{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261024-r02-measurable-function-0048",
  "pilot_number": "R0021-11473",
  "number": 11473,
  "family_id": "qf-mod24-measurable-map-audit",
  "title": {
    "zh-Hant": "第 11473 題：有限可測函數",
    "en": "Question 11473: Audit Every Target Measurable Set by Preimage"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261024-r02-measurable-function-0048",
    "batch_id": "batch-20261013-advance-20261024-r02-measurable-function-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-measurable-function"
    ],
    "title_zh": "有限可測函數",
    "stem_zh": "完整有限來源與目標sigma代數。操作=measurable-map-audit；資料={\"source_n\":4,\"target_n\":4,\"source_sigma\":[[],[1],[0,2,3],[0,1,2,3]],\"target_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"mapping\":[2,1,1,0]}。請依 [source_n,target_n,mapping,preimage_rows,measurable] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(4, 4, [2, 1, 1, 0], [[[], [], 是], [[0], [3], 否], [[1], [1, 2], 否], [[0, 1], [1, 2, 3], 否], [[2], [0], 否], [[0, 2], [0, 3], 否], [[1, 2], [0, 1, 2], 否], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [3], 否], [[1, 3], [1, 2], 否], [[0, 1, 3], [1, 2, 3], 否], [[2, 3], [0], 否], [[0, 2, 3], [0, 3], 否], [[1, 2, 3], [0, 1, 2], 否], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 否)",
      "explanation_zh": "依賴結構核對=(4, 4, [2, 1, 1, 0], [[[], [], 是], [[0], [3], 否], [[1], [1, 2], 否], [[0, 1], [1, 2, 3], 否], [[2], [0], 否], [[0, 2], [0, 3], 否], [[1, 2], [0, 1, 2], 否], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [3], 否], [[1, 3], [1, 2], 否], [[0, 1, 3], [1, 2, 3], 否], [[2, 3], [0], 否], [[0, 2, 3], [0, 3], 否], [[1, 2, 3], [0, 1, 2], 否], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；逐一核目標可測集的逆像；可測不推出連續可積或保測。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 3,
      "skills": [
        "有限可測函數"
      ]
    },
    "variant": {
      "family_id": "qf-mod24-measurable-map-audit",
      "seed": 202610242048,
      "parameters": {
        "source_n": 4,
        "target_n": 4,
        "source_sigma": [
          [],
          [
            1
          ],
          [
            0,
            2,
            3
          ],
          [
            0,
            1,
            2,
            3
          ]
        ],
        "target_sigma": [
          [],
          [
            0
          ],
          [
            1
          ],
          [
            0,
            1
          ],
          [
            2
          ],
          [
            0,
            2
          ],
          [
            1,
            2
          ],
          [
            0,
            1,
            2
          ],
          [
            3
          ],
          [
            0,
            3
          ],
          [
            1,
            3
          ],
          [
            0,
            1,
            3
          ],
          [
            2,
            3
          ],
          [
            0,
            2,
            3
          ],
          [
            1,
            2,
            3
          ],
          [
            0,
            1,
            2,
            3
          ]
        ],
        "mapping": [
          2,
          1,
          1,
          0
        ],
        "domain": "Complete_finite_measurable_spaces",
        "operation": "measurable-map-audit",
        "semantic_template": "qf-mod24-measurable-map-audit/{\"mapping\":[2,1,1,0],\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,2,3],[1]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]]}"
      }
    },
    "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長期性、PDE適定性、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-20261024-r02-measurable-function-0048",
    "batch_id": "batch-20261013-advance-20261024-r02-measurable-function-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "5fe351da30720f3434fa94e3a4fc54dcf1e1ab137b5b28a8b3c2421b971fc4c3",
      "family_id": "qf-mod24-measurable-map-audit",
      "seed": 202610242048,
      "semantic_template": "qf-mod24-measurable-map-audit/{\"mapping\":[2,1,1,0],\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,2,3],[1]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]]}"
    },
    "fields": {
      "title": "Audit Every Target Measurable Set by Preimage",
      "stem": "Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [1], [0, 2, 3], [0, 1, 2, 3]], target size=4, target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[2, 1, 1, 0]. Give (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).",
      "answer": {
        "kind": "expression",
        "value": "(4, 4, [2, 1, 1, 0], [[[], [], Yes], [[0], [3], No], [[1], [1, 2], No], [[0, 1], [1, 2, 3], No], [[2], [0], No], [[0, 2], [0, 3], No], [[1, 2], [0, 1, 2], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [3], No], [[1, 3], [1, 2], No], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0], No], [[0, 2, 3], [0, 3], No], [[1, 2, 3], [0, 1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)",
        "explanation": "Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [2, 1, 1, 0], [[[], [], Yes], [[0], [3], No], [[1], [1, 2], No], [[0, 1], [1, 2, 3], No], [[2], [0], No], [[0, 2], [0, 3], No], [[1, 2], [0, 1, 2], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [3], No], [[1, 3], [1, 2], No], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0], No], [[0, 2, 3], [0, 3], No], [[1, 2, 3], [0, 1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)."
      },
      "skills": [
        "Audit Every Target Measurable Set by Preimage"
      ],
      "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": "完整有限來源與目標sigma代數。操作=measurable-map-audit；資料={\"source_n\":4,\"target_n\":4,\"source_sigma\":[[],[1],[0,2,3],[0,1,2,3]],\"target_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"mapping\":[2,1,1,0]}。請依 [source_n,target_n,mapping,preimage_rows,measurable] 欄序回答。",
        "en": "Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [1], [0, 2, 3], [0, 1, 2, 3]], target size=4, target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[2, 1, 1, 0]. Give (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "given map source:carrier Fin(4);complete canonical sigma family=[[], [1], [0, 2, 3], [0, 1, 2, 3]]",
      "after": "empty member=是;whole member=是;missing complements=[];missing pair unions=[]",
      "relation": "implies",
      "rule_id": "qualify_complete_finite_sigma_algebra",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先核完整且按位元遮罩排序的集合族：空集、全集、補集與所有兩兩聯集。有限族的可數聯集只含有限種不同成員，可化為有限聯集，故這些封閉條件足夠。",
        "en": "Qualify the complete bit-mask-ordered family:empty set,universe,complements and all pair unions.In a finite family a countable union has only finitely many distinct members,so finite-union closure suffices."
      },
      "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": "given map target:carrier Fin(4);complete canonical sigma family=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]",
      "after": "empty member=是;whole member=是;missing complements=[];missing pair unions=[]",
      "relation": "implies",
      "rule_id": "qualify_complete_finite_sigma_algebra",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先核完整且按位元遮罩排序的集合族：空集、全集、補集與所有兩兩聯集。有限族的可數聯集只含有限種不同成員，可化為有限聯集，故這些封閉條件足夠。",
        "en": "Qualify the complete bit-mask-ordered family:empty set,universe,complements and all pair unions.In a finite family a countable union has only finitely many distinct members,so finite-union closure suffices."
      },
      "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": "given map:mapping=[2, 1, 1, 0];source Fin(4);target Fin(4)",
      "after": "complete total map qualified;preimages,not forward images",
      "relation": "implies",
      "rule_id": "measurable_total_map_and_direction",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個來源標籤都恰有一個合法目標值；可測性檢查方向是目標可測集的逆像屬來源sigma。",
        "en": "Every source label has one permitted target value.Measurability checks that inverse images of target measurable sets belong to the source sigma-algebra."
      },
      "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": "given map:target event index=0;event=[];mapping=[2, 1, 1, 0]",
      "after": "source membership tests=[否, 否, 否, 否];preimage=[];in source sigma=是",
      "relation": "implies",
      "rule_id": "every_target_event_preimage_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依來源標籤逐項判f(x)是否在此目標集合，收集所有成立的x，再查完整來源集合族；不只查單點或非空集合。",
        "en": "For each source label test whether f(x) lies in this target event.Collect all passing labels and test the complete source family;do not inspect only singletons or nonempty events."
      },
      "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": "given map:target event index=1;event=[0];mapping=[2, 1, 1, 0]",
      "after": "source membership tests=[否, 否, 否, 是];preimage=[3];in source sigma=否",
      "relation": "implies",
      "rule_id": "every_target_event_preimage_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依來源標籤逐項判f(x)是否在此目標集合，收集所有成立的x，再查完整來源集合族；不只查單點或非空集合。",
        "en": "For each source label test whether f(x) lies in this target event.Collect all passing labels and test the complete source family;do not inspect only singletons or nonempty events."
      },
      "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": "given map:target event index=2;event=[1];mapping=[2, 1, 1, 0]",
      "after": "source membership tests=[否, 是, 是, 否];preimage=[1, 2];in source sigma=否",
      "relation": "implies",
      "rule_id": "every_target_event_preimage_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依來源標籤逐項判f(x)是否在此目標集合，收集所有成立的x，再查完整來源集合族；不只查單點或非空集合。",
        "en": "For each source label test whether f(x) lies in this target event.Collect all passing labels and test the complete source family;do not inspect only singletons or nonempty events."
      },
      "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": "given map:target event index=3;event=[0, 1];mapping=[2, 1, 1, 0]",
      "after": "source membership tests=[否, 是, 是, 是];preimage=[1, 2, 3];in source sigma=否",
      "relation": "implies",
      "rule_id": "every_target_event_preimage_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依來源標籤逐項判f(x)是否在此目標集合，收集所有成立的x，再查完整來源集合族；不只查單點或非空集合。",
        "en": "For each source label test whether f(x) lies in this target event.Collect all passing labels and test the complete source family;do not inspect only singletons or nonempty events."
      },
      "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": "given map:target event index=4;event=[2];mapping=[2, 1, 1, 0]",
      "after": "source membership tests=[是, 否, 否, 否];preimage=[0];in source sigma=否",
      "relation": "implies",
      "rule_id": "every_target_event_preimage_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依來源標籤逐項判f(x)是否在此目標集合，收集所有成立的x，再查完整來源集合族；不只查單點或非空集合。",
        "en": "For each source label test whether f(x) lies in this target event.Collect all passing labels and test the complete source family;do not inspect only singletons or nonempty events."
      },
      "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": "given map:target event index=5;event=[0, 2];mapping=[2, 1, 1, 0]",
      "after": "source membership tests=[是, 否, 否, 是];preimage=[0, 3];in source sigma=否",
      "relation": "implies",
      "rule_id": "every_target_event_preimage_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依來源標籤逐項判f(x)是否在此目標集合，收集所有成立的x，再查完整來源集合族；不只查單點或非空集合。",
        "en": "For each source label test whether f(x) lies in this target event.Collect all passing labels and test the complete source family;do not inspect only singletons or nonempty events."
      },
      "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": "given map:target event index=6;event=[1, 2];mapping=[2, 1, 1, 0]",
      "after": "source membership tests=[是, 是, 是, 否];preimage=[0, 1, 2];in source sigma=否",
      "relation": "implies",
      "rule_id": "every_target_event_preimage_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依來源標籤逐項判f(x)是否在此目標集合，收集所有成立的x，再查完整來源集合族；不只查單點或非空集合。",
        "en": "For each source label test whether f(x) lies in this target event.Collect all passing labels and test the complete source family;do not inspect only singletons or nonempty events."
      },
      "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": "given map:target event index=7;event=[0, 1, 2];mapping=[2, 1, 1, 0]",
      "after": "source membership tests=[是, 是, 是, 是];preimage=[0, 1, 2, 3];in source sigma=是",
      "relation": "implies",
      "rule_id": "every_target_event_preimage_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依來源標籤逐項判f(x)是否在此目標集合，收集所有成立的x，再查完整來源集合族；不只查單點或非空集合。",
        "en": "For each source label test whether f(x) lies in this target event.Collect all passing labels and test the complete source family;do not inspect only singletons or nonempty events."
      },
      "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": "given map:target event index=8;event=[3];mapping=[2, 1, 1, 0]",
      "after": "source membership tests=[否, 否, 否, 否];preimage=[];in source sigma=是",
      "relation": "implies",
      "rule_id": "every_target_event_preimage_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依來源標籤逐項判f(x)是否在此目標集合，收集所有成立的x，再查完整來源集合族；不只查單點或非空集合。",
        "en": "For each source label test whether f(x) lies in this target event.Collect all passing labels and test the complete source family;do not inspect only singletons or nonempty events."
      },
      "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": "given map:target event index=9;event=[0, 3];mapping=[2, 1, 1, 0]",
      "after": "source membership tests=[否, 否, 否, 是];preimage=[3];in source sigma=否",
      "relation": "implies",
      "rule_id": "every_target_event_preimage_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依來源標籤逐項判f(x)是否在此目標集合，收集所有成立的x，再查完整來源集合族；不只查單點或非空集合。",
        "en": "For each source label test whether f(x) lies in this target event.Collect all passing labels and test the complete source family;do not inspect only singletons or nonempty events."
      },
      "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": "given map:target event index=10;event=[1, 3];mapping=[2, 1, 1, 0]",
      "after": "source membership tests=[否, 是, 是, 否];preimage=[1, 2];in source sigma=否",
      "relation": "implies",
      "rule_id": "every_target_event_preimage_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依來源標籤逐項判f(x)是否在此目標集合，收集所有成立的x，再查完整來源集合族；不只查單點或非空集合。",
        "en": "For each source label test whether f(x) lies in this target event.Collect all passing labels and test the complete source family;do not inspect only singletons or nonempty events."
      },
      "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": "given map:target event index=11;event=[0, 1, 3];mapping=[2, 1, 1, 0]",
      "after": "source membership tests=[否, 是, 是, 是];preimage=[1, 2, 3];in source sigma=否",
      "relation": "implies",
      "rule_id": "every_target_event_preimage_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依來源標籤逐項判f(x)是否在此目標集合，收集所有成立的x，再查完整來源集合族；不只查單點或非空集合。",
        "en": "For each source label test whether f(x) lies in this target event.Collect all passing labels and test the complete source family;do not inspect only singletons or nonempty events."
      },
      "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": "given map:target event index=12;event=[2, 3];mapping=[2, 1, 1, 0]",
      "after": "source membership tests=[是, 否, 否, 否];preimage=[0];in source sigma=否",
      "relation": "implies",
      "rule_id": "every_target_event_preimage_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依來源標籤逐項判f(x)是否在此目標集合，收集所有成立的x，再查完整來源集合族；不只查單點或非空集合。",
        "en": "For each source label test whether f(x) lies in this target event.Collect all passing labels and test the complete source family;do not inspect only singletons or nonempty events."
      },
      "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": "given map:target event index=13;event=[0, 2, 3];mapping=[2, 1, 1, 0]",
      "after": "source membership tests=[是, 否, 否, 是];preimage=[0, 3];in source sigma=否",
      "relation": "implies",
      "rule_id": "every_target_event_preimage_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依來源標籤逐項判f(x)是否在此目標集合，收集所有成立的x，再查完整來源集合族；不只查單點或非空集合。",
        "en": "For each source label test whether f(x) lies in this target event.Collect all passing labels and test the complete source family;do not inspect only singletons or nonempty events."
      },
      "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": "given map:target event index=14;event=[1, 2, 3];mapping=[2, 1, 1, 0]",
      "after": "source membership tests=[是, 是, 是, 否];preimage=[0, 1, 2];in source sigma=否",
      "relation": "implies",
      "rule_id": "every_target_event_preimage_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依來源標籤逐項判f(x)是否在此目標集合，收集所有成立的x，再查完整來源集合族；不只查單點或非空集合。",
        "en": "For each source label test whether f(x) lies in this target event.Collect all passing labels and test the complete source family;do not inspect only singletons or nonempty events."
      },
      "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": "given map:target event index=15;event=[0, 1, 2, 3];mapping=[2, 1, 1, 0]",
      "after": "source membership tests=[是, 是, 是, 是];preimage=[0, 1, 2, 3];in source sigma=是",
      "relation": "implies",
      "rule_id": "every_target_event_preimage_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依來源標籤逐項判f(x)是否在此目標集合，收集所有成立的x，再查完整來源集合族；不只查單點或非空集合。",
        "en": "For each source label test whether f(x) lies in this target event.Collect all passing labels and test the complete source family;do not inspect only singletons or nonempty events."
      },
      "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": "given map:all target preimage rows=[[[], [], 是], [[0], [3], 否], [[1], [1, 2], 否], [[0, 1], [1, 2, 3], 否], [[2], [0], 否], [[0, 2], [0, 3], 否], [[1, 2], [0, 1, 2], 否], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [3], 否], [[1, 3], [1, 2], 否], [[0, 1, 3], [1, 2, 3], 否], [[2, 3], [0], 否], [[0, 2, 3], [0, 3], 否], [[1, 2, 3], [0, 1, 2], 否], [[0, 1, 2, 3], [0, 1, 2, 3], 是]]",
      "after": "measurable=否",
      "relation": "implies",
      "rule_id": "measurability_all_target_events_conjunction",
      "conditions": [],
      "explanation": {
        "zh-Hant": "全部目標可測集都通過才可測；任一失敗給精確否證。這個有限空間完整核查不附帶連續、可積或保測結論。",
        "en": "Every target measurable set must pass.One failure is an exact counterexample.This complete finite-space check adds no continuity,integrability or measure-preservation claim."
      },
      "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,
      4,
      [
        2,
        1,
        1,
        0
      ],
      [
        [
          [],
          [],
          true
        ],
        [
          [
            0
          ],
          [
            3
          ],
          false
        ],
        [
          [
            1
          ],
          [
            1,
            2
          ],
          false
        ],
        [
          [
            0,
            1
          ],
          [
            1,
            2,
            3
          ],
          false
        ],
        [
          [
            2
          ],
          [
            0
          ],
          false
        ],
        [
          [
            0,
            2
          ],
          [
            0,
            3
          ],
          false
        ],
        [
          [
            1,
            2
          ],
          [
            0,
            1,
            2
          ],
          false
        ],
        [
          [
            0,
            1,
            2
          ],
          [
            0,
            1,
            2,
            3
          ],
          true
        ],
        [
          [
            3
          ],
          [],
          true
        ],
        [
          [
            0,
            3
          ],
          [
            3
          ],
          false
        ],
        [
          [
            1,
            3
          ],
          [
            1,
            2
          ],
          false
        ],
        [
          [
            0,
            1,
            3
          ],
          [
            1,
            2,
            3
          ],
          false
        ],
        [
          [
            2,
            3
          ],
          [
            0
          ],
          false
        ],
        [
          [
            0,
            2,
            3
          ],
          [
            0,
            3
          ],
          false
        ],
        [
          [
            1,
            2,
            3
          ],
          [
            0,
            1,
            2
          ],
          false
        ],
        [
          [
            0,
            1,
            2,
            3
          ],
          [
            0,
            1,
            2,
            3
          ],
          true
        ]
      ],
      false
    ]
  },
  "conclusion": {
    "canonical_value": "(4, 4, [2, 1, 1, 0], [[[], [], 是], [[0], [3], 否], [[1], [1, 2], 否], [[0, 1], [1, 2, 3], 否], [[2], [0], 否], [[0, 2], [0, 3], 否], [[1, 2], [0, 1, 2], 否], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [3], 否], [[1, 3], [1, 2], 否], [[0, 1, 3], [1, 2, 3], 否], [[2, 3], [0], 否], [[0, 2, 3], [0, 3], 否], [[1, 2, 3], [0, 1, 2], 否], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 否)",
    "explanation": {
      "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。",
      "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged."
    }
  },
  "definition_note": {
    "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。",
    "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate."
  },
  "source": {
    "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db",
    "question_binding": {
      "batch_id": "batch-20261013-advance-20261024-r02-measurable-function-foundations",
      "batch_path": "public/data/questions/batches/batch-20261013-advance-20261024-r02-measurable-function-foundations/questions.jsonl",
      "batch_sha256": "cc58ac5d745b4f3e9c1baadc64646f9363c8af604e4f8951f11b67dc2e22759a",
      "original_line": 48,
      "original_line_sha256": "e1f786a1ab336c7a1d677851e412c4dacda300682071f5a29f20920bd4df30e2",
      "question_compact_sha256": "5fe351da30720f3434fa94e3a4fc54dcf1e1ab137b5b28a8b3c2421b971fc4c3",
      "english_row_compact_sha256": "566fcf4f627a05fffdf3f6958a7acca6b357812359976ffc2248a51b8d07f495",
      "english_manifest_path": "public/data/i18n/editions/batch-20261013-advance-20261024-r02-measurable-function-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261013-advance-20261024-r02-measurable-function-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-measurable-function": {
        "sha256": "0c21c3c35f2f6b0451adf573f920b3dd8e797c631bcae8c14bf6bd959dc1571b",
        "english_sha256": "92b1662eff5366dc439172033e87ad4948ab0cbef9b4f22c08d2efd3e2310267",
        "original_path": "public/data/mko/measurable-function.json",
        "english_path": "public/data/i18n/editions/mko-measurable-function.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "18bbb89127e14ce3466c1a8730f1c3c6e22a8f59a9148433b2abc7f6ea274ee5",
    "original_candidate_sha256": "c1ac1e0fd2af55ec03a043d6ccca568acc5fcdef9aa9caaad1873025c01be2f2",
    "original_candidate_line_sha256": "c1ac1e0fd2af55ec03a043d6ccca568acc5fcdef9aa9caaad1873025c01be2f2"
  },
  "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-p22",
    "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": "47d23d3735987d68e27733b051bdfa80a3616d44fb223b82bf2e6c8e46d49258",
      "question_first_witness_sha256": "a857db558000724020d2d5e815cb6bc3e2c9e2933df295bf33eb478d3c695139"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 20,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "66303ec72f693c9317116f7312f7016af2bc62df0e3eecef223643dd3c37c8f9",
      "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."
  }
}