{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261019-r01-open-closed-set-0090",
  "pilot_number": "R0018-09865",
  "number": 9865,
  "family_id": "qf-ext19-openclosed-family-closure",
  "title": {
    "zh-Hant": "第 9865 題：開閉集有限核對",
    "en": "Question 9865: Empty Families and Qualified Closure Premises"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261019-r01-open-closed-set-0090",
    "batch_id": "batch-20261013-advance-20261019-r01-open-closed-set-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-open-closed-set"
    ],
    "title_zh": "開閉集有限核對",
    "stem_zh": "完整有限拓撲。操作=openclosed-family-closure；資料={\"n\":4,\"opens\":[[],[2],[0,2],[0,2,3],[0,1,2,3]],\"subsets\":[[],[1,3],[0,1,2,3]],\"mode\":\"closed_intersection\"}。請依 [n,mode,selected,all_qualified,result,result_qualified] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(4, closed_intersection, [[], [1, 3], [0, 1, 2, 3]], 是, [], 是)",
      "explanation_zh": "第二批擴展核對=(4, closed_intersection, [[], [1, 3], [0, 1, 2, 3]], 是, [], 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "開閉集有限核對"
      ]
    },
    "variant": {
      "family_id": "qf-ext19-openclosed-family-closure",
      "seed": 202610191090,
      "parameters": {
        "n": 4,
        "opens": [
          [],
          [
            2
          ],
          [
            0,
            2
          ],
          [
            0,
            2,
            3
          ],
          [
            0,
            1,
            2,
            3
          ]
        ],
        "subsets": [
          [],
          [
            1,
            3
          ],
          [
            0,
            1,
            2,
            3
          ]
        ],
        "mode": "closed_intersection",
        "domain": "Fin_topology_open_closed",
        "operation": "openclosed-family-closure",
        "semantic_template": "qf-ext19-openclosed-family-closure/{\"mode\":\"closed_intersection\",\"n\":4,\"opens\":[[],[2],[0,2],[0,2,3],[0,1,2,3]],\"subsets\":[[],[1,3],[0,1,2,3]]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-13",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261013-advance-20261019-r01-open-closed-set-0090",
    "batch_id": "batch-20261013-advance-20261019-r01-open-closed-set-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "affb09996f00f7bd75387ed7135a96cee04393105276c3554a51678cde29feab",
      "family_id": "qf-ext19-openclosed-family-closure",
      "seed": 202610191090,
      "semantic_template": "qf-ext19-openclosed-family-closure/{\"mode\":\"closed_intersection\",\"n\":4,\"opens\":[[],[2],[0,2],[0,2,3],[0,1,2,3]],\"subsets\":[[],[1,3],[0,1,2,3]]}"
    },
    "fields": {
      "title": "Empty Families and Qualified Closure Premises",
      "stem": "Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [0, 2], [0, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. mode=closed_intersection, selected=[[], [1, 3], [0, 1, 2, 3]]. Modes are open_union, open_finite_intersection, closed_intersection, closed_finite_union. Give (n, mode, normalized selected family, all selected sets have the required open/closed property?, union/intersection result, result has that property?). Empty union is ∅ and empty intersection is X. Compute the result even if a premise fails.",
      "answer": {
        "kind": "expression",
        "value": "(4, closed_intersection, [[], [1, 3], [0, 1, 2, 3]], Yes, [], Yes)",
        "explanation": "Keep premise qualification separate from the resulting set’s property; a false premise does not prevent an accidentally qualified result. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, closed_intersection, [[], [1, 3], [0, 1, 2, 3]], Yes, [], Yes)."
      },
      "skills": [
        "Empty Families and Qualified Closure Premises"
      ],
      "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": "完整有限拓撲。操作=openclosed-family-closure；資料={\"n\":4,\"opens\":[[],[2],[0,2],[0,2,3],[0,1,2,3]],\"subsets\":[[],[1,3],[0,1,2,3]],\"mode\":\"closed_intersection\"}。請依 [n,mode,selected,all_qualified,result,result_qualified] 欄序回答。",
        "en": "Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [0, 2], [0, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. mode=closed_intersection, selected=[[], [1, 3], [0, 1, 2, 3]]. Modes are open_union, open_finite_intersection, closed_intersection, closed_finite_union. Give (n, mode, normalized selected family, all selected sets have the required open/closed property?, union/intersection result, result has that property?). Empty union is ∅ and empty intersection is X. Compute the result even if a premise fails."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "n=4;raw open family=[[], [2], [0, 2], [0, 2, 3], [0, 1, 2, 3]]",
      "after": "universe=[0, 1, 2, 3];normalized opens=[[], [2], [0, 2], [0, 2, 3], [0, 1, 2, 3]];empty=是;full=是;union closed=是;intersection closed=是",
      "relation": "implies",
      "rule_id": "given_complete_finite_topology_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "子集與族去重後按mask排序，完整核空集、全集及二元聯交閉合。有限底集合使任意聯集只需有限不同成員；本題要求真拓撲，非法資料不拿來算後續性質。",
        "en": "Deduplicate subsets/families and order by masks.Check empty/whole membership and all binary union/intersection closure.On a finite underlying set,arbitrary unions reduce to finitely many distinct members.These tasks require an actual topology;invalid data is not used for later properties."
      },
      "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": "mode=closed_intersection;selected member=[]",
      "after": "required property qualified=是",
      "relation": "implies",
      "rule_id": "closure_each_selected_property",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個選定集合先依模式分別核開或閉；只保留真假，不因有一個失格就停止求聯交結果。",
        "en": "Audit the required open/closed property of every selected set by the chosen mode.A failure does not stop computation of the union/intersection."
      },
      "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": "mode=closed_intersection;selected member=[1, 3]",
      "after": "required property qualified=是",
      "relation": "implies",
      "rule_id": "closure_each_selected_property",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個選定集合先依模式分別核開或閉；只保留真假，不因有一個失格就停止求聯交結果。",
        "en": "Audit the required open/closed property of every selected set by the chosen mode.A failure does not stop computation of the union/intersection."
      },
      "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": "mode=closed_intersection;selected member=[0, 1, 2, 3]",
      "after": "required property qualified=是",
      "relation": "implies",
      "rule_id": "closure_each_selected_property",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個選定集合先依模式分別核開或閉；只保留真假，不因有一個失格就停止求聯交結果。",
        "en": "Audit the required open/closed property of every selected set by the chosen mode.A failure does not stop computation of the union/intersection."
      },
      "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": "mode=closed_intersection;selected normalized family=[[], [1, 3], [0, 1, 2, 3]]",
      "after": "initial accumulator=[0, 1, 2, 3]",
      "relation": "implies",
      "rule_id": "empty_union_intersection_identity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "空聯集是空集，空交集是同一全集；先定此單位元，才依序折疊選定集合。空選定族的all-qualified為真。",
        "en": "The empty union is empty and the empty intersection is the same whole universe.Start from that identity and fold the selected sets.The all-qualified condition is true for an empty family."
      },
      "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": "mode=closed_intersection;accumulator=[0, 1, 2, 3];next member=[]",
      "after": "next accumulator=[]",
      "relation": "implies",
      "rule_id": "complete_selected_family_set_fold",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只按指定聯集／交集運算更新，原集合及模式不被偷偷替換。",
        "en": "Update using exactly the specified union/intersection operation without substituting a different set or mode."
      },
      "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": "mode=closed_intersection;accumulator=[];next member=[1, 3]",
      "after": "next accumulator=[]",
      "relation": "implies",
      "rule_id": "complete_selected_family_set_fold",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只按指定聯集／交集運算更新，原集合及模式不被偷偷替換。",
        "en": "Update using exactly the specified union/intersection operation without substituting a different set or mode."
      },
      "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": "mode=closed_intersection;accumulator=[];next member=[0, 1, 2, 3]",
      "after": "next accumulator=[]",
      "relation": "implies",
      "rule_id": "complete_selected_family_set_fold",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只按指定聯集／交集運算更新，原集合及模式不被偷偷替換。",
        "en": "Update using exactly the specified union/intersection operation without substituting a different set or mode."
      },
      "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": "all selected properties qualified=是;result=[]",
      "after": "result property qualified=是",
      "relation": "implies",
      "rule_id": "closure_premises_and_result_property_separate",
      "conditions": [],
      "explanation": {
        "zh-Hant": "閉合定理的前提全合格時保證結果資格；前提不合格時，結果仍可能碰巧合格，所以兩欄分開計算。",
        "en": "Qualified premises guarantee the corresponding closure property.If premises fail,the result may still qualify accidentally,so compute the two fields separately."
      },
      "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": [
      4,
      "closed_intersection",
      [
        [],
        [
          1,
          3
        ],
        [
          0,
          1,
          2,
          3
        ]
      ],
      true,
      [],
      true
    ]
  },
  "conclusion": {
    "canonical_value": "(4, closed_intersection, [[], [1, 3], [0, 1, 2, 3]], 是, [], 是)",
    "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-20261013-advance-20261019-r01-open-closed-set-foundations",
      "batch_path": "public/data/questions/batches/batch-20261013-advance-20261019-r01-open-closed-set-foundations/questions.jsonl",
      "batch_sha256": "043794a3a296b6f0436386fb98351c9abd056b196f271a2e35df448d3a44b021",
      "original_line": 90,
      "original_line_sha256": "94d05e6c08991e40385c0390afce9581ecfc75c3fb12f1d92708a995f008ffbb",
      "question_compact_sha256": "affb09996f00f7bd75387ed7135a96cee04393105276c3554a51678cde29feab",
      "english_row_compact_sha256": "f9eb834aafa91675c1cc2a990aafde11f91bfe8cc8c0368d8aef352cf40976b7",
      "english_manifest_path": "public/data/i18n/editions/batch-20261013-advance-20261019-r01-open-closed-set-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261013-advance-20261019-r01-open-closed-set-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-open-closed-set": {
        "sha256": "78fdc40df7a4161259566db404d47fa805aa334d6f661420675bbc8c139c6508",
        "english_sha256": "54fd15465e9b24d23c770140d38653759ffbdbae2087c9777b70d3cf4211da31",
        "original_path": "public/data/mko/open-closed-set.json",
        "english_path": "public/data/i18n/editions/mko-open-closed-set.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "2cf5df7bf9c3d48dfd41d968723b60d910c4ecfec9c2089d22ac1cda059a6086",
    "original_candidate_sha256": "acde93dec41a278f50206d5eb48fec1d4729b7f74bebc56a189deca8b66bf59d",
    "original_candidate_line_sha256": "acde93dec41a278f50206d5eb48fec1d4729b7f74bebc56a189deca8b66bf59d"
  },
  "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-20261026-006-p19",
    "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": "bd35da62087a15d6b89b45585dc46e89b9efbe097cba7b3c707b5030496f013c",
      "question_first_witness_sha256": "fab45a10d8ac95c7d61d32e47d1bdd3ca862c17c3289a6bf4b01d4dc9492fbb6"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 9,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "65c41808e103d72681c08d097c1e3a494e5f44d7a11f32a96e86a021989cb887",
      "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."
  }
}