{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0098",
  "pilot_number": "R0019-10473",
  "number": 10473,
  "family_id": "qf-mod23-measure-probability-measure-audit",
  "title": {
    "zh-Hant": "第 10473 題：有限測度空間",
    "en": "Question 10473: A Finite Measure Need Not Have Total Mass One"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0098",
    "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-probability-measure-audit；資料={\"n\":7,\"family\":[[],[0,1,2,4],[3,5,6],[0,1,2,3,4,5,6]],\"values\":[[0,1],[23,47],[24,47],[1,1]]}。請依 [n,family,values,empty_zero,bad_disjoint_pairs,total,finite_countable_reduction,probability_measure] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(7, [[], [0, 1, 2, 4], [3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 23/47, 24/47, 1], 是, [], 1, 是, 是)",
      "explanation_zh": "現代橋梁核對=(7, [[], [0, 1, 2, 4], [3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 23/47, 24/47, 1], 是, [], 1, 是, 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": 3,
      "skills": [
        "有限測度空間"
      ]
    },
    "variant": {
      "family_id": "qf-mod23-measure-probability-measure-audit",
      "seed": 202610235098,
      "parameters": {
        "n": 7,
        "family": [
          [],
          [
            0,
            1,
            2,
            4
          ],
          [
            3,
            5,
            6
          ],
          [
            0,
            1,
            2,
            3,
            4,
            5,
            6
          ]
        ],
        "values": [
          [
            0,
            1
          ],
          [
            23,
            47
          ],
          [
            24,
            47
          ],
          [
            1,
            1
          ]
        ],
        "domain": "Complete_finite_measure_space",
        "operation": "measure-probability-measure-audit",
        "semantic_template": "qf-mod23-measure-probability-measure-audit/{\"family\":[[],[0,1,2,3,4,5,6],[0,1,2,4],[3,5,6]],\"n\":7,\"values\":[[0,1],[23,47],[24,47],[1,1]]}"
      }
    },
    "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-0098",
    "batch_id": "batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "8ce70b99f6733995f9157c5e6404016ecd9251abcfd047e2cfe665f8b8936fc9",
      "family_id": "qf-mod23-measure-probability-measure-audit",
      "seed": 202610235098,
      "semantic_template": "qf-mod23-measure-probability-measure-audit/{\"family\":[[],[0,1,2,3,4,5,6],[0,1,2,4],[3,5,6]],\"n\":7,\"values\":[[0,1],[23,47],[24,47],[1,1]]}"
    },
    "fields": {
      "title": "A Finite Measure Need Not Have Total Mass One",
      "stem": "Use the explicit finite carrier Fin(7) and complete set family=[[], [0, 1, 2, 4], [3, 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. Require a qualified sigma-algebra; aligned nonnegative rational values=[0/1, 23/47, 24/47, 1/1]. Give (size,family,values,empty-zero?,all bad disjoint-pair records,total carrier mass,finite countable-additivity reduction?,probability measure?). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.",
      "answer": {
        "kind": "expression",
        "value": "(7, [[], [0, 1, 2, 4], [3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 23/47, 24/47, 1], Yes, [], 1, Yes, Yes)",
        "explanation": "First check empty-zero and every disjoint pair on the finite carrier. A probability measure additionally requires total mass exactly 1; a valid measure of total 2 is not a probability measure. 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, 4], [3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 23/47, 24/47, 1], Yes, [], 1, Yes, Yes)."
      },
      "skills": [
        "A Finite Measure Need Not Have Total Mass One"
      ],
      "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-probability-measure-audit；資料={\"n\":7,\"family\":[[],[0,1,2,4],[3,5,6],[0,1,2,3,4,5,6]],\"values\":[[0,1],[23,47],[24,47],[1,1]]}。請依 [n,family,values,empty_zero,bad_disjoint_pairs,total,finite_countable_reduction,probability_measure] 欄序回答。",
        "en": "Use the explicit finite carrier Fin(7) and complete set family=[[], [0, 1, 2, 4], [3, 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. Require a qualified sigma-algebra; aligned nonnegative rational values=[0/1, 23/47, 24/47, 1/1]. Give (size,family,values,empty-zero?,all bad disjoint-pair records,total carrier mass,finite countable-additivity reduction?,probability measure?). Use exact reduced rational outputs with positive denominators, omitting /1 for integers."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "carrier Fin(7);raw family=[[], [0, 1, 2, 4], [3, 5, 6], [0, 1, 2, 3, 4, 5, 6]]",
      "after": "already canonical family=[[], [0, 1, 2, 4], [3, 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, 4]",
      "after": "union=[0, 1, 2, 4];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, 5, 6]",
      "after": "union=[3, 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, 4];fixed carrier=[0, 1, 2, 3, 4, 5, 6]",
      "after": "complement=[3, 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, 4];ordered right=[]",
      "after": "union=[0, 1, 2, 4];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, 4];ordered right=[0, 1, 2, 4]",
      "after": "union=[0, 1, 2, 4];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, 4];ordered right=[3, 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, 4];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, 5, 6];fixed carrier=[0, 1, 2, 3, 4, 5, 6]",
      "after": "complement=[0, 1, 2, 4];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, 5, 6];ordered right=[]",
      "after": "union=[3, 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, 5, 6];ordered right=[0, 1, 2, 4]",
      "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, 5, 6];ordered right=[3, 5, 6]",
      "after": "union=[3, 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, 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, 4]",
      "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, 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
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "canonical row index=0;set=[];raw rational=[0, 1]",
      "after": "aligned normalized measure=0;nonnegative=是",
      "relation": "implies",
      "rule_id": "measure_original_family_value_pairing",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原分母非零，正負號正規化後須非負；數值綁原canonical行，不能排序數值或拿另一集合的質量替代。",
        "en": "The original denominator is nonzero and the normalized value must be nonnegative.Bind each value to its original canonical row,not a sorted mass list or another set's mass."
      },
      "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": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "canonical row index=1;set=[0, 1, 2, 4];raw rational=[23, 47]",
      "after": "aligned normalized measure=23/47;nonnegative=是",
      "relation": "implies",
      "rule_id": "measure_original_family_value_pairing",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原分母非零，正負號正規化後須非負；數值綁原canonical行，不能排序數值或拿另一集合的質量替代。",
        "en": "The original denominator is nonzero and the normalized value must be nonnegative.Bind each value to its original canonical row,not a sorted mass list or another set's mass."
      },
      "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": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "canonical row index=2;set=[3, 5, 6];raw rational=[24, 47]",
      "after": "aligned normalized measure=24/47;nonnegative=是",
      "relation": "implies",
      "rule_id": "measure_original_family_value_pairing",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原分母非零，正負號正規化後須非負；數值綁原canonical行，不能排序數值或拿另一集合的質量替代。",
        "en": "The original denominator is nonzero and the normalized value must be nonnegative.Bind each value to its original canonical row,not a sorted mass list or another set's mass."
      },
      "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": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "canonical row index=3;set=[0, 1, 2, 3, 4, 5, 6];raw rational=[1, 1]",
      "after": "aligned normalized measure=1;nonnegative=是",
      "relation": "implies",
      "rule_id": "measure_original_family_value_pairing",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原分母非零，正負號正規化後須非負；數值綁原canonical行，不能排序數值或拿另一集合的質量替代。",
        "en": "The original denominator is nonzero and the normalized value must be nonnegative.Bind each value to its original canonical row,not a sorted mass list or another set's mass."
      },
      "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": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "empty-set measure=0",
      "after": "empty-set zero=是",
      "relation": "implies",
      "rule_id": "measure_empty_zero_separate_obligation",
      "conditions": [],
      "explanation": {
        "zh-Hant": "空集零測度是獨立輸出義務，不能只靠總量或某些可加性樣本略過。",
        "en": "Zero empty-set measure is a separate output obligation,not waived by total mass or selected additivity samples."
      },
      "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": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "measure sum rows0,0 := sum(0, 0)",
      "after": "measure sum rows0,0 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。",
        "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 measure sum rows0,0，不以小數近似替代。",
          "en": "Recompute measure sum rows0,0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "measure sum rows0,0"
      }
    },
    {
      "id": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "disjoint ordered pair=[[], []];union=[];mu(union)=0;mu(A)=0;mu(B)=0",
      "after": "sum=0;additivity=是",
      "relation": "implies",
      "rule_id": "measure_all_disjoint_ordered_pairs",
      "conditions": [],
      "explanation": {
        "zh-Hant": "含空集的有序對也要核；只對互斥集合用二元可加性，所有失敗都保留。",
        "en": "Include ordered pairs involving the empty set.Apply binary additivity to disjoint sets only and retain every failure."
      },
      "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": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "measure sum rows0,1 := sum(0, 23/47)",
      "after": "measure sum rows0,1 = 23/47",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。",
        "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 measure sum rows0,1，不以小數近似替代。",
          "en": "Recompute measure sum rows0,1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "23/47"
        ],
        "output": "23/47",
        "label": "measure sum rows0,1"
      }
    },
    {
      "id": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "disjoint ordered pair=[[], [0, 1, 2, 4]];union=[0, 1, 2, 4];mu(union)=23/47;mu(A)=0;mu(B)=23/47",
      "after": "sum=23/47;additivity=是",
      "relation": "implies",
      "rule_id": "measure_all_disjoint_ordered_pairs",
      "conditions": [],
      "explanation": {
        "zh-Hant": "含空集的有序對也要核；只對互斥集合用二元可加性，所有失敗都保留。",
        "en": "Include ordered pairs involving the empty set.Apply binary additivity to disjoint sets only and retain every failure."
      },
      "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": "S032",
      "uses": [
        "G001",
        "S031"
      ],
      "before": "measure sum rows0,2 := sum(0, 24/47)",
      "after": "measure sum rows0,2 = 24/47",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。",
        "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 measure sum rows0,2，不以小數近似替代。",
          "en": "Recompute measure sum rows0,2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "24/47"
        ],
        "output": "24/47",
        "label": "measure sum rows0,2"
      }
    },
    {
      "id": "S033",
      "uses": [
        "G001",
        "S032"
      ],
      "before": "disjoint ordered pair=[[], [3, 5, 6]];union=[3, 5, 6];mu(union)=24/47;mu(A)=0;mu(B)=24/47",
      "after": "sum=24/47;additivity=是",
      "relation": "implies",
      "rule_id": "measure_all_disjoint_ordered_pairs",
      "conditions": [],
      "explanation": {
        "zh-Hant": "含空集的有序對也要核；只對互斥集合用二元可加性，所有失敗都保留。",
        "en": "Include ordered pairs involving the empty set.Apply binary additivity to disjoint sets only and retain every failure."
      },
      "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": "S034",
      "uses": [
        "G001",
        "S033"
      ],
      "before": "measure sum rows0,3 := sum(0, 1)",
      "after": "measure sum rows0,3 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。",
        "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 measure sum rows0,3，不以小數近似替代。",
          "en": "Recompute measure sum rows0,3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "1"
        ],
        "output": "1",
        "label": "measure sum rows0,3"
      }
    },
    {
      "id": "S035",
      "uses": [
        "G001",
        "S034"
      ],
      "before": "disjoint ordered pair=[[], [0, 1, 2, 3, 4, 5, 6]];union=[0, 1, 2, 3, 4, 5, 6];mu(union)=1;mu(A)=0;mu(B)=1",
      "after": "sum=1;additivity=是",
      "relation": "implies",
      "rule_id": "measure_all_disjoint_ordered_pairs",
      "conditions": [],
      "explanation": {
        "zh-Hant": "含空集的有序對也要核；只對互斥集合用二元可加性，所有失敗都保留。",
        "en": "Include ordered pairs involving the empty set.Apply binary additivity to disjoint sets only and retain every failure."
      },
      "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": "S036",
      "uses": [
        "G001",
        "S035"
      ],
      "before": "measure sum rows1,0 := sum(23/47, 0)",
      "after": "measure sum rows1,0 = 23/47",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。",
        "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 measure sum rows1,0，不以小數近似替代。",
          "en": "Recompute measure sum rows1,0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "23/47",
          "0"
        ],
        "output": "23/47",
        "label": "measure sum rows1,0"
      }
    },
    {
      "id": "S037",
      "uses": [
        "G001",
        "S036"
      ],
      "before": "disjoint ordered pair=[[0, 1, 2, 4], []];union=[0, 1, 2, 4];mu(union)=23/47;mu(A)=23/47;mu(B)=0",
      "after": "sum=23/47;additivity=是",
      "relation": "implies",
      "rule_id": "measure_all_disjoint_ordered_pairs",
      "conditions": [],
      "explanation": {
        "zh-Hant": "含空集的有序對也要核；只對互斥集合用二元可加性，所有失敗都保留。",
        "en": "Include ordered pairs involving the empty set.Apply binary additivity to disjoint sets only and retain every failure."
      },
      "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": "S038",
      "uses": [
        "G001",
        "S037"
      ],
      "before": "measure sum rows1,2 := sum(23/47, 24/47)",
      "after": "measure sum rows1,2 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。",
        "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 measure sum rows1,2，不以小數近似替代。",
          "en": "Recompute measure sum rows1,2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "23/47",
          "24/47"
        ],
        "output": "1",
        "label": "measure sum rows1,2"
      }
    },
    {
      "id": "S039",
      "uses": [
        "G001",
        "S038"
      ],
      "before": "disjoint ordered pair=[[0, 1, 2, 4], [3, 5, 6]];union=[0, 1, 2, 3, 4, 5, 6];mu(union)=1;mu(A)=23/47;mu(B)=24/47",
      "after": "sum=1;additivity=是",
      "relation": "implies",
      "rule_id": "measure_all_disjoint_ordered_pairs",
      "conditions": [],
      "explanation": {
        "zh-Hant": "含空集的有序對也要核；只對互斥集合用二元可加性，所有失敗都保留。",
        "en": "Include ordered pairs involving the empty set.Apply binary additivity to disjoint sets only and retain every failure."
      },
      "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": "S040",
      "uses": [
        "G001",
        "S039"
      ],
      "before": "measure sum rows2,0 := sum(24/47, 0)",
      "after": "measure sum rows2,0 = 24/47",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。",
        "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 measure sum rows2,0，不以小數近似替代。",
          "en": "Recompute measure sum rows2,0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "24/47",
          "0"
        ],
        "output": "24/47",
        "label": "measure sum rows2,0"
      }
    },
    {
      "id": "S041",
      "uses": [
        "G001",
        "S040"
      ],
      "before": "disjoint ordered pair=[[3, 5, 6], []];union=[3, 5, 6];mu(union)=24/47;mu(A)=24/47;mu(B)=0",
      "after": "sum=24/47;additivity=是",
      "relation": "implies",
      "rule_id": "measure_all_disjoint_ordered_pairs",
      "conditions": [],
      "explanation": {
        "zh-Hant": "含空集的有序對也要核；只對互斥集合用二元可加性，所有失敗都保留。",
        "en": "Include ordered pairs involving the empty set.Apply binary additivity to disjoint sets only and retain every failure."
      },
      "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": "S042",
      "uses": [
        "G001",
        "S041"
      ],
      "before": "measure sum rows2,1 := sum(24/47, 23/47)",
      "after": "measure sum rows2,1 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。",
        "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 measure sum rows2,1，不以小數近似替代。",
          "en": "Recompute measure sum rows2,1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "24/47",
          "23/47"
        ],
        "output": "1",
        "label": "measure sum rows2,1"
      }
    },
    {
      "id": "S043",
      "uses": [
        "G001",
        "S042"
      ],
      "before": "disjoint ordered pair=[[3, 5, 6], [0, 1, 2, 4]];union=[0, 1, 2, 3, 4, 5, 6];mu(union)=1;mu(A)=24/47;mu(B)=23/47",
      "after": "sum=1;additivity=是",
      "relation": "implies",
      "rule_id": "measure_all_disjoint_ordered_pairs",
      "conditions": [],
      "explanation": {
        "zh-Hant": "含空集的有序對也要核；只對互斥集合用二元可加性，所有失敗都保留。",
        "en": "Include ordered pairs involving the empty set.Apply binary additivity to disjoint sets only and retain every failure."
      },
      "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": "S044",
      "uses": [
        "G001",
        "S043"
      ],
      "before": "measure sum rows3,0 := sum(1, 0)",
      "after": "measure sum rows3,0 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。",
        "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 measure sum rows3,0，不以小數近似替代。",
          "en": "Recompute measure sum rows3,0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "0"
        ],
        "output": "1",
        "label": "measure sum rows3,0"
      }
    },
    {
      "id": "S045",
      "uses": [
        "G001",
        "S044"
      ],
      "before": "disjoint ordered pair=[[0, 1, 2, 3, 4, 5, 6], []];union=[0, 1, 2, 3, 4, 5, 6];mu(union)=1;mu(A)=1;mu(B)=0",
      "after": "sum=1;additivity=是",
      "relation": "implies",
      "rule_id": "measure_all_disjoint_ordered_pairs",
      "conditions": [],
      "explanation": {
        "zh-Hant": "含空集的有序對也要核；只對互斥集合用二元可加性，所有失敗都保留。",
        "en": "Include ordered pairs involving the empty set.Apply binary additivity to disjoint sets only and retain every failure."
      },
      "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": "S046",
      "uses": [
        "G001",
        "S045"
      ],
      "before": "sigma qualified=是;empty zero=是;all bad disjoint pairs=[]",
      "after": "finite countable-additivity reduction=是",
      "relation": "implies",
      "rule_id": "finite_measure_countable_disjoint_reduction",
      "conditions": [],
      "explanation": {
        "zh-Hant": "有限載體的兩兩互斥非空集合至多n個，剩下可數多項皆空集且質量0；完整有限互斥可加性因此足以處理此載體的可數情形，不是任意無限測度空間的證明。",
        "en": "On a finite carrier there are at most n pairwise disjoint nonempty sets;all remaining countable terms are empty with mass0.Complete finite disjoint additivity therefore handles this carrier's countable case,not arbitrary infinite measure spaces."
      },
      "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": "S047",
      "uses": [
        "G001",
        "S046"
      ],
      "before": "finite measure qualified=是;total carrier measure=1",
      "after": "probability measure=是",
      "relation": "implies",
      "rule_id": "probability_normalization_after_measure_obligations",
      "conditions": [],
      "explanation": {
        "zh-Hant": "機率測度還要全載體質量恰1；合法總量2的測度不是機率，總量1也不會彌補不可加或空集質量非零。",
        "en": "A probability measure additionally needs total mass exactly1.A valid measure of total2 is not a probability measure,and total1 does not repair failed additivity or nonzero empty mass."
      },
      "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,
          4
        ],
        [
          3,
          5,
          6
        ],
        [
          0,
          1,
          2,
          3,
          4,
          5,
          6
        ]
      ],
      [
        "0",
        "23/47",
        "24/47",
        "1"
      ],
      true,
      [],
      "1",
      true,
      true
    ]
  },
  "conclusion": {
    "canonical_value": "(7, [[], [0, 1, 2, 4], [3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 23/47, 24/47, 1], 是, [], 1, 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-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": 98,
      "original_line_sha256": "f85508594487464d214e96e7be46fe34d92eacbbf04cff213744e0aef45f65e6",
      "question_compact_sha256": "8ce70b99f6733995f9157c5e6404016ecd9251abcfd047e2cfe665f8b8936fc9",
      "english_row_compact_sha256": "99026b29ae87434c95abca6b7f2fa50351451b3c729dc985bb0119daca0576a2",
      "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": "c5b4fbf742900c547c14775b1f8d5dab39223bfff8fbf33be0344b9ebbdd4bcd",
    "original_candidate_line_sha256": "c5b4fbf742900c547c14775b1f8d5dab39223bfff8fbf33be0344b9ebbdd4bcd"
  },
  "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": 47,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "deb8e5bbf9a4d74fa0da624bbb8ea709d344113f443420855b77960683df6b98",
      "arithmetic_steps": 9
    },
    "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."
  }
}