{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261024-r02-measurable-function-0013",
  "pilot_number": "R0021-11488",
  "number": 11488,
  "family_id": "qf-mod24-measurable-preimage",
  "title": {
    "zh-Hant": "第 11488 題：有限可測函數",
    "en": "Question 11488: The Inverse Image under a Complete Finite Total Map"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261024-r02-measurable-function-0013",
    "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-preimage；資料={\"source_n\":4,\"target_n\":4,\"mapping\":[0,3,0,0],\"target_event\":[0,1]}。請依 [source_n,target_n,mapping,target_event,preimage] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(4, 4, [0, 3, 0, 0], [0, 1], [0, 2, 3])",
      "explanation_zh": "依賴結構核對=(4, 4, [0, 3, 0, 0], [0, 1], [0, 2, 3])；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-preimage",
      "seed": 202610242013,
      "parameters": {
        "source_n": 4,
        "target_n": 4,
        "mapping": [
          0,
          3,
          0,
          0
        ],
        "target_event": [
          0,
          1
        ],
        "domain": "Complete_finite_measurable_spaces",
        "operation": "measurable-preimage",
        "semantic_template": "qf-mod24-measurable-preimage/{\"mapping\":[0,3,0,0],\"source_n\":4,\"target_event\":[0,1],\"target_n\":4}"
      }
    },
    "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-0013",
    "batch_id": "batch-20261013-advance-20261024-r02-measurable-function-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "3614f29ad647a94a918424fb0fd1a87ec09bc19991402ee318e777be00205f11",
      "family_id": "qf-mod24-measurable-preimage",
      "seed": 202610242013,
      "semantic_template": "qf-mod24-measurable-preimage/{\"mapping\":[0,3,0,0],\"source_n\":4,\"target_event\":[0,1],\"target_n\":4}"
    },
    "fields": {
      "title": "The Inverse Image under a Complete Finite Total Map",
      "stem": "Use carriers Fin(4) and Fin(4), complete total mapping=[0, 3, 0, 0] indexed by source labels, and target event=[0, 1] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.",
      "answer": {
        "kind": "expression",
        "value": "(4, 4, [0, 3, 0, 0], [0, 1], [0, 2, 3])",
        "explanation": "Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [0, 3, 0, 0], [0, 1], [0, 2, 3])."
      },
      "skills": [
        "The Inverse Image under a Complete Finite Total Map"
      ],
      "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-preimage；資料={\"source_n\":4,\"target_n\":4,\"mapping\":[0,3,0,0],\"target_event\":[0,1]}。請依 [source_n,target_n,mapping,target_event,preimage] 欄序回答。",
        "en": "Use carriers Fin(4) and Fin(4), complete total mapping=[0, 3, 0, 0] indexed by source labels, and target event=[0, 1] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "raw target event=[0, 1];mapping=[0, 3, 0, 0];source Fin(4);target Fin(4)",
      "after": "sorted distinct target event=[0, 1]",
      "relation": "implies",
      "rule_id": "preimage_event_set_normalization",
      "conditions": [],
      "explanation": {
        "zh-Hant": "目標事件只去重排序；總映射依來源索引保留。這題沒有sigma代數前提，只求集合逆像，不宣稱可測。",
        "en": "Normalize only the target event as a sorted set.Keep the total map indexed by source.This exercise has no sigma-algebra premise and computes a set preimage only,not measurability."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "source label=0;image=0;target event=[0, 1]",
      "after": "image in event=是;include source label=是",
      "relation": "implies",
      "rule_id": "preimage_each_source_label",
      "conditions": [],
      "explanation": {
        "zh-Hant": "判斷的是像是否在目標事件，但加入的是原來源標籤，不是把像值再加入。",
        "en": "Test image membership in the target event,but include the original source label,not the image value."
      },
      "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": "source label=1;image=3;target event=[0, 1]",
      "after": "image in event=否;include source label=否",
      "relation": "implies",
      "rule_id": "preimage_each_source_label",
      "conditions": [],
      "explanation": {
        "zh-Hant": "判斷的是像是否在目標事件，但加入的是原來源標籤，不是把像值再加入。",
        "en": "Test image membership in the target event,but include the original source label,not the image value."
      },
      "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": "source label=2;image=0;target event=[0, 1]",
      "after": "image in event=是;include source label=是",
      "relation": "implies",
      "rule_id": "preimage_each_source_label",
      "conditions": [],
      "explanation": {
        "zh-Hant": "判斷的是像是否在目標事件，但加入的是原來源標籤，不是把像值再加入。",
        "en": "Test image membership in the target event,but include the original source label,not the image value."
      },
      "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": "source label=3;image=0;target event=[0, 1]",
      "after": "image in event=是;include source label=是",
      "relation": "implies",
      "rule_id": "preimage_each_source_label",
      "conditions": [],
      "explanation": {
        "zh-Hant": "判斷的是像是否在目標事件，但加入的是原來源標籤，不是把像值再加入。",
        "en": "Test image membership in the target event,but include the original source label,not the image value."
      },
      "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": "complete source carrier=[0, 1, 2, 3];target event=[0, 1]",
      "after": "preimage=[0, 2, 3]",
      "relation": "implies",
      "rule_id": "preimage_complete_source_inventory",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整來源載體已遍歷，空逆像或整個來源都合法，來源與目標大小不必相同。",
        "en": "The complete source carrier has been checked.Empty and full preimages are valid;source and target sizes need not match."
      },
      "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,
      [
        0,
        3,
        0,
        0
      ],
      [
        0,
        1
      ],
      [
        0,
        2,
        3
      ]
    ]
  },
  "conclusion": {
    "canonical_value": "(4, 4, [0, 3, 0, 0], [0, 1], [0, 2, 3])",
    "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": 13,
      "original_line_sha256": "49872bafbdc2ab4f9c3dc5092b3bae15e47e87c8f8a31c01c6868bb9cc33f9cd",
      "question_compact_sha256": "3614f29ad647a94a918424fb0fd1a87ec09bc19991402ee318e777be00205f11",
      "english_row_compact_sha256": "23bc0f7eb5c7f1b3457ca55f87699990b712ce4ab5372bc1a020e91e40ce27d1",
      "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": "d20a12d934a11a3919844397a38b89809c691f05def71a8fb57fc7aa87aee9f6",
    "original_candidate_line_sha256": "d20a12d934a11a3919844397a38b89809c691f05def71a8fb57fc7aa87aee9f6"
  },
  "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": 6,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "0f9af1d4167af76e278aadf41322a2cba0c35dc1347382f2815f5f869cf22e71",
      "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."
  }
}