{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261031-r05-graph-coloring-0045",
  "pilot_number": "R0030-15620",
  "number": 15620,
  "family_id": "qf-mod31-coloring-conflict",
  "title": {
    "zh-Hant": "第 15620 題：圖著色",
    "en": "Question 15620: List Every Monochromatic Edge, Not Every Repeated Color"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261031-r05-graph-coloring-0045",
    "batch_id": "batch-20261014-advance-20261031-r05-graph-coloring-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-graph-coloring"
    ],
    "title_zh": "圖著色",
    "stem_zh": "完整有限無向簡單圖與每頂點顏色。操作=coloring-conflict；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[19,19,20,19,20,19]}。請依 [conflicting_edges,count,proper] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([[0, 1]], 1, 否)",
      "explanation_zh": "Wave4收束核對=([[0, 1]], 1, 否)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 3,
      "skills": [
        "圖著色"
      ]
    },
    "variant": {
      "family_id": "qf-mod31-coloring-conflict",
      "seed": 202610315045,
      "parameters": {
        "n": 6,
        "edges": [
          [
            0,
            1
          ],
          [
            1,
            2
          ],
          [
            2,
            3
          ],
          [
            3,
            4
          ],
          [
            4,
            5
          ]
        ],
        "colors": [
          19,
          19,
          20,
          19,
          20,
          19
        ],
        "domain": "Complete_finite_simple_graph_vertex_coloring",
        "operation": "coloring-conflict",
        "semantic_template": "qf-mod31-coloring-conflict/{\"colors\":[19,19,20,19,20,19],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-14",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261014-advance-20261031-r05-graph-coloring-0045",
    "batch_id": "batch-20261014-advance-20261031-r05-graph-coloring-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "d3c04f17d18063795c20ea41485e4f6c32ae1dd44b462f4c2f57da185cd098bc",
      "family_id": "qf-mod31-coloring-conflict",
      "seed": 202610315045,
      "semantic_template": "qf-mod31-coloring-conflict/{\"colors\":[19,19,20,19,20,19],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6}"
    },
    "fields": {
      "title": "List Every Monochromatic Edge, Not Every Repeated Color",
      "stem": "Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[19, 19, 20, 19, 20, 19] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Give (conflicting_edges,count,proper?), with edges in normalized sorted order.",
      "answer": {
        "kind": "expression",
        "value": "([[0, 1]], 1, No)",
        "explanation": "Only same-colored endpoints joined by an edge are conflicts. Reusing a color on nonadjacent vertices is allowed. The exact result, retaining the requested order, is ([[0, 1]], 1, No)."
      },
      "skills": [
        "List Every Monochromatic Edge, Not Every Repeated Color"
      ],
      "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."
    }
  },
  "givens": [
    {
      "id": "G001",
      "text": {
        "zh-Hant": "完整有限無向簡單圖與每頂點顏色。操作=coloring-conflict；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[19,19,20,19,20,19]}。請依 [conflicting_edges,count,proper] 欄序回答。",
        "en": "Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[19, 19, 20, 19, 20, 19] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Give (conflicting_edges,count,proper?), with edges in normalized sorted order."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "complete undirected vertex carrier Fin(6);original edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]",
      "after": "normalized sorted simple edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]];all vertices including isolates=[0, 1, 2, 3, 4, 5]",
      "relation": "implies",
      "rule_id": "graph_complete_simple_carrier_normalized_edge_order",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐邊先排端點再把邊作字典序排序，反向重複仍是同一邊而須拒絕；loop亦拒絕。完整載體包括沒有邊的頂點，不能從邊表遺漏它們。",
        "en": "Normalize endpoints and then sort edges lexicographically.Reversed duplicates are still duplicates and are rejected,as are loops.The complete carrier includes vertices absent from the edge list."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "colors:normalized edge index=0;edge=[0, 1];endpoint labels=[19, 19]",
      "after": "different endpoints=否;same-label conflict=是",
      "relation": "implies",
      "rule_id": "graph_every_edge_endpoint_constraint",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只檢同一條實際邊的兩端；不相鄰頂點重用標記不算衝突，某一條跨部亦不代表其他邊都通過。",
        "en": "Check the endpoints of each actual edge.Reused labels at nonadjacent vertices are not conflicts,and one crossing edge does not certify all other edges."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "colors:normalized edge index=1;edge=[1, 2];endpoint labels=[19, 20]",
      "after": "different endpoints=是;same-label conflict=否",
      "relation": "implies",
      "rule_id": "graph_every_edge_endpoint_constraint",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只檢同一條實際邊的兩端；不相鄰頂點重用標記不算衝突，某一條跨部亦不代表其他邊都通過。",
        "en": "Check the endpoints of each actual edge.Reused labels at nonadjacent vertices are not conflicts,and one crossing edge does not certify all other edges."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "colors:normalized edge index=2;edge=[2, 3];endpoint labels=[20, 19]",
      "after": "different endpoints=是;same-label conflict=否",
      "relation": "implies",
      "rule_id": "graph_every_edge_endpoint_constraint",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只檢同一條實際邊的兩端；不相鄰頂點重用標記不算衝突，某一條跨部亦不代表其他邊都通過。",
        "en": "Check the endpoints of each actual edge.Reused labels at nonadjacent vertices are not conflicts,and one crossing edge does not certify all other edges."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "colors:normalized edge index=3;edge=[3, 4];endpoint labels=[19, 20]",
      "after": "different endpoints=是;same-label conflict=否",
      "relation": "implies",
      "rule_id": "graph_every_edge_endpoint_constraint",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只檢同一條實際邊的兩端；不相鄰頂點重用標記不算衝突，某一條跨部亦不代表其他邊都通過。",
        "en": "Check the endpoints of each actual edge.Reused labels at nonadjacent vertices are not conflicts,and one crossing edge does not certify all other edges."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "colors:normalized edge index=4;edge=[4, 5];endpoint labels=[20, 19]",
      "after": "different endpoints=是;same-label conflict=否",
      "relation": "implies",
      "rule_id": "graph_every_edge_endpoint_constraint",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只檢同一條實際邊的兩端；不相鄰頂點重用標記不算衝突，某一條跨部亦不代表其他邊都通過。",
        "en": "Check the endpoints of each actual edge.Reused labels at nonadjacent vertices are not conflicts,and one crossing edge does not certify all other edges."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "total original vertex colors=[19, 19, 20, 19, 20, 19];all normalized conflicts=[[0, 1]]",
      "after": "increasing distinct used labels=[19, 20];used count=2;all edges proper=否",
      "relation": "implies",
      "rule_id": "coloring_distinct_label_count_not_numeric_magnitude",
      "conditions": [],
      "explanation": {
        "zh-Hant": "顏色是名字，使用幾種要數distinct，不用最大標籤加1。只有相鄰同色才衝突，非相鄰重用同色合法。",
        "en": "Colors are names:count distinct used labels rather than maximum label plus1.Only adjacent equal colors conflict;nonadjacent color reuse is allowed."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      [
        [
          0,
          1
        ]
      ],
      1,
      false
    ]
  },
  "conclusion": {
    "canonical_value": "([[0, 1]], 1, 否)",
    "explanation": {
      "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。",
      "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged."
    }
  },
  "definition_note": {
    "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。",
    "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate."
  },
  "source": {
    "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db",
    "question_binding": {
      "batch_id": "batch-20261014-advance-20261031-r05-graph-coloring-foundations",
      "batch_path": "public/data/questions/batches/batch-20261014-advance-20261031-r05-graph-coloring-foundations/questions.jsonl",
      "batch_sha256": "eab8b1754ce3dd63e89131a108810395caffcea2b8903cfec26847fe776c97a0",
      "original_line": 45,
      "original_line_sha256": "45c6d7e3e06e8c8380054878676a71ba4b29b68c6ec22ffced2da4dd71100b97",
      "question_compact_sha256": "d3c04f17d18063795c20ea41485e4f6c32ae1dd44b462f4c2f57da185cd098bc",
      "english_row_compact_sha256": "ef51a5a2ce29b91e1a6fb41720905de58dedf7beb3f0ddd7ffe64206d7903b00",
      "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261031-r05-graph-coloring-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261031-r05-graph-coloring-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-graph-coloring": {
        "sha256": "6c2084e53867b83ec1c3c9f3a21f1023e1d3b0910c9b23c6f774aa05a08ff54d",
        "english_sha256": "d5c203e54e6a16948a99c1802bfc693db9999eafa424314ef0841cca900e9476",
        "original_path": "public/data/mko/graph-coloring.json",
        "english_path": "public/data/i18n/editions/mko-graph-coloring.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "d85118d74859ec7e59736714f767732c6ec092eaa0bf5279abeb28e4a628e3c3",
    "original_candidate_sha256": "7b999e23a5162de3a3d980a34ab089fce64a8d800aa660c76149d083f6bebda1",
    "original_candidate_line_sha256": "7b999e23a5162de3a3d980a34ab089fce64a8d800aa660c76149d083f6bebda1"
  },
  "historical_source_state": {
    "review_state": {
      "derivation": "candidate_authored_and_locally_checked",
      "source_question_state": "candidate",
      "independent_AI_or_human_review": "NOT_PERFORMED",
      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
    },
    "scope_note": {
      "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。",
      "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged."
    },
    "label": "Frozen producer metadata; current per-question review is separate below."
  },
  "revision": {
    "id": "review-20261026-007-p31",
    "review_date": "2026-10-26",
    "owner_authorized_reader_publication": true,
    "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.",
    "review_receipts": {
      "semantic_review_sha256": "73bc08066ffb1a81eacb2d4e288d03f522aeefd3949fa0d97da41b484ed737bc",
      "question_first_witness_sha256": "ac120452a208b257b8270f38351a09dcd832cdbc8d9db6b69933f83e43cc9336"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 7,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "d89616f9f4dfaa45313bfa012ca5b7a91f8f6b0072218d822a7e7f21086fe45b",
      "arithmetic_steps": 0
    },
    "uses_answer_as_premise": false,
    "human_review": "NOT_PERFORMED",
    "formalization": "not_formalized",
    "Lean": "NOT_RUN",
    "Canonical_truth_promotion": false,
    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
  }
}