{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261103-r06-maximum-matching-0059",
  "pilot_number": "R0033-17484",
  "number": 17484,
  "family_id": "qf-mod34-maxmatching-multiple-maxima",
  "title": {
    "zh-Hant": "第 17484 題：最大匹配",
    "en": "Question 17484: Distinct Maximum Matchings Need Not Be Perfect"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261103-r06-maximum-matching-0059",
    "batch_id": "batch-20261014-advance-20261103-r06-maximum-matching-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-maximum-matching"
    ],
    "title_zh": "最大匹配",
    "stem_zh": "edge count有界的完整有限簡單圖與全edge-subset枚舉。操作=maxmatching-multiple-maxima；資料={\"n\":6,\"edges\":[[0,1],[1,3],[2,3]]}。請依 [maximum_size,maximum_matchings,count,multiple] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(2, [[0, 2]], 1, 否)",
      "explanation_zh": "Wave5第三層核對=(2, [[0, 2]], 1, 否)；依完整明示資料重算；每個候選先核matching，maximum需全域比較；maximum不等於maximal、perfect或unique。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "最大匹配"
      ]
    },
    "variant": {
      "family_id": "qf-mod34-maxmatching-multiple-maxima",
      "seed": 202611036059,
      "parameters": {
        "n": 6,
        "edges": [
          [
            0,
            1
          ],
          [
            1,
            3
          ],
          [
            2,
            3
          ]
        ],
        "domain": "Exhaustive_finite_graph_maximum_matching",
        "operation": "maxmatching-multiple-maxima",
        "semantic_template": "qf-mod34-maxmatching-multiple-maxima/{\"edges\":[[0,1],[1,3],[2,3]],\"n\":6}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_or_symbolic_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、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、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-20261103-r06-maximum-matching-0059",
    "batch_id": "batch-20261014-advance-20261103-r06-maximum-matching-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "2c323ef808c5a8b81cd74a48e1d545538592f9d63f520545b2c3a8b28f1400ce",
      "family_id": "qf-mod34-maxmatching-multiple-maxima",
      "seed": 202611036059,
      "semantic_template": "qf-mod34-maxmatching-multiple-maxima/{\"edges\":[[0,1],[1,3],[2,3]],\"n\":6}"
    },
    "fields": {
      "title": "Distinct Maximum Matchings Need Not Be Perfect",
      "stem": "Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (maximum_size,maximum_matchings,count,multiple?), listing every maximum in the declared order.",
      "answer": {
        "kind": "expression",
        "value": "(2, [[0, 2]], 1, No)",
        "explanation": "Exhaustively collect valid matchings of global maximum size and count them. multiple means more than one such matching,not more than one edge or a perfect-matching certificate. The exact result, retaining the requested order, is (2, [[0, 2]], 1, No)."
      },
      "skills": [
        "Distinct Maximum Matchings Need Not Be Perfect"
      ],
      "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": "edge count有界的完整有限簡單圖與全edge-subset枚舉。操作=maxmatching-multiple-maxima；資料={\"n\":6,\"edges\":[[0,1],[1,3],[2,3]]}。請依 [maximum_size,maximum_matchings,count,multiple] 欄序回答。",
        "en": "Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (maximum_size,maximum_matchings,count,multiple?), listing every maximum in the declared order."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "complete original carrier Fin(6);ALREADY canonical original edge list=[[0, 1], [1, 3], [2, 3]]",
      "after": "loop-free unique original edges=3<=10;all edge subsets have mask0 through7",
      "relation": "implies",
      "rule_id": "maxmatching_complete_bounded_original_graph_mask_domain",
      "conditions": [],
      "explanation": {
        "zh-Hant": "n上界12、邊數上界10是本題可完整枚舉的域，不套前輪n30的guard。端點與邊表須原本已排序，索引固定原邊；無loop或重邊。",
        "en": "This family's bounds are n≤12 and at most10 edges,permitting exhaustive enumeration.Do not reuse the earlier n30 guard.Input endpoints and edges must already be canonical,with indices bound to original edges and no loops or duplicates."
      },
      "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": "original edge-subset integer mask=0;increasing original indices=[];actual selected edges=[]",
      "after": "used endpoint union=[];repeated endpoints in traversal order=[];valid matching=是;size=0",
      "relation": "implies",
      "rule_id": "maxmatching_every_mask_validity_before_global_comparison",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按整數遮罩遞增含空集合逐一核端點衝突；不合格族不能拿邊數參加最優值比較。每個原邊子集恰對應一遮罩，沒有漏掉替換已有邊的方案。",
        "en": "Enumerate integer masks in order,including the empty subset,and check endpoint conflicts first.Invalid collections cannot enter the optimum comparison.Every original edge subset has one mask,including alternatives that replace selected 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": "original edge-subset integer mask=1;increasing original indices=[0];actual selected edges=[[0, 1]]",
      "after": "used endpoint union=[0, 1];repeated endpoints in traversal order=[];valid matching=是;size=1",
      "relation": "implies",
      "rule_id": "maxmatching_every_mask_validity_before_global_comparison",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按整數遮罩遞增含空集合逐一核端點衝突；不合格族不能拿邊數參加最優值比較。每個原邊子集恰對應一遮罩，沒有漏掉替換已有邊的方案。",
        "en": "Enumerate integer masks in order,including the empty subset,and check endpoint conflicts first.Invalid collections cannot enter the optimum comparison.Every original edge subset has one mask,including alternatives that replace selected 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": "original edge-subset integer mask=2;increasing original indices=[1];actual selected edges=[[1, 3]]",
      "after": "used endpoint union=[1, 3];repeated endpoints in traversal order=[];valid matching=是;size=1",
      "relation": "implies",
      "rule_id": "maxmatching_every_mask_validity_before_global_comparison",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按整數遮罩遞增含空集合逐一核端點衝突；不合格族不能拿邊數參加最優值比較。每個原邊子集恰對應一遮罩，沒有漏掉替換已有邊的方案。",
        "en": "Enumerate integer masks in order,including the empty subset,and check endpoint conflicts first.Invalid collections cannot enter the optimum comparison.Every original edge subset has one mask,including alternatives that replace selected 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": "original edge-subset integer mask=3;increasing original indices=[0, 1];actual selected edges=[[0, 1], [1, 3]]",
      "after": "used endpoint union=[0, 1, 3];repeated endpoints in traversal order=[1];valid matching=否;size=2",
      "relation": "implies",
      "rule_id": "maxmatching_every_mask_validity_before_global_comparison",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按整數遮罩遞增含空集合逐一核端點衝突；不合格族不能拿邊數參加最優值比較。每個原邊子集恰對應一遮罩，沒有漏掉替換已有邊的方案。",
        "en": "Enumerate integer masks in order,including the empty subset,and check endpoint conflicts first.Invalid collections cannot enter the optimum comparison.Every original edge subset has one mask,including alternatives that replace selected 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": "original edge-subset integer mask=4;increasing original indices=[2];actual selected edges=[[2, 3]]",
      "after": "used endpoint union=[2, 3];repeated endpoints in traversal order=[];valid matching=是;size=1",
      "relation": "implies",
      "rule_id": "maxmatching_every_mask_validity_before_global_comparison",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按整數遮罩遞增含空集合逐一核端點衝突；不合格族不能拿邊數參加最優值比較。每個原邊子集恰對應一遮罩，沒有漏掉替換已有邊的方案。",
        "en": "Enumerate integer masks in order,including the empty subset,and check endpoint conflicts first.Invalid collections cannot enter the optimum comparison.Every original edge subset has one mask,including alternatives that replace selected 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 edge-subset integer mask=5;increasing original indices=[0, 2];actual selected edges=[[0, 1], [2, 3]]",
      "after": "used endpoint union=[0, 1, 2, 3];repeated endpoints in traversal order=[];valid matching=是;size=2",
      "relation": "implies",
      "rule_id": "maxmatching_every_mask_validity_before_global_comparison",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按整數遮罩遞增含空集合逐一核端點衝突；不合格族不能拿邊數參加最優值比較。每個原邊子集恰對應一遮罩，沒有漏掉替換已有邊的方案。",
        "en": "Enumerate integer masks in order,including the empty subset,and check endpoint conflicts first.Invalid collections cannot enter the optimum comparison.Every original edge subset has one mask,including alternatives that replace selected 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": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "original edge-subset integer mask=6;increasing original indices=[1, 2];actual selected edges=[[1, 3], [2, 3]]",
      "after": "used endpoint union=[1, 2, 3];repeated endpoints in traversal order=[3];valid matching=否;size=2",
      "relation": "implies",
      "rule_id": "maxmatching_every_mask_validity_before_global_comparison",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按整數遮罩遞增含空集合逐一核端點衝突；不合格族不能拿邊數參加最優值比較。每個原邊子集恰對應一遮罩，沒有漏掉替換已有邊的方案。",
        "en": "Enumerate integer masks in order,including the empty subset,and check endpoint conflicts first.Invalid collections cannot enter the optimum comparison.Every original edge subset has one mask,including alternatives that replace selected 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": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "original edge-subset integer mask=7;increasing original indices=[0, 1, 2];actual selected edges=[[0, 1], [1, 3], [2, 3]]",
      "after": "used endpoint union=[0, 1, 2, 3];repeated endpoints in traversal order=[1, 3];valid matching=否;size=3",
      "relation": "implies",
      "rule_id": "maxmatching_every_mask_validity_before_global_comparison",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按整數遮罩遞增含空集合逐一核端點衝突；不合格族不能拿邊數參加最優值比較。每個原邊子集恰對應一遮罩，沒有漏掉替換已有邊的方案。",
        "en": "Enumerate integer masks in order,including the empty subset,and check endpoint conflicts first.Invalid collections cannot enter the optimum comparison.Every original edge subset has one mask,including alternatives that replace selected 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": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "exhaustive matching validity indicator count := sum(1, 1, 1, 0, 1, 1, 0, 0)",
      "after": "exhaustive matching validity indicator count = 5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依同一原始角色與已核中間量精確計算，保留分母、正負號、座標與索引。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 exhaustive matching validity indicator count，不以小數近似替代。",
          "en": "Recompute exhaustive matching validity indicator count with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "1",
          "1",
          "0",
          "1",
          "1",
          "0",
          "0"
        ],
        "output": "5",
        "label": "exhaustive matching validity indicator count"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "all8 original masks examined;valid matching count=5;all valid sizes=[0, 1, 1, 1, 2]",
      "after": "global maximum size=2;ALL maxima in ORIGINAL MASK order=[[0, 2]]",
      "relation": "implies",
      "rule_id": "maxmatching_exhaustive_upper_bound_and_attaining_all_witnesses",
      "conditions": [],
      "explanation": {
        "zh-Hant": "有效候選全列後每一個大小都≤此最大值，且保留所有達到者，才同時有全域上界及達到見證。最大解依遮罩順序，不再按索引清單字典序排序，也不宣稱一般複雜度定理。",
        "en": "Every valid candidate has size at most this maximum,and every attaining witness is retained.This supplies both a global upper bound and attainment.Keep mask order rather than lexicographic list order,and infer no general complexity theorem."
      },
      "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": [
      2,
      [
        [
          0,
          2
        ]
      ],
      1,
      false
    ]
  },
  "conclusion": {
    "canonical_value": "(2, [[0, 2]], 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-20261103-r06-maximum-matching-foundations",
      "batch_path": "public/data/questions/revisions/ocme-reference-revision-20261017/batch-20261014-advance-20261103-r06-maximum-matching-foundations/questions.jsonl",
      "batch_sha256": "36d36d687ea400d9490dcdc67eaace4ab92c6337297da67df193b946fbd81506",
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      "original_line_sha256": "f1876ea64ec78e75d48ad32fc4fe34635c8066c92c8ed6a7de4f7a31852d72c5",
      "question_compact_sha256": "2c323ef808c5a8b81cd74a48e1d545538592f9d63f520545b2c3a8b28f1400ce",
      "english_row_compact_sha256": "4b8e156840147176b5e1f33bc1289a32c513218e4ada855a574368dfbec6fc45",
      "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261103-r06-maximum-matching-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261103-r06-maximum-matching-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-maximum-matching": {
        "sha256": "bb97ee67fcffa420b79baaaf56c7ebabe2977fb5d14f5826d1f1c0452f4ed7f9",
        "english_sha256": "62ea84a565cbbc9bfd9f80ccaf5cdd5474c7027eca349c25cded547f5fbe5886",
        "original_path": "public/data/mko/maximum-matching.json",
        "english_path": "public/data/i18n/editions/mko-maximum-matching.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "c6f506c0e91ce68861b828c110bfee6002681c5a6e691f1ba5ae296376b30ad5",
    "original_candidate_sha256": "11c8d81690cabdb06cefd5b8b682ecc8fbce85456b7171d170f79bbe639b16d3",
    "original_candidate_line_sha256": "11c8d81690cabdb06cefd5b8b682ecc8fbce85456b7171d170f79bbe639b16d3"
  },
  "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-20261026-008-p34",
    "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": "4a0957a15bb33fbe5f663e37a8babf46ab5a291eb5d1dfa3bf9045204cd43ec3",
      "question_first_witness_sha256": "22a8506bcfa34c503216083d1a8e813d61aad22dbc81589c0d41089c528a7f93"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 11,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "0efca9d9ae129a13d5542478839f52f44194e9892020e83d342340523e951fcd",
      "arithmetic_steps": 1
    },
    "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."
  }
}