi=3。",
          "en": "A three-coloring alone proves chi=3."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "它只給上界；八案排除才給下界。",
          "en": "It gives only an upper bound; the eight-case exclusion supplies the lower bound."
        }
      },
      {
        "claim": {
          "zh-Hant": "010在三角形proper。",
          "en": "010 is proper on the triangle."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "邊02兩端都是0。",
          "en": "Edge 02 has color0 at both ends."
        }
      },
      {
        "claim": {
          "zh-Hant": "非空無邊圖chi=0。",
          "en": "A nonempty edgeless graph has chi=0."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "每頂點仍須有一色，最小為1。",
          "en": "Each vertex still needs a color, so the minimum is 1."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "mko_definition",
        "kind": "cited",
        "statement": {
          "zh-Hant": "一個proper k-coloring只證chi(G)<=k；等號還須排除(k-1)-coloring。小型完整圖以exhaustive search給下界並用明示著色給上界；greedy結果不當作最優性。空頂點圖採chi=0，非空無邊圖採chi=1。",
          "en": "One proper k-coloring proves chi(G)≤k; equality also needs a valid exclusion of (k−1)-colorings. Complete small graphs are checked exhaustively for a lower bound and given an explicit coloring for an upper bound. A greedy result alone is not optimality evidence. The empty vertex graph has chi=0 and a nonempty edgeless graph has chi=1."
        },
        "source": {
          "mko_id": "mko-chromatic-number",
          "sha256": "c6461c8fe8f1d88f27aaea185ce8ec6d8d3e4318855946ae3b5ef9daf6f5f109",
          "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。 proper coloring使每邊兩端異色；χ為全體total proper colorings的最小使用色數，k-colorable表示最多k色。",
          "en": "A finite simple undirected graph has loop-free duplicate-free unordered edges; graph properties are audited on all declared V and E. A proper coloring gives distinct colors across every edge; chi is the minimum used-color count over all total proper colorings, and k-colorable means using at most k colors."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "N01",
        "lhs": "2**3",
        "rhs": "8"
      },
      {
        "id": "N02",
        "lhs": "2+2+2+2",
        "rhs": "8"
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "n=3 exact枚舉；不以greedy最優性代替。",
      "en": "Exact enumeration at n=3, with no greedy-optimality assumption."
    },
    "md_symbols": [
      [
        "V,E",
        "完整頂點與邊集合",
        "complete vertex and edge sets"
      ],
      [
        "χ(G)",
        "最小色數",
        "minimum number of colors"
      ]
    ],
    "md_error_controls": [
      {
        "title": "枚舉完整 / Complete enumeration",
        "zh": "八個bit配置各出現一次。",
        "en": "Each of the eight bit assignments appears once."
      },
      {
        "title": "上下界 / Bounds",
        "zh": "先三色見證，再排所有二色，最後用整數性。",
        "en": "Give three colors, exclude every two-color assignment, then use integrality."
      }
    ],
    "md_open_items": []
  },
  "revision": {
    "id": "ocme-new142-reader-adoption-20261021",
    "collection": "coverage-20261021",
    "source_commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
    "adopted_source_sha256": "e067a4fb1baa6f694ba27da6d3b5d80c12c74434d72f796ad27dd14c604b48ad",
    "original_source_sha256": "1ca25d8841495fe4e298468a1912e58c28402ed9745e1fee11300a1b7c39f7b5",
    "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-S098",
      "BR142-S101",
      "BR142-S102",
      "BR142-S099",
      "BR142-S100"
    ],
    "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."
  }
}