{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261007-advance-20261008-r06-antiderivative-0059",
  "pilot_number": "R0001-01359",
  "number": 1359,
  "family_id": "qf-antiderivative-candidate",
  "title": {
    "zh-Hant": "第 1359 題：候選原函數的係數資格",
    "en": "Question 1359: Coefficient Eligibility of a Candidate Antiderivative"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261007-advance-20261008-r06-antiderivative-0059",
    "batch_id": "batch-20261007-advance-20261008-r06-antiderivative-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-antiderivative"
    ],
    "title_zh": "候選原函數的係數資格",
    "stem_zh": "全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[-4, 9, 1, 1]；候選 F 的升冪有理係數為 [[-5,1],[-4,1],[9,2],[1,3],[1,4]]，每對是 [分子,正分母]。是否有 F′=f 的多項式恆等？",
    "answer": {
      "kind": "short_text",
      "value": "是",
      "explanation_zh": "候選導數係數=[-4/1, 9/1, 1/1, 1/1]；目標係數=[-4/1, 9/1, 1/1, 1/1]。按係數恆等逐項核對，不在單點或有限點試值；F 的常數項不影響資格。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "候選原函數的係數資格"
      ]
    },
    "variant": {
      "family_id": "qf-antiderivative-candidate",
      "seed": 202610086059,
      "parameters": {
        "f": [
          -4,
          9,
          1,
          1
        ],
        "F": [
          [
            -5,
            1
          ],
          [
            -4,
            1
          ],
          [
            9,
            2
          ],
          [
            1,
            3
          ],
          [
            1,
            4
          ]
        ],
        "operation": "primitive_candidate",
        "domain": "reals",
        "representation": "ascending_powers",
        "semantic_template": "qf-antiderivative-candidate/{\"inputs\":[[\"-4\",\"9\",\"1\",\"1\"],[\"-4/1\",\"9/2\",\"1/3\",\"1/4\"]],\"operation\":\"primitive_candidate\"}"
      }
    },
    "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": "2026-10-07",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261007-advance-20261008-r06-antiderivative-0059",
    "batch_id": "batch-20261007-advance-20261008-r06-antiderivative-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "2033042f6d7485fcea728c2e9c0c4daec5e9072bf43ffa58f223e3ece1d712a6",
      "family_id": "qf-antiderivative-candidate",
      "seed": 202610086059,
      "semantic_template": "qf-antiderivative-candidate/{\"inputs\":[[\"-4\",\"9\",\"1\",\"1\"],[\"-4/1\",\"9/2\",\"1/3\",\"1/4\"]],\"operation\":\"primitive_candidate\"}"
    },
    "fields": {
      "title": "Coefficient Eligibility of a Candidate Antiderivative",
      "stem": "On all R, f has ascending coefficients [-4, 9, 1, 1], starting with its constant term. The candidate F has ascending rational coefficient pairs [[-5, 1], [-4, 1], [9, 2], [1, 3], [1, 4]], with positive denominators. Is F′=f as a coefficient identity?",
      "answer": {
        "kind": "short_text",
        "value": "Yes",
        "explanation": "Differentiate each positive-degree rational coefficient and compare the full normalized coefficient lists, giving Yes. This is coefficient identity, not finite point sampling; the candidate constant is free."
      },
      "skills": [
        "Coefficient Eligibility of a Candidate Antiderivative"
      ],
      "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": "全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[-4, 9, 1, 1]；候選 F 的升冪有理係數為 [[-5,1],[-4,1],[9,2],[1,3],[1,4]]，每對是 [分子,正分母]。是否有 F′=f 的多項式恆等？",
        "en": "On all R, f has ascending coefficients [-4, 9, 1, 1], starting with its constant term. The candidate F has ascending rational coefficient pairs [[-5, 1], [-4, 1], [9, 2], [1, 3], [1, 4]], with positive denominators. Is F′=f as a coefficient identity?"
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "domain=R; coefficient lists are in ascending powers",
      "after": "constant term first; rational pairs=[numerator,denominator]",
      "relation": "implies",
      "rule_id": "polynomial_representation",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先固定係數次序，不能把升冪清單當降冪，也不能把分子分母交換。",
        "en": "Fix coefficient order;do not read ascending coefficients as descending or swap numerator and denominator."
      },
      "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": "candidate F coefficients=[-5, -4, 9/2, 1/3, 1/4]",
      "after": "target f coefficients=[-4, 9, 1, 1]",
      "relation": "implies",
      "rule_id": "candidate_vs_target",
      "conditions": [],
      "explanation": {
        "zh-Hant": "候選可帶任意常數；微分後常數消失。",
        "en": "A candidate may contain any constant,which disappears under differentiation."
      },
      "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": "candidate derivative coefficient0 := mul(1, -4)",
      "after": "candidate derivative coefficient0 = -4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每一項的次數乘係數，指數降1。",
        "en": "Multiply each coefficient by its degree and lower the exponent by1."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 candidate derivative coefficient0，不以小數近似替代。",
          "en": "Recompute candidate derivative coefficient0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "-4"
        ],
        "output": "-4",
        "label": "candidate derivative coefficient0"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "candidate derivative coefficient1 := mul(2, 9/2)",
      "after": "candidate derivative coefficient1 = 9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每一項的次數乘係數，指數降1。",
        "en": "Multiply each coefficient by its degree and lower the exponent by1."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 candidate derivative coefficient1，不以小數近似替代。",
          "en": "Recompute candidate derivative coefficient1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "9/2"
        ],
        "output": "9",
        "label": "candidate derivative coefficient1"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "candidate derivative coefficient2 := mul(3, 1/3)",
      "after": "candidate derivative coefficient2 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每一項的次數乘係數，指數降1。",
        "en": "Multiply each coefficient by its degree and lower the exponent by1."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 candidate derivative coefficient2，不以小數近似替代。",
          "en": "Recompute candidate derivative coefficient2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3",
          "1/3"
        ],
        "output": "1",
        "label": "candidate derivative coefficient2"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "candidate derivative coefficient3 := mul(4, 1/4)",
      "after": "candidate derivative coefficient3 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每一項的次數乘係數，指數降1。",
        "en": "Multiply each coefficient by its degree and lower the exponent by1."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 candidate derivative coefficient3，不以小數近似替代。",
          "en": "Recompute candidate derivative coefficient3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "4",
          "1/4"
        ],
        "output": "1",
        "label": "candidate derivative coefficient3"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "differentiate all positive-degree terms",
      "after": "F′ coefficients=[-4, 9, 1, 1]",
      "relation": "implies",
      "rule_id": "constant_derivative_zero",
      "conditions": [],
      "explanation": {
        "zh-Hant": "若只有常數，導數清單正規化為單一零係數。",
        "en": "For a constant polynomial,normalize the derivative list to one zero coefficient."
      },
      "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": "F′ coefficients=[-4, 9, 1, 1]; f coefficients=[-4, 9, 1, 1]",
      "after": "F′=f identically: 是",
      "relation": "implies",
      "rule_id": "primitive_candidate_verdict",
      "conditions": [],
      "explanation": {
        "zh-Hant": "真例依完整係數恆等，假例另有明示反例點。",
        "en": "Positive cases use full coefficient identity;negative cases also have an explicit counterexample point."
      },
      "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": {
    "target": [
      "-4",
      "9",
      "1",
      "1"
    ],
    "derivative": [
      "-4",
      "9",
      "1",
      "1"
    ],
    "identity": true
  },
  "conclusion": {
    "canonical_value": "是",
    "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": {
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      "english_rows_path": "public/data/i18n/questions/batch-20261007-advance-20261008-r06-antiderivative-foundations.en.jsonl"
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        "original_path": "public/data/mko/antiderivative.json",
        "english_path": "public/data/i18n/editions/mko-antiderivative.en.json"
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    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "b9fe70a7a264c90c20349be0449b2bd1a1b4d0e0a97092bc9fbcb7ca36ce0694",
    "original_candidate_sha256": "f26955c16ce6623a9855d695ed544c29a666ae9c1a93d62a9d072adec9276af9",
    "original_candidate_line_sha256": "f26955c16ce6623a9855d695ed544c29a666ae9c1a93d62a9d072adec9276af9"
  },
  "historical_source_state": {
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      "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-p2",
    "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": {
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    },
    "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": "efeb4ac71e35620c991a512b160c72f2911291416987e268c48358dcaeeec46e",
      "arithmetic_steps": 4
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    "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."
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}