{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20260930-r04-composition-domain-set-0013",
  "pilot_number": "R0005-03063",
  "number": 3063,
  "family_id": "qf-composition-domain-set",
  "title": {
    "zh-Hant": "第 3063 題：受限合成定義域",
    "en": "Question 3063: Permitted Domain of a Restricted Composition"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20260930-r04-composition-domain-set-0013",
    "batch_id": "batch-20260930-r04-composition-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-composition"
    ],
    "title_zh": "受限合成定義域",
    "stem_zh": "明示總函數 f:A→B 與 g:D→C；A={0, 1, 2}、B={-14, 0, 15}、D={-14, 0, 15}、C={-24, 1, 25}；f 表=[(0, -14), (1, 0), (2, 15)]，g 表=[(-14, -24), (0, 1), (15, 25)]。列出受限合成的允許輸入集合 {x∈A:f(x)∈D}。",
    "answer": {
      "kind": "short_text",
      "value": "{0, 1, 2}",
      "explanation_zh": "逐個檢查 A 中輸入的 f 值是否落入 D；保留下來的集合為 {0, 1, 2}。沒有相容中間值時就是空集合。"
    },
    "difficulty": {
      "band": "upper_secondary",
      "level": 1,
      "skills": [
        "中間值相容",
        "空集合邊界"
      ]
    },
    "variant": {
      "family_id": "qf-composition-domain-set",
      "seed": 202609304213,
      "parameters": {
        "operation": "restricted_domain",
        "A": [
          0,
          1,
          2
        ],
        "B": [
          -14,
          0,
          15
        ],
        "D": [
          -14,
          0,
          15
        ],
        "C": [
          -24,
          1,
          25
        ],
        "F": [
          [
            0,
            -14
          ],
          [
            1,
            0
          ],
          [
            2,
            15
          ]
        ],
        "G": [
          [
            -14,
            -24
          ],
          [
            0,
            1
          ],
          [
            15,
            25
          ]
        ],
        "semantic_template": "qf-composition-domain-set/{\"A\":[0,1,2],\"B\":[-14,0,15],\"C\":[-24,1,25],\"D\":[-14,0,15],\"F\":[[0,-14],[1,0],[2,15]],\"G\":[[-14,-24],[0,1],[15,25]],\"operation\":\"restricted_domain\"}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "json_schema",
        "producer_local_answer_invariants",
        "explanation_witness_accuracy"
      ],
      "warning_zh": "本題保持 candidate；有限精確計算與產生者自檢不構成人工審查、形式證明或主架構者驗收。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "Codex GPT-6 question producer",
      "generated_at": "2026-09-30",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20260930-r04-composition-domain-set-0013",
    "batch_id": "batch-20260930-r04-composition-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "6a3bc6e511d2b619800ea370105563e883ff365461acca4b030eace48705b9b5",
      "family_id": "qf-composition-domain-set",
      "seed": 202609304213,
      "semantic_template": "qf-composition-domain-set/{\"A\":[0,1,2],\"B\":[-14,0,15],\"C\":[-24,1,25],\"D\":[-14,0,15],\"F\":[[0,-14],[1,0],[2,15]],\"G\":[[-14,-24],[0,1],[15,25]],\"operation\":\"restricted_domain\"}"
    },
    "fields": {
      "title": "Permitted Domain of a Restricted Composition",
      "stem": "Let f:A→B and g:D→C be the explicit total functions A={0, 1, 2}, B={-14, 0, 15}, D={-14, 0, 15}, C={-24, 1, 25}, f=[(0, -14), (1, 0), (2, 15)], and g=[(-14, -24), (0, 1), (15, 25)]. List {x∈A:f(x)∈D}.",
      "answer": {
        "kind": "short_text",
        "value": "{0, 1, 2}",
        "explanation": "Check each input in A and retain it exactly when its f-value lies in D. The permitted composition domain is {0, 1, 2}; it may be empty."
      },
      "skills": [
        "intermediate-value compatibility",
        "empty-set boundary"
      ],
      "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": "明示總函數 f:A→B 與 g:D→C；A={0, 1, 2}、B={-14, 0, 15}、D={-14, 0, 15}、C={-24, 1, 25}；f 表=[(0, -14), (1, 0), (2, 15)]，g 表=[(-14, -24), (0, 1), (15, 25)]。列出受限合成的允許輸入集合 {x∈A:f(x)∈D}。",
        "en": "Let f:A→B and g:D→C be the explicit total functions A={0, 1, 2}, B={-14, 0, 15}, D={-14, 0, 15}, C={-24, 1, 25}, f=[(0, -14), (1, 0), (2, 15)], and g=[(-14, -24), (0, 1), (15, 25)]. List {x∈A:f(x)∈D}."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "(g∘f)(x)",
      "after": "g(f(x)):apply f first,then g",
      "relation": "implies",
      "rule_id": "composition_order",
      "conditions": [],
      "explanation": {
        "zh-Hant": "合成符號的右邊函數先作用，不是把兩個函數值相乘。",
        "en": "The function on the right acts first;composition is not multiplication of two function values."
      },
      "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": "f:A→B;g:D→C;D={-14, 0, 15}",
      "after": "both tables are total on their declared domains",
      "relation": "implies",
      "rule_id": "composition_totality_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐表確認每個來源輸入恰有一值且輸出合法；g的定義域D不必等於f的陪域B。",
        "en": "Audit totality and codomain membership for each table;g's domain D need not equal f's codomain B."
      },
      "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": "x=0;f(x)=-14;D={-14, 0, 15}",
      "after": "f(x)∈D=是",
      "relation": "implies",
      "rule_id": "composition_domain_row",
      "conditions": [],
      "explanation": {
        "zh-Hant": "輸入可用當且僅當f的中間值落在g的定義域；不能只看x是否在B。",
        "en": "An input is permitted exactly when its intermediate f-value lies in g's domain;membership of x in B is not the criterion."
      },
      "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": "x=1;f(x)=0;D={-14, 0, 15}",
      "after": "f(x)∈D=是",
      "relation": "implies",
      "rule_id": "composition_domain_row",
      "conditions": [],
      "explanation": {
        "zh-Hant": "輸入可用當且僅當f的中間值落在g的定義域；不能只看x是否在B。",
        "en": "An input is permitted exactly when its intermediate f-value lies in g's domain;membership of x in B is not the criterion."
      },
      "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": "x=2;f(x)=15;D={-14, 0, 15}",
      "after": "f(x)∈D=是",
      "relation": "implies",
      "rule_id": "composition_domain_row",
      "conditions": [],
      "explanation": {
        "zh-Hant": "輸入可用當且僅當f的中間值落在g的定義域；不能只看x是否在B。",
        "en": "An input is permitted exactly when its intermediate f-value lies in g's domain;membership of x in B is not the criterion."
      },
      "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": "{x∈A:f(x)∈D}",
      "after": "{0, 1, 2}",
      "relation": "implies",
      "rule_id": "restricted_composition_domain",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留全部且僅有合法輸入；A為空或沒有相容中間值時，允許域為空。",
        "en": "Retain all and only legal inputs;if A is empty or no intermediate value is compatible,the permitted domain is empty."
      },
      "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": [
      0,
      1,
      2
    ],
    "domain_rows": [
      [
        0,
        -14,
        true
      ],
      [
        1,
        0,
        true
      ],
      [
        2,
        15,
        true
      ]
    ]
  },
  "conclusion": {
    "canonical_value": "{0, 1, 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-20260930-r04-composition-foundations",
      "batch_path": "public/data/questions/batches/batch-20260930-r04-composition-foundations/questions.jsonl",
      "batch_sha256": "36c06ad1401a2402e83fc4db2843422ed626fdaa20ea488f30a0832aaf4c781f",
      "original_line": 63,
      "original_line_sha256": "8dac539eff11affe7561be2438854073651d5e6eef96eae6b50fe1bd4cf66671",
      "question_compact_sha256": "6a3bc6e511d2b619800ea370105563e883ff365461acca4b030eace48705b9b5",
      "english_row_compact_sha256": "e4d9556ae839f2f1dee3bf46ac6e0c0d859f68215ff87a2741d25ca893997582",
      "english_manifest_path": "public/data/i18n/editions/batch-20260930-r04-composition-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20260930-r04-composition-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-composition": {
        "sha256": "a915b980c52ed5cfa4fd9ef59e2a23ef237a0eebbe262cd6f1a67b15f052bfa9",
        "english_sha256": "702841fe3d6cacc8f4b48338fbd4fc96731540292253b3030c43dce8e1cf1979",
        "original_path": "public/data/mko/composition.json",
        "english_path": "public/data/i18n/editions/mko-composition.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "66b95c054a23815e6aa9999f9605667abcb96e0c5a97b71cd67b67041065878a",
    "original_candidate_sha256": "56865d0895f3c5552d50dd30ec6ff081e51368f62fa5dac6cb34d68d5a630afc",
    "original_candidate_line_sha256": "56865d0895f3c5552d50dd30ec6ff081e51368f62fa5dac6cb34d68d5a630afc"
  },
  "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-20261026-004-p6",
    "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": "5d74de11610ebc6037a4fc9946d77154d4ed18389f8f0feeb823148b6320c4c2",
      "question_first_witness_sha256": "1fba4ab0f13c2cfcc4d9fdebe492c442d43dd45700394a443daa28f333355ed9"
    },
    "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": "2646f3712a94db39a5c28a4ab59f8c7b53aad3d1b745a84000e6c18011fb9c4b",
      "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."
  }
}