{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261024-r02-measurable-function-0092",
  "pilot_number": "R0021-11417",
  "number": 11417,
  "family_id": "qf-mod24-measurable-composition-audit",
  "title": {
    "zh-Hant": "第 11417 題：有限可測函數",
    "en": "Question 11417: Composition: Individual Checks, Composite Check and Their Conjunction"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261024-r02-measurable-function-0092",
    "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-composition-audit；資料={\"source_n\":4,\"middle_n\":4,\"target_n\":4,\"source_sigma\":[[],[0,1],[2,3],[0,1,2,3]],\"middle_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]],\"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]],\"first\":[0,0,1,0],\"second\":[1,1,3,0]}。請依 [first_measurable,second_measurable,composite_mapping,composite_rows,composite_measurable,composition_theorem_consistent] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(否, 是, [1, 1, 1, 1], [[[], [], 是], [[0], [], 是], [[1], [0, 1, 2, 3], 是], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [], 是], [[1, 2], [0, 1, 2, 3], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [], 是], [[1, 3], [0, 1, 2, 3], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [], 是], [[1, 2, 3], [0, 1, 2, 3], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是, 否)",
      "explanation_zh": "依賴結構核對=(否, 是, [1, 1, 1, 1], [[[], [], 是], [[0], [], 是], [[1], [0, 1, 2, 3], 是], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [], 是], [[1, 2], [0, 1, 2, 3], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [], 是], [[1, 3], [0, 1, 2, 3], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [], 是], [[1, 2, 3], [0, 1, 2, 3], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 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": 2,
      "skills": [
        "有限可測函數"
      ]
    },
    "variant": {
      "family_id": "qf-mod24-measurable-composition-audit",
      "seed": 202610242092,
      "parameters": {
        "source_n": 4,
        "middle_n": 4,
        "target_n": 4,
        "source_sigma": [
          [],
          [
            0,
            1
          ],
          [
            2,
            3
          ],
          [
            0,
            1,
            2,
            3
          ]
        ],
        "middle_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
          ]
        ],
        "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
          ]
        ],
        "first": [
          0,
          0,
          1,
          0
        ],
        "second": [
          1,
          1,
          3,
          0
        ],
        "domain": "Complete_finite_measurable_spaces",
        "operation": "measurable-composition-audit",
        "semantic_template": "qf-mod24-measurable-composition-audit/{\"first\":[0,0,1,0],\"middle_n\":4,\"middle_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]],\"second\":[1,1,3,0],\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,1],[2,3]],\"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-0092",
    "batch_id": "batch-20261013-advance-20261024-r02-measurable-function-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "472253395459dc88328f6a49cb9ba54f10e059b16b79340b570537a977956844",
      "family_id": "qf-mod24-measurable-composition-audit",
      "seed": 202610242092,
      "semantic_template": "qf-mod24-measurable-composition-audit/{\"first\":[0,0,1,0],\"middle_n\":4,\"middle_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]],\"second\":[1,1,3,0],\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,1],[2,3]],\"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": "Composition: Individual Checks, Composite Check and Their Conjunction",
      "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, middle size=4, target size=4; source sigma=[[], [0, 1], [2, 3], [0, 1, 2, 3]], middle 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]], 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]]. First map=[0, 0, 1, 0], second=[1, 1, 3, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.",
      "answer": {
        "kind": "expression",
        "value": "(No, Yes, [1, 1, 1, 1], [[[], [], Yes], [[0], [], Yes], [[1], [0, 1, 2, 3], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [], Yes], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [0, 1, 2, 3], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, No)",
        "explanation": "Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (No, Yes, [1, 1, 1, 1], [[[], [], Yes], [[0], [], Yes], [[1], [0, 1, 2, 3], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [], Yes], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [0, 1, 2, 3], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, No)."
      },
      "skills": [
        "Composition: Individual Checks, Composite Check and Their Conjunction"
      ],
      "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-composition-audit；資料={\"source_n\":4,\"middle_n\":4,\"target_n\":4,\"source_sigma\":[[],[0,1],[2,3],[0,1,2,3]],\"middle_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]],\"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]],\"first\":[0,0,1,0],\"second\":[1,1,3,0]}。請依 [first_measurable,second_measurable,composite_mapping,composite_rows,composite_measurable,composition_theorem_consistent] 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, middle size=4, target size=4; source sigma=[[], [0, 1], [2, 3], [0, 1, 2, 3]], middle 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]], 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]]. First map=[0, 0, 1, 0], second=[1, 1, 3, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "first map source:carrier Fin(4);complete canonical sigma family=[[], [0, 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": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "first 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": "first map:mapping=[0, 0, 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": "first map:target event index=0;event=[];mapping=[0, 0, 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": "first map:target event index=1;event=[0];mapping=[0, 0, 1, 0]",
      "after": "source membership tests=[是, 是, 否, 是];preimage=[0, 1, 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": "first map:target event index=2;event=[1];mapping=[0, 0, 1, 0]",
      "after": "source membership tests=[否, 否, 是, 否];preimage=[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": "first map:target event index=3;event=[0, 1];mapping=[0, 0, 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": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "first map:target event index=4;event=[2];mapping=[0, 0, 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": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "first map:target event index=5;event=[0, 2];mapping=[0, 0, 1, 0]",
      "after": "source membership tests=[是, 是, 否, 是];preimage=[0, 1, 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": "first map:target event index=6;event=[1, 2];mapping=[0, 0, 1, 0]",
      "after": "source membership tests=[否, 否, 是, 否];preimage=[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": "first map:target event index=7;event=[0, 1, 2];mapping=[0, 0, 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": "first map:target event index=8;event=[3];mapping=[0, 0, 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": "first map:target event index=9;event=[0, 3];mapping=[0, 0, 1, 0]",
      "after": "source membership tests=[是, 是, 否, 是];preimage=[0, 1, 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": "first map:target event index=10;event=[1, 3];mapping=[0, 0, 1, 0]",
      "after": "source membership tests=[否, 否, 是, 否];preimage=[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": "first map:target event index=11;event=[0, 1, 3];mapping=[0, 0, 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": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "first map:target event index=12;event=[2, 3];mapping=[0, 0, 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": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "first map:target event index=13;event=[0, 2, 3];mapping=[0, 0, 1, 0]",
      "after": "source membership tests=[是, 是, 否, 是];preimage=[0, 1, 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": "first map:target event index=14;event=[1, 2, 3];mapping=[0, 0, 1, 0]",
      "after": "source membership tests=[否, 否, 是, 否];preimage=[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": "first map:target event index=15;event=[0, 1, 2, 3];mapping=[0, 0, 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": "first map:all target preimage rows=[[[], [], 是], [[0], [0, 1, 3], 否], [[1], [2], 否], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [0, 1, 3], 否], [[1, 2], [2], 否], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [0, 1, 3], 否], [[1, 3], [2], 否], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [0, 1, 3], 否], [[1, 2, 3], [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
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "second map source: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": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "second 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": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "second map:mapping=[1, 1, 3, 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": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "second map:target event index=0;event=[];mapping=[1, 1, 3, 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": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "second map:target event index=1;event=[0];mapping=[1, 1, 3, 0]",
      "after": "source membership tests=[否, 否, 否, 是];preimage=[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": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "second map:target event index=2;event=[1];mapping=[1, 1, 3, 0]",
      "after": "source membership tests=[是, 是, 否, 否];preimage=[0, 1];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": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "second map:target event index=3;event=[0, 1];mapping=[1, 1, 3, 0]",
      "after": "source membership tests=[是, 是, 否, 是];preimage=[0, 1, 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": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "second map:target event index=4;event=[2];mapping=[1, 1, 3, 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": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "second map:target event index=5;event=[0, 2];mapping=[1, 1, 3, 0]",
      "after": "source membership tests=[否, 否, 否, 是];preimage=[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": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "second map:target event index=6;event=[1, 2];mapping=[1, 1, 3, 0]",
      "after": "source membership tests=[是, 是, 否, 否];preimage=[0, 1];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": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "second map:target event index=7;event=[0, 1, 2];mapping=[1, 1, 3, 0]",
      "after": "source membership tests=[是, 是, 否, 是];preimage=[0, 1, 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": "S032",
      "uses": [
        "G001",
        "S031"
      ],
      "before": "second map:target event index=8;event=[3];mapping=[1, 1, 3, 0]",
      "after": "source membership tests=[否, 否, 是, 否];preimage=[2];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": "S033",
      "uses": [
        "G001",
        "S032"
      ],
      "before": "second map:target event index=9;event=[0, 3];mapping=[1, 1, 3, 0]",
      "after": "source membership tests=[否, 否, 是, 是];preimage=[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": "S034",
      "uses": [
        "G001",
        "S033"
      ],
      "before": "second map:target event index=10;event=[1, 3];mapping=[1, 1, 3, 0]",
      "after": "source membership tests=[是, 是, 是, 否];preimage=[0, 1, 2];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": "S035",
      "uses": [
        "G001",
        "S034"
      ],
      "before": "second map:target event index=11;event=[0, 1, 3];mapping=[1, 1, 3, 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": "S036",
      "uses": [
        "G001",
        "S035"
      ],
      "before": "second map:target event index=12;event=[2, 3];mapping=[1, 1, 3, 0]",
      "after": "source membership tests=[否, 否, 是, 否];preimage=[2];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": "S037",
      "uses": [
        "G001",
        "S036"
      ],
      "before": "second map:target event index=13;event=[0, 2, 3];mapping=[1, 1, 3, 0]",
      "after": "source membership tests=[否, 否, 是, 是];preimage=[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": "S038",
      "uses": [
        "G001",
        "S037"
      ],
      "before": "second map:target event index=14;event=[1, 2, 3];mapping=[1, 1, 3, 0]",
      "after": "source membership tests=[是, 是, 是, 否];preimage=[0, 1, 2];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": "S039",
      "uses": [
        "G001",
        "S038"
      ],
      "before": "second map:target event index=15;event=[0, 1, 2, 3];mapping=[1, 1, 3, 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": "S040",
      "uses": [
        "G001",
        "S039"
      ],
      "before": "second map:all target preimage rows=[[[], [], 是], [[0], [3], 是], [[1], [0, 1], 是], [[0, 1], [0, 1, 3], 是], [[2], [], 是], [[0, 2], [3], 是], [[1, 2], [0, 1], 是], [[0, 1, 2], [0, 1, 3], 是], [[3], [2], 是], [[0, 3], [2, 3], 是], [[1, 3], [0, 1, 2], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [2], 是], [[0, 2, 3], [2, 3], 是], [[1, 2, 3], [0, 1, 2], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]]",
      "after": "measurable=是",
      "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
    },
    {
      "id": "S041",
      "uses": [
        "G001",
        "S040"
      ],
      "before": "source label=0;first image=0;second table=[1, 1, 3, 0]",
      "after": "second-after-first image=1",
      "relation": "implies",
      "rule_id": "composition_declared_order_lookup",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先以first取中間值，再將該值當索引查second；不是把兩張表逐項相乘或反向合成。",
        "en": "Apply first to get the middle label,then use it as the index into second.Do not multiply the tables pointwise or reverse composition."
      },
      "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": "source label=1;first image=0;second table=[1, 1, 3, 0]",
      "after": "second-after-first image=1",
      "relation": "implies",
      "rule_id": "composition_declared_order_lookup",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先以first取中間值，再將該值當索引查second；不是把兩張表逐項相乘或反向合成。",
        "en": "Apply first to get the middle label,then use it as the index into second.Do not multiply the tables pointwise or reverse composition."
      },
      "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": "S043",
      "uses": [
        "G001",
        "S042"
      ],
      "before": "source label=2;first image=1;second table=[1, 1, 3, 0]",
      "after": "second-after-first image=1",
      "relation": "implies",
      "rule_id": "composition_declared_order_lookup",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先以first取中間值，再將該值當索引查second；不是把兩張表逐項相乘或反向合成。",
        "en": "Apply first to get the middle label,then use it as the index into second.Do not multiply the tables pointwise or reverse composition."
      },
      "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": "source label=3;first image=0;second table=[1, 1, 3, 0]",
      "after": "second-after-first image=1",
      "relation": "implies",
      "rule_id": "composition_declared_order_lookup",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先以first取中間值，再將該值當索引查second；不是把兩張表逐項相乘或反向合成。",
        "en": "Apply first to get the middle label,then use it as the index into second.Do not multiply the tables pointwise or reverse composition."
      },
      "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": "S045",
      "uses": [
        "G001",
        "S044"
      ],
      "before": "composite map source:carrier Fin(4);complete canonical sigma family=[[], [0, 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": "S046",
      "uses": [
        "G001",
        "S045"
      ],
      "before": "composite 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": "S047",
      "uses": [
        "G001",
        "S046"
      ],
      "before": "composite map:mapping=[1, 1, 1, 1];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": "S048",
      "uses": [
        "G001",
        "S047"
      ],
      "before": "composite map:target event index=0;event=[];mapping=[1, 1, 1, 1]",
      "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": "S049",
      "uses": [
        "G001",
        "S048"
      ],
      "before": "composite map:target event index=1;event=[0];mapping=[1, 1, 1, 1]",
      "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": "S050",
      "uses": [
        "G001",
        "S049"
      ],
      "before": "composite map:target event index=2;event=[1];mapping=[1, 1, 1, 1]",
      "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": "S051",
      "uses": [
        "G001",
        "S050"
      ],
      "before": "composite map:target event index=3;event=[0, 1];mapping=[1, 1, 1, 1]",
      "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": "S052",
      "uses": [
        "G001",
        "S051"
      ],
      "before": "composite map:target event index=4;event=[2];mapping=[1, 1, 1, 1]",
      "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": "S053",
      "uses": [
        "G001",
        "S052"
      ],
      "before": "composite map:target event index=5;event=[0, 2];mapping=[1, 1, 1, 1]",
      "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": "S054",
      "uses": [
        "G001",
        "S053"
      ],
      "before": "composite map:target event index=6;event=[1, 2];mapping=[1, 1, 1, 1]",
      "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": "S055",
      "uses": [
        "G001",
        "S054"
      ],
      "before": "composite map:target event index=7;event=[0, 1, 2];mapping=[1, 1, 1, 1]",
      "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": "S056",
      "uses": [
        "G001",
        "S055"
      ],
      "before": "composite map:target event index=8;event=[3];mapping=[1, 1, 1, 1]",
      "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": "S057",
      "uses": [
        "G001",
        "S056"
      ],
      "before": "composite map:target event index=9;event=[0, 3];mapping=[1, 1, 1, 1]",
      "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": "S058",
      "uses": [
        "G001",
        "S057"
      ],
      "before": "composite map:target event index=10;event=[1, 3];mapping=[1, 1, 1, 1]",
      "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": "S059",
      "uses": [
        "G001",
        "S058"
      ],
      "before": "composite map:target event index=11;event=[0, 1, 3];mapping=[1, 1, 1, 1]",
      "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": "S060",
      "uses": [
        "G001",
        "S059"
      ],
      "before": "composite map:target event index=12;event=[2, 3];mapping=[1, 1, 1, 1]",
      "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": "S061",
      "uses": [
        "G001",
        "S060"
      ],
      "before": "composite map:target event index=13;event=[0, 2, 3];mapping=[1, 1, 1, 1]",
      "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": "S062",
      "uses": [
        "G001",
        "S061"
      ],
      "before": "composite map:target event index=14;event=[1, 2, 3];mapping=[1, 1, 1, 1]",
      "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": "S063",
      "uses": [
        "G001",
        "S062"
      ],
      "before": "composite map:target event index=15;event=[0, 1, 2, 3];mapping=[1, 1, 1, 1]",
      "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": "S064",
      "uses": [
        "G001",
        "S063"
      ],
      "before": "composite map:all target preimage rows=[[[], [], 是], [[0], [], 是], [[1], [0, 1, 2, 3], 是], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [], 是], [[1, 2], [0, 1, 2, 3], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [], 是], [[1, 3], [0, 1, 2, 3], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [], 是], [[1, 2, 3], [0, 1, 2, 3], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]]",
      "after": "measurable=是",
      "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
    },
    {
      "id": "S065",
      "uses": [
        "G001",
        "S064"
      ],
      "before": "first measurable=否;second measurable=是;composite measurable=是",
      "after": "final source flag is conjunction=否,not implication",
      "relation": "implies",
      "rule_id": "composition_conjunction_not_theorem_implication",
      "conditions": [],
      "explanation": {
        "zh-Hant": "(g∘f)⁻¹(E)=f⁻¹(g⁻¹(E))，所以兩因子可測足以推出合成可測；本題最後欄是三者同真，不是這個蘊涵的真值。合成可測不倒推兩因子皆可測。",
        "en": "The identity(g∘f)^−1(E)=f^−1(g^−1(E)) proves the theorem when both factors are measurable.The requested last field is all three checks together,not the truth value of that implication.A measurable composite need not have two measurable factors."
      },
      "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": [
      false,
      true,
      [
        1,
        1,
        1,
        1
      ],
      [
        [
          [],
          [],
          true
        ],
        [
          [
            0
          ],
          [],
          true
        ],
        [
          [
            1
          ],
          [
            0,
            1,
            2,
            3
          ],
          true
        ],
        [
          [
            0,
            1
          ],
          [
            0,
            1,
            2,
            3
          ],
          true
        ],
        [
          [
            2
          ],
          [],
          true
        ],
        [
          [
            0,
            2
          ],
          [],
          true
        ],
        [
          [
            1,
            2
          ],
          [
            0,
            1,
            2,
            3
          ],
          true
        ],
        [
          [
            0,
            1,
            2
          ],
          [
            0,
            1,
            2,
            3
          ],
          true
        ],
        [
          [
            3
          ],
          [],
          true
        ],
        [
          [
            0,
            3
          ],
          [],
          true
        ],
        [
          [
            1,
            3
          ],
          [
            0,
            1,
            2,
            3
          ],
          true
        ],
        [
          [
            0,
            1,
            3
          ],
          [
            0,
            1,
            2,
            3
          ],
          true
        ],
        [
          [
            2,
            3
          ],
          [],
          true
        ],
        [
          [
            0,
            2,
            3
          ],
          [],
          true
        ],
        [
          [
            1,
            2,
            3
          ],
          [
            0,
            1,
            2,
            3
          ],
          true
        ],
        [
          [
            0,
            1,
            2,
            3
          ],
          [
            0,
            1,
            2,
            3
          ],
          true
        ]
      ],
      true,
      false
    ]
  },
  "conclusion": {
    "canonical_value": "(否, 是, [1, 1, 1, 1], [[[], [], 是], [[0], [], 是], [[1], [0, 1, 2, 3], 是], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [], 是], [[1, 2], [0, 1, 2, 3], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [], 是], [[1, 3], [0, 1, 2, 3], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [], 是], [[1, 2, 3], [0, 1, 2, 3], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 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": 92,
      "original_line_sha256": "89df266294b6e48b6136fc2c9f0504288784d521d9c14facf90d30eb43db4426",
      "question_compact_sha256": "472253395459dc88328f6a49cb9ba54f10e059b16b79340b570537a977956844",
      "english_row_compact_sha256": "5f2e7c40230302de1d0dfed7760144fbbe2baae0571f3728de7826b973aec8d0",
      "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": "c479abc38dfd7a5bc080c6bdecba5c08c0a207b37941f4b36b4309ff45091948",
    "original_candidate_line_sha256": "c479abc38dfd7a5bc080c6bdecba5c08c0a207b37941f4b36b4309ff45091948"
  },
  "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": 65,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "1da22d8c5da5a8d392f5d85f69e2d6636a1ae4aae66616c738b66ee4fc903efa",
      "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."
  }
}