{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261011-advance-20261015-r03-domain-codomain-range-0041",
  "pilot_number": "R0013-07266",
  "number": 7266,
  "family_id": "qf-density15-typed-mapping-preimage-two-values",
  "title": {
    "zh-Hant": "第 7266 題：陪域候選與完整原像",
    "en": "Question 7266: Two Typed Codomain Candidates and Their Preimages"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261011-advance-20261015-r03-domain-codomain-range-0041",
    "batch_id": "batch-20261011-advance-20261015-r03-domain-codomain-range-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-domain-codomain-range"
    ],
    "title_zh": "陪域候選與完整原像",
    "stem_zh": "Fin型別從0起，Fin(5)=[0, 1, 2, 3, 4]，Fin(3)=[0, 1, 2]。完整合型images表=[0, 0, 1, 1, 0]，有序陪域候選=[0, 1]。求 (候選型別資格, 各原像索引列或不合陪域, 各大小或不合陪域)。域外候選不作合法空原像。",
    "answer": {
      "kind": "expression",
      "value": "([是, 是], [[0, 1, 4], [2, 3]], [3, 2])",
      "explanation_zh": "陪域候選資格與原像列=([是, 是], [[0, 1, 4], [2, 3]], [3, 2])。先核候選是否屬陪域；合型才遍歷全部來源列找輸出相等的位置。完整總表非部分程式函數，有限表非一般無限集合證明。"
    },
    "difficulty": {
      "band": "upper_secondary",
      "level": 2,
      "skills": [
        "陪域候選與完整原像"
      ]
    },
    "variant": {
      "family_id": "qf-density15-typed-mapping-preimage-two-values",
      "seed": 202610153041,
      "parameters": {
        "n": 5,
        "m": 3,
        "images": [
          0,
          0,
          1,
          1,
          0
        ],
        "queries": [
          0,
          1
        ],
        "domain": "finite_typed_function_data",
        "operation": "density15_typed_mapping_preimage_two_values",
        "semantic_template": "qf-density15-typed-mapping-preimage-two-values/[[\"typed\",[0,1,4]],[\"typed\",[2,3]]]"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_integer_rational_arithmetic",
        "producer_finite_enumeration",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-11",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261011-advance-20261015-r03-domain-codomain-range-0041",
    "batch_id": "batch-20261011-advance-20261015-r03-domain-codomain-range-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "3c71d699271ce019c4c321eccda1796f62795dc6ea89381cd2a676dbe8e03a4a",
      "family_id": "qf-density15-typed-mapping-preimage-two-values",
      "seed": 202610153041,
      "semantic_template": "qf-density15-typed-mapping-preimage-two-values/[[\"typed\",[0,1,4]],[\"typed\",[2,3]]]"
    },
    "fields": {
      "title": "Two Typed Codomain Candidates and Their Preimages",
      "stem": "Fin(5)={0,…,4} is the source and Fin(3)={0,…,2} the codomain; complete typed images=[0, 0, 1, 1, 0] are indexed from 0. All entries must be typed, even unused ones. Ordered codomain candidates=[0, 1]. Give (candidate-type eligibility, preimage index lists or outside-codomain, sizes or outside-codomain). An outside value is not a legal empty preimage.",
      "answer": {
        "kind": "expression",
        "value": "([Yes, Yes], [[0, 1, 4], [2, 3]], [3, 2])",
        "explanation": "Guard codomain typing before traversing every source output for equality; a legal unused codomain value has an empty preimage, while an outside candidate is ill-typed. The exact result, retaining the requested order, is ([Yes, Yes], [[0, 1, 4], [2, 3]], [3, 2])."
      },
      "skills": [
        "Two Typed Codomain Candidates and Their Preimages"
      ],
      "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": "Fin型別從0起，Fin(5)=[0, 1, 2, 3, 4]，Fin(3)=[0, 1, 2]。完整合型images表=[0, 0, 1, 1, 0]，有序陪域候選=[0, 1]。求 (候選型別資格, 各原像索引列或不合陪域, 各大小或不合陪域)。域外候選不作合法空原像。",
        "en": "Fin(5)={0,…,4} is the source and Fin(3)={0,…,2} the codomain; complete typed images=[0, 0, 1, 1, 0] are indexed from 0. All entries must be typed, even unused ones. Ordered codomain candidates=[0, 1]. Give (candidate-type eligibility, preimage index lists or outside-codomain, sizes or outside-codomain). An outside value is not a legal empty preimage."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "declared types Fin(5)→Fin(3);0-based source labels",
      "after": "domain,codomain and actual range remain distinct",
      "relation": "implies",
      "rule_id": "typed_domain_codomain_data_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "Fin(0)是空型別；完整總表不是部分程式函數，實際值域也不自動等於宣告陪域。此處核對有限資料，不冒稱一般無限集合定理。",
        "en": "Fin(0) is empty;a complete total table is not a partial program function,and its actual range need not equal its declared codomain.This audits finite data,not general infinite-set theorems."
      },
      "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 index=0∈Fin(5)",
      "after": "image[0]=0∈Fin(3)",
      "relation": "implies",
      "rule_id": "typed_complete_original_table_row",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整原表的每格都先合型，不只核對稍後會查詢的格。",
        "en": "First type-check every cell of the complete original table,not only cells queried later."
      },
      "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 index=1∈Fin(5)",
      "after": "image[1]=0∈Fin(3)",
      "relation": "implies",
      "rule_id": "typed_complete_original_table_row",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整原表的每格都先合型，不只核對稍後會查詢的格。",
        "en": "First type-check every cell of the complete original table,not only cells queried later."
      },
      "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 index=2∈Fin(5)",
      "after": "image[2]=1∈Fin(3)",
      "relation": "implies",
      "rule_id": "typed_complete_original_table_row",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整原表的每格都先合型，不只核對稍後會查詢的格。",
        "en": "First type-check every cell of the complete original table,not only cells queried later."
      },
      "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 index=3∈Fin(5)",
      "after": "image[3]=1∈Fin(3)",
      "relation": "implies",
      "rule_id": "typed_complete_original_table_row",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整原表的每格都先合型，不只核對稍後會查詢的格。",
        "en": "First type-check every cell of the complete original table,not only cells queried later."
      },
      "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": "source index=4∈Fin(5)",
      "after": "image[4]=0∈Fin(3)",
      "relation": "implies",
      "rule_id": "typed_complete_original_table_row",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整原表的每格都先合型，不只核對稍後會查詢的格。",
        "en": "First type-check every cell of the complete original table,not only cells queried later."
      },
      "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": "target candidate=0;inside Fin(3)=是",
      "after": "preimage=[0, 1, 4];size/status=3",
      "relation": "implies",
      "rule_id": "typed_preimage_codomain_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "合法但未被命中的陪域值有空原像且大小0；陪域外候選在此型別接口不作合法空原像。",
        "en": "A legal but unhit codomain value has empty preimage of size0.An outside-codomain candidate is not a legal empty-preimage query in this typed interface."
      },
      "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": "target candidate=1;inside Fin(3)=是",
      "after": "preimage=[2, 3];size/status=2",
      "relation": "implies",
      "rule_id": "typed_preimage_codomain_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "合法但未被命中的陪域值有空原像且大小0；陪域外候選在此型別接口不作合法空原像。",
        "en": "A legal but unhit codomain value has empty preimage of size0.An outside-codomain candidate is not a legal empty-preimage query in this typed interface."
      },
      "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": [
      [
        true,
        true
      ],
      [
        [
          0,
          1,
          4
        ],
        [
          2,
          3
        ]
      ],
      [
        3,
        2
      ]
    ]
  },
  "conclusion": {
    "canonical_value": "([是, 是], [[0, 1, 4], [2, 3]], [3, 2])",
    "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-20261011-advance-20261015-r03-domain-codomain-range-foundations",
      "batch_path": "public/data/questions/batches/batch-20261011-advance-20261015-r03-domain-codomain-range-foundations/questions.jsonl",
      "batch_sha256": "4c652a7166afbd0ec4ff7932d1aedd85130eb2e1c5b132fd1a6d51ee0824c47e",
      "original_line": 41,
      "original_line_sha256": "948dad53d79bf7b29448f81f5defe91a5f987a69092219a5d96e21bed67774f0",
      "question_compact_sha256": "3c71d699271ce019c4c321eccda1796f62795dc6ea89381cd2a676dbe8e03a4a",
      "english_row_compact_sha256": "821c00e5d00a18b3a833f3c99811da9d4dd3644aeca449ef7b6af188fa835a22",
      "english_manifest_path": "public/data/i18n/editions/batch-20261011-advance-20261015-r03-domain-codomain-range-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261011-advance-20261015-r03-domain-codomain-range-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-domain-codomain-range": {
        "sha256": "94e55c31535caafd763f4e1f805a7da15848c153c844351ec087944dea68d451",
        "english_sha256": "1e8e497051747d1f535313e0ce98ad171f6ab459c72f0d46af1d1592c667bb58",
        "original_path": "public/data/mko/domain-codomain-range.json",
        "english_path": "public/data/i18n/editions/mko-domain-codomain-range.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "944e5a6e7ddc77eb28f9280c1b90da6c7083069b4790cbeb7488d009328efce8",
    "original_candidate_sha256": "7852ced98d17ee64238a8230961108d44ebd2d0095010a4a9368cfad83770000",
    "original_candidate_line_sha256": "7852ced98d17ee64238a8230961108d44ebd2d0095010a4a9368cfad83770000"
  },
  "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-20261026-005-p14",
    "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": "e89fc6d4a50a4a7fc942c6c8939a92f835e30105a9a5850e37dd198f31c1ed59",
      "question_first_witness_sha256": "0e1039f4cf022bf35af4e5531651c9a2b7ca7a5fdacac34fb1e4bcd9a81f6299"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 8,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "bcc4c84e52a8b6ab261b2ba1ce306b055651b2c9a7a77c5e8540c20d2052197a",
      "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."
  }
}