{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261031-r02-bipartite-graph-0089",
  "pilot_number": "R0030-15539",
  "number": 15539,
  "family_id": "qf-mod31-bipartite-isolated-boundary",
  "title": {
    "zh-Hant": "第 15539 題：二分圖",
    "en": "Question 15539: Isolated Vertices Still Belong to Exactly One Part"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261031-r02-bipartite-graph-0089",
    "batch_id": "batch-20261014-advance-20261031-r02-bipartite-graph-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-bipartite-graph"
    ],
    "title_zh": "二分圖",
    "stem_zh": "完整有限無向無loop簡單圖。操作=bipartite-isolated-boundary；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[1,0,1,1,0,0],\"vertex\":5}。請依 [vertex,incident_edges,assigned_part,whole_partition_proper,boundary] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(5, [], 0, 否, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)",
      "explanation_zh": "Wave4收束核對=(5, [], 0, 否, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "二分圖"
      ]
    },
    "variant": {
      "family_id": "qf-mod31-bipartite-isolated-boundary",
      "seed": 202610312089,
      "parameters": {
        "n": 6,
        "edges": [
          [
            0,
            1
          ],
          [
            1,
            2
          ],
          [
            2,
            3
          ],
          [
            3,
            4
          ]
        ],
        "parts": [
          1,
          0,
          1,
          1,
          0,
          0
        ],
        "vertex": 5,
        "domain": "Complete_finite_simple_graph_bipartition",
        "operation": "bipartite-isolated-boundary",
        "semantic_template": "qf-mod31-bipartite-isolated-boundary/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,0,1,1,0,0],\"vertex\":5}"
      }
    },
    "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-r02-bipartite-graph-0089",
    "batch_id": "batch-20261014-advance-20261031-r02-bipartite-graph-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "b63905e2f95e5ca292763a871513225c46ba601722e9e2e569754b63a37c8828",
      "family_id": "qf-mod31-bipartite-isolated-boundary",
      "seed": 202610312089,
      "semantic_template": "qf-mod31-bipartite-isolated-boundary/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,0,1,1,0,0],\"vertex\":5}"
    },
    "fields": {
      "title": "Isolated Vertices Still Belong to Exactly One Part",
      "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]]. 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. The total part assignment=[1, 0, 1, 1, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. vertex=5 must be isolated. Give (vertex,incident_edges,assigned_part,whole_partition_proper?,isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).",
      "answer": {
        "kind": "expression",
        "value": "(5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)",
        "explanation": "Confirm there are no incident edges before reading the part of this vertex. It can be assigned either part without affecting edge constraints, but it must not be omitted from the total assignment. The exact result, retaining the requested order, is (5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)."
      },
      "skills": [
        "Isolated Vertices Still Belong to Exactly One Part"
      ],
      "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": "完整有限無向無loop簡單圖。操作=bipartite-isolated-boundary；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[1,0,1,1,0,0],\"vertex\":5}。請依 [vertex,incident_edges,assigned_part,whole_partition_proper,boundary] 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]]. 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. The total part assignment=[1, 0, 1, 1, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. vertex=5 must be isolated. Give (vertex,incident_edges,assigned_part,whole_partition_proper?,isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "complete undirected vertex carrier Fin(6);original edges=[[0, 1], [1, 2], [2, 3], [3, 4]]",
      "after": "normalized sorted simple edges=[[0, 1], [1, 2], [2, 3], [3, 4]];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": "total original part labels=[1, 0, 1, 1, 0, 0];complete carrier Fin(6)",
      "after": "part0=[1, 4, 5];part1=[0, 2, 3];disjoint full cover=是",
      "relation": "implies",
      "rule_id": "bipartition_total_inverse_image_parts",
      "conditions": [],
      "explanation": {
        "zh-Hant": "0、1標記的兩個逆像自動互斥，且每個原頂點恰有一標記；空部在無邊圖可以，不漏孤立點。這只核分部，還須逐邊跨部。",
        "en": "The two inverse images of0 and1 are disjoint and cover every originally labeled vertex.An empty part is allowed for an edgeless graph;isolates remain included.Edge crossing is still a separate obligation."
      },
      "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": "parts:normalized edge index=0;edge=[0, 1];endpoint labels=[1, 0]",
      "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": "parts:normalized edge index=1;edge=[1, 2];endpoint labels=[0, 1]",
      "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": "parts:normalized edge index=2;edge=[2, 3];endpoint labels=[1, 1]",
      "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": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "parts:normalized edge index=3;edge=[3, 4];endpoint labels=[1, 0]",
      "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": "original vertex=5;all complete graph edges=[[0, 1], [1, 2], [2, 3], [3, 4]];assigned part=0",
      "after": "all incident edges=[];whole partition proper=否;isolate stays in exactly one part",
      "relation": "implies",
      "rule_id": "bipartite_isolated_vertex_not_omitted_or_whole_graph_fixed",
      "conditions": [],
      "explanation": {
        "zh-Hant": "沒有incident邊才允許任選該孤立點的部；它仍屬完整載體且需恰有一標記。改它不影響邊約束，但也不修正其他邊的原有衝突。",
        "en": "Only after checking no incident edges may this isolate be put in either part.It remains in the complete carrier with exactly one label.Changing it neither affects edge constraints nor repairs unrelated conflicts."
      },
      "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": [
      5,
      [],
      0,
      false,
      "isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph"
    ]
  },
  "conclusion": {
    "canonical_value": "(5, [], 0, 否, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)",
    "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-r02-bipartite-graph-foundations",
      "batch_path": "public/data/questions/batches/batch-20261014-advance-20261031-r02-bipartite-graph-foundations/questions.jsonl",
      "batch_sha256": "013fee61d2719d2970c6b2a7d34b0bd208896e85fff9c06a97c22ed5aaf25113",
      "original_line": 89,
      "original_line_sha256": "ce238c6f43a454acf4581d0bc17560e4edac256aaf549c1f2e1489d4b68b0a3e",
      "question_compact_sha256": "b63905e2f95e5ca292763a871513225c46ba601722e9e2e569754b63a37c8828",
      "english_row_compact_sha256": "e0f061a68a12c572d2f89e272def9253abe80cc91356f98dfa846b2d1ed6563b",
      "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261031-r02-bipartite-graph-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261031-r02-bipartite-graph-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-bipartite-graph": {
        "sha256": "c54116e1bcf2bbb83b39260399271673ba3ad43ec2e0b27732fd148d843f7aa1",
        "english_sha256": "472cf9fff25f9a4141e6af6495349bcca9b7d0bc995a018899ee403da6dc3ab1",
        "original_path": "public/data/mko/bipartite-graph.json",
        "english_path": "public/data/i18n/editions/mko-bipartite-graph.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "d85118d74859ec7e59736714f767732c6ec092eaa0bf5279abeb28e4a628e3c3",
    "original_candidate_sha256": "adb5e80282fda59a84612f9bc5df66c5f4c0bffb9d13755f1377efaccd7dbb54",
    "original_candidate_line_sha256": "adb5e80282fda59a84612f9bc5df66c5f4c0bffb9d13755f1377efaccd7dbb54"
  },
  "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": "b5648286756bc9e70e050ef8ba4a1e05bb9a355536324d1d9bfab7778cb9826c",
      "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."
  }
}