{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0032",
  "pilot_number": "R0019-10407",
  "number": 10407,
  "family_id": "qf-mod23-measure-closure-inventory",
  "title": {
    "zh-Hant": "第 10407 題：有限測度空間",
    "en": "Question 10407: List Missing Closure Obligations without Guessing a Sigma-Algebra"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0032",
    "batch_id": "batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-sigma-algebra-measure-space"
    ],
    "title_zh": "有限測度空間",
    "stem_zh": "完整有限載體與canonical集合族。操作=measure-closure-inventory；資料={\"n\":7,\"family\":[[],[0,1,2],[3,4,5,6],[0,1,2,3,4,5,6]]}。請依 [n,family,missing_complements,missing_unions,finite_countable_reduction] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(7, [[], [0, 1, 2], [3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], 是)",
      "explanation_zh": "現代橋梁核對=(7, [[], [0, 1, 2], [3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；可數語義只由指定有限載體歸約；不外推任意無限測度空間。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "有限測度空間"
      ]
    },
    "variant": {
      "family_id": "qf-mod23-measure-closure-inventory",
      "seed": 202610235032,
      "parameters": {
        "n": 7,
        "family": [
          [],
          [
            0,
            1,
            2
          ],
          [
            3,
            4,
            5,
            6
          ],
          [
            0,
            1,
            2,
            3,
            4,
            5,
            6
          ]
        ],
        "domain": "Complete_finite_measure_space",
        "operation": "measure-closure-inventory",
        "semantic_template": "qf-mod23-measure-closure-inventory/{\"family\":[[],[0,1,2,3,4,5,6],[0,1,2],[3,4,5,6]],\"n\":7}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、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-20261023-r05-sigma-algebra-measure-space-0032",
    "batch_id": "batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "06abcb29dd57246e6274f22cec789fa7b18906954dea25367cd0b1f04e2786f8",
      "family_id": "qf-mod23-measure-closure-inventory",
      "seed": 202610235032,
      "semantic_template": "qf-mod23-measure-closure-inventory/{\"family\":[[],[0,1,2,3,4,5,6],[0,1,2],[3,4,5,6]],\"n\":7}"
    },
    "fields": {
      "title": "List Missing Closure Obligations without Guessing a Sigma-Algebra",
      "stem": "Use the explicit finite carrier Fin(7) and complete set family=[[], [0, 1, 2], [3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Give (size,canonical family,missing [set,complement] records,missing [left,right,union] records,empty/full and both closure checks pass?).",
      "answer": {
        "kind": "expression",
        "value": "(7, [[], [0, 1, 2], [3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)",
        "explanation": "Enumerate every declared family member and ordered pair. The finite-carrier reduction is not a general infinite-carrier closure theorem. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [0, 1, 2], [3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)."
      },
      "skills": [
        "List Missing Closure Obligations without Guessing a Sigma-Algebra"
      ],
      "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": "完整有限載體與canonical集合族。操作=measure-closure-inventory；資料={\"n\":7,\"family\":[[],[0,1,2],[3,4,5,6],[0,1,2,3,4,5,6]]}。請依 [n,family,missing_complements,missing_unions,finite_countable_reduction] 欄序回答。",
        "en": "Use the explicit finite carrier Fin(7) and complete set family=[[], [0, 1, 2], [3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Give (size,canonical family,missing [set,complement] records,missing [left,right,union] records,empty/full and both closure checks pass?)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "carrier Fin(7);raw family=[[], [0, 1, 2], [3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]",
      "after": "already canonical family=[[], [0, 1, 2], [3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]",
      "relation": "implies",
      "rule_id": "finite_sigma_canonical_alignment_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個集合須已遞增且相異，族須已依mask排序且無重複。此協定拒絕非canonical資料，不重排後面的測度值來洗白對齊。",
        "en": "Each set must already be sorted/distinct,and the family mask-ordered and duplicate-free.This protocol rejects noncanonical data rather than reordering aligned measure values to repair it."
      },
      "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": "source member=[];fixed carrier=[0, 1, 2, 3, 4, 5, 6]",
      "after": "complement=[0, 1, 2, 3, 4, 5, 6];present=是",
      "relation": "implies",
      "rule_id": "sigma_every_relative_complement",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐一取同一載體的補集，保留所有缺補集記錄。",
        "en": "Take the complement in this same carrier for every member and retain every missing-complement record."
      },
      "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": "ordered left=[];ordered right=[]",
      "after": "union=[];present=是",
      "relation": "implies",
      "rule_id": "sigma_every_ordered_binary_union",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整核所有有序對，反向對也依協定保留；不只列第一個反例或抽樣聯集。",
        "en": "Check every ordered pair,retaining reverse-pair records as specified.Do not stop at a first failure or sample unions."
      },
      "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": "ordered left=[];ordered right=[0, 1, 2]",
      "after": "union=[0, 1, 2];present=是",
      "relation": "implies",
      "rule_id": "sigma_every_ordered_binary_union",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整核所有有序對，反向對也依協定保留；不只列第一個反例或抽樣聯集。",
        "en": "Check every ordered pair,retaining reverse-pair records as specified.Do not stop at a first failure or sample unions."
      },
      "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": "ordered left=[];ordered right=[3, 4, 5, 6]",
      "after": "union=[3, 4, 5, 6];present=是",
      "relation": "implies",
      "rule_id": "sigma_every_ordered_binary_union",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整核所有有序對，反向對也依協定保留；不只列第一個反例或抽樣聯集。",
        "en": "Check every ordered pair,retaining reverse-pair records as specified.Do not stop at a first failure or sample unions."
      },
      "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": "ordered left=[];ordered right=[0, 1, 2, 3, 4, 5, 6]",
      "after": "union=[0, 1, 2, 3, 4, 5, 6];present=是",
      "relation": "implies",
      "rule_id": "sigma_every_ordered_binary_union",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整核所有有序對，反向對也依協定保留；不只列第一個反例或抽樣聯集。",
        "en": "Check every ordered pair,retaining reverse-pair records as specified.Do not stop at a first failure or sample unions."
      },
      "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": "source member=[0, 1, 2];fixed carrier=[0, 1, 2, 3, 4, 5, 6]",
      "after": "complement=[3, 4, 5, 6];present=是",
      "relation": "implies",
      "rule_id": "sigma_every_relative_complement",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐一取同一載體的補集，保留所有缺補集記錄。",
        "en": "Take the complement in this same carrier for every member and retain every missing-complement record."
      },
      "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": "ordered left=[0, 1, 2];ordered right=[]",
      "after": "union=[0, 1, 2];present=是",
      "relation": "implies",
      "rule_id": "sigma_every_ordered_binary_union",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整核所有有序對，反向對也依協定保留；不只列第一個反例或抽樣聯集。",
        "en": "Check every ordered pair,retaining reverse-pair records as specified.Do not stop at a first failure or sample unions."
      },
      "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": "ordered left=[0, 1, 2];ordered right=[0, 1, 2]",
      "after": "union=[0, 1, 2];present=是",
      "relation": "implies",
      "rule_id": "sigma_every_ordered_binary_union",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整核所有有序對，反向對也依協定保留；不只列第一個反例或抽樣聯集。",
        "en": "Check every ordered pair,retaining reverse-pair records as specified.Do not stop at a first failure or sample unions."
      },
      "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": "ordered left=[0, 1, 2];ordered right=[3, 4, 5, 6]",
      "after": "union=[0, 1, 2, 3, 4, 5, 6];present=是",
      "relation": "implies",
      "rule_id": "sigma_every_ordered_binary_union",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整核所有有序對，反向對也依協定保留；不只列第一個反例或抽樣聯集。",
        "en": "Check every ordered pair,retaining reverse-pair records as specified.Do not stop at a first failure or sample unions."
      },
      "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": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "ordered left=[0, 1, 2];ordered right=[0, 1, 2, 3, 4, 5, 6]",
      "after": "union=[0, 1, 2, 3, 4, 5, 6];present=是",
      "relation": "implies",
      "rule_id": "sigma_every_ordered_binary_union",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整核所有有序對，反向對也依協定保留；不只列第一個反例或抽樣聯集。",
        "en": "Check every ordered pair,retaining reverse-pair records as specified.Do not stop at a first failure or sample unions."
      },
      "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": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "source member=[3, 4, 5, 6];fixed carrier=[0, 1, 2, 3, 4, 5, 6]",
      "after": "complement=[0, 1, 2];present=是",
      "relation": "implies",
      "rule_id": "sigma_every_relative_complement",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐一取同一載體的補集，保留所有缺補集記錄。",
        "en": "Take the complement in this same carrier for every member and retain every missing-complement record."
      },
      "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": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "ordered left=[3, 4, 5, 6];ordered right=[]",
      "after": "union=[3, 4, 5, 6];present=是",
      "relation": "implies",
      "rule_id": "sigma_every_ordered_binary_union",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整核所有有序對，反向對也依協定保留；不只列第一個反例或抽樣聯集。",
        "en": "Check every ordered pair,retaining reverse-pair records as specified.Do not stop at a first failure or sample unions."
      },
      "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": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "ordered left=[3, 4, 5, 6];ordered right=[0, 1, 2]",
      "after": "union=[0, 1, 2, 3, 4, 5, 6];present=是",
      "relation": "implies",
      "rule_id": "sigma_every_ordered_binary_union",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整核所有有序對，反向對也依協定保留；不只列第一個反例或抽樣聯集。",
        "en": "Check every ordered pair,retaining reverse-pair records as specified.Do not stop at a first failure or sample unions."
      },
      "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": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "ordered left=[3, 4, 5, 6];ordered right=[3, 4, 5, 6]",
      "after": "union=[3, 4, 5, 6];present=是",
      "relation": "implies",
      "rule_id": "sigma_every_ordered_binary_union",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整核所有有序對，反向對也依協定保留；不只列第一個反例或抽樣聯集。",
        "en": "Check every ordered pair,retaining reverse-pair records as specified.Do not stop at a first failure or sample unions."
      },
      "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": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "ordered left=[3, 4, 5, 6];ordered right=[0, 1, 2, 3, 4, 5, 6]",
      "after": "union=[0, 1, 2, 3, 4, 5, 6];present=是",
      "relation": "implies",
      "rule_id": "sigma_every_ordered_binary_union",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整核所有有序對，反向對也依協定保留；不只列第一個反例或抽樣聯集。",
        "en": "Check every ordered pair,retaining reverse-pair records as specified.Do not stop at a first failure or sample unions."
      },
      "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": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "source member=[0, 1, 2, 3, 4, 5, 6];fixed carrier=[0, 1, 2, 3, 4, 5, 6]",
      "after": "complement=[];present=是",
      "relation": "implies",
      "rule_id": "sigma_every_relative_complement",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐一取同一載體的補集，保留所有缺補集記錄。",
        "en": "Take the complement in this same carrier for every member and retain every missing-complement record."
      },
      "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": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "ordered left=[0, 1, 2, 3, 4, 5, 6];ordered right=[]",
      "after": "union=[0, 1, 2, 3, 4, 5, 6];present=是",
      "relation": "implies",
      "rule_id": "sigma_every_ordered_binary_union",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整核所有有序對，反向對也依協定保留；不只列第一個反例或抽樣聯集。",
        "en": "Check every ordered pair,retaining reverse-pair records as specified.Do not stop at a first failure or sample unions."
      },
      "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": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "ordered left=[0, 1, 2, 3, 4, 5, 6];ordered right=[0, 1, 2]",
      "after": "union=[0, 1, 2, 3, 4, 5, 6];present=是",
      "relation": "implies",
      "rule_id": "sigma_every_ordered_binary_union",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整核所有有序對，反向對也依協定保留；不只列第一個反例或抽樣聯集。",
        "en": "Check every ordered pair,retaining reverse-pair records as specified.Do not stop at a first failure or sample unions."
      },
      "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": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "ordered left=[0, 1, 2, 3, 4, 5, 6];ordered right=[3, 4, 5, 6]",
      "after": "union=[0, 1, 2, 3, 4, 5, 6];present=是",
      "relation": "implies",
      "rule_id": "sigma_every_ordered_binary_union",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整核所有有序對，反向對也依協定保留；不只列第一個反例或抽樣聯集。",
        "en": "Check every ordered pair,retaining reverse-pair records as specified.Do not stop at a first failure or sample unions."
      },
      "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": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "ordered left=[0, 1, 2, 3, 4, 5, 6];ordered right=[0, 1, 2, 3, 4, 5, 6]",
      "after": "union=[0, 1, 2, 3, 4, 5, 6];present=是",
      "relation": "implies",
      "rule_id": "sigma_every_ordered_binary_union",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整核所有有序對，反向對也依協定保留；不只列第一個反例或抽樣聯集。",
        "en": "Check every ordered pair,retaining reverse-pair records as specified.Do not stop at a first failure or sample unions."
      },
      "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": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "empty present=是;full present=是;missing complements=[];missing unions=[]",
      "after": "finite countable reduction=是;sigma qualified=是",
      "relation": "implies",
      "rule_id": "finite_sigma_countable_union_reduction",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定有限載體只有有限多個不同子集，可數族的聯集等於有限不同成員的聯集，所以空／全集、補集與二元聯集檢查在此足夠。不能外推任意無限載體。",
        "en": "A fixed finite carrier has finitely many distinct subsets,so a countable family's union equals a finite distinct-member union.Empty/whole membership,complements and binary unions therefore suffice here,not for arbitrary infinite carriers."
      },
      "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": [
      7,
      [
        [],
        [
          0,
          1,
          2
        ],
        [
          3,
          4,
          5,
          6
        ],
        [
          0,
          1,
          2,
          3,
          4,
          5,
          6
        ]
      ],
      [],
      [],
      true
    ]
  },
  "conclusion": {
    "canonical_value": "(7, [[], [0, 1, 2], [3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], 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-20261023-r05-sigma-algebra-measure-space-foundations",
      "batch_path": "public/data/questions/batches/batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations/questions.jsonl",
      "batch_sha256": "04fbc4beaedc039f6ef83416753cabb1a18688ec3adae59ec64ae687867a8394",
      "original_line": 32,
      "original_line_sha256": "6263ab97056a683192cf4a06ec9838e762ba80b8df6f7a1dcd573987fc7611c4",
      "question_compact_sha256": "06abcb29dd57246e6274f22cec789fa7b18906954dea25367cd0b1f04e2786f8",
      "english_row_compact_sha256": "41e405c2cac1d107c4dde0f48c965bf230e903fc4973824663c53c6d036b6b22",
      "english_manifest_path": "public/data/i18n/editions/batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-sigma-algebra-measure-space": {
        "sha256": "36f29056738113842fd5fc437c89ae6f234815fa3abaea4cc68e968a358d6650",
        "english_sha256": "2148bebcaf753b4b69f42f056cf1cca5abc628d871406923f55af3da86c876b6",
        "original_path": "public/data/mko/sigma-algebra-measure-space.json",
        "english_path": "public/data/i18n/editions/mko-sigma-algebra-measure-space.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "7359cb2346bdb6bf38b7ca4a5fd5b5a99c3007a711f34346754d57b77b2dfcc8",
    "original_candidate_sha256": "b0d26265cee00fd1fa28980131d9b2f1ca083fed86540a5a2d26fed04df1e257",
    "original_candidate_line_sha256": "b0d26265cee00fd1fa28980131d9b2f1ca083fed86540a5a2d26fed04df1e257"
  },
  "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-p20",
    "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": "eeb35dc5daab92234bf72827627392b47f38ead4cfb574d4373bebc43c09fbf5",
      "question_first_witness_sha256": "3cf8e0d3a00b0df1994db2bfde81f4a31b382e170bdda83ea3d0b68da64e8419"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 22,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "bb511744cf65669095aa67d22f6a3189008ed1cca6c508030cf2b43f59885706",
      "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."
  }
}