{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261007-advance-20261009-r01-fundamental-theorem-calculus-0098",
  "pilot_number": "R0019-10198",
  "number": 10198,
  "family_id": "qf-ftc-primitive-hypothesis",
  "title": {
    "zh-Hant": "第 10198 題：端點公式的候選原函數前提",
    "en": "Question 10198: Candidate Antiderivative Premise for the Endpoint Formula"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261007-advance-20261009-r01-fundamental-theorem-calculus-0098",
    "batch_id": "batch-20261007-advance-20261009-r01-fundamental-theorem-calculus-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-fundamental-theorem-calculus"
    ],
    "title_zh": "端點公式的候選原函數前提",
    "stem_zh": "全實數論域；連續多項式被積函數，FTC的明示充分前提。升冪係數 f=[10, 23, 0, 3]；候選 F 的升冪有理係數為 [[10,1],[10,1],[23,2],[0,1],[7,4]]，每對為[分子,正分母]。是否滿足 F′=f 的係數恆等，以使此候選適用FTC端點差？",
    "answer": {
      "kind": "short_text",
      "value": "否",
      "explanation_zh": "候選導數係數=[10/1, 23/1, 0/1, 7/1]；FTC前提=否。逐係數求導核對；候選失敗不否定其他原函數存在，加常數不影響此資格，有限核對不是一般定理的形式證明。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 3,
      "skills": [
        "端點公式的候選原函數前提"
      ]
    },
    "variant": {
      "family_id": "qf-ftc-primitive-hypothesis",
      "seed": 202610091098,
      "parameters": {
        "f": [
          10,
          23,
          0,
          3
        ],
        "F": [
          [
            10,
            1
          ],
          [
            10,
            1
          ],
          [
            23,
            2
          ],
          [
            0,
            1
          ],
          [
            7,
            4
          ]
        ],
        "operation": "ftc_primitive_hypothesis",
        "domain": "reals",
        "representation": "ascending_powers",
        "semantic_template": "qf-ftc-primitive-hypothesis/[[10,23,0,3],[\"10/1\",\"23/2\",\"0/1\",\"7/4\"]]"
      }
    },
    "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-20261009-r01-fundamental-theorem-calculus-0098",
    "batch_id": "batch-20261007-advance-20261009-r01-fundamental-theorem-calculus-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "7513a4cdfa03cc624f99c5adddb6bc3d209699ba3ea7ae205a2ccc9938ee9f7d",
      "family_id": "qf-ftc-primitive-hypothesis",
      "seed": 202610091098,
      "semantic_template": "qf-ftc-primitive-hypothesis/[[10,23,0,3],[\"10/1\",\"23/2\",\"0/1\",\"7/4\"]]"
    },
    "fields": {
      "title": "Candidate Antiderivative Premise for the Endpoint Formula",
      "stem": "On all R, continuous f has ascending coefficients [10, 23, 0, 3], constant term first. Candidate F has ascending rational coefficients [10/1, 10/1, 23/2, 0/1, 7/4]. Does F′=f satisfy the antiderivative premise for the endpoint formula?",
      "answer": {
        "kind": "short_text",
        "value": "No",
        "explanation": "Compare complete derivative coefficients with f, giving No, not a finite-sample equality claim. Rejecting this candidate does not reject other possible primitives; its constant term does not affect eligibility."
      },
      "skills": [
        "Candidate Antiderivative Premise for the Endpoint Formula"
      ],
      "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；連續多項式被積函數，FTC的明示充分前提。升冪係數 f=[10, 23, 0, 3]；候選 F 的升冪有理係數為 [[10,1],[10,1],[23,2],[0,1],[7,4]]，每對為[分子,正分母]。是否滿足 F′=f 的係數恆等，以使此候選適用FTC端點差？",
        "en": "On all R, continuous f has ascending coefficients [10, 23, 0, 3], constant term first. Candidate F has ascending rational coefficients [10/1, 10/1, 23/2, 0/1, 7/4]. Does F′=f satisfy the antiderivative premise for the endpoint formula?"
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "raw integrand f=[10, 23, 0, 3];ascending powers on all R",
      "after": "normalized coefficients=[10/1, 23/1, 0/1, 3/1];declared continuous real polynomial",
      "relation": "implies",
      "rule_id": "declared_FTC_continuity_hypothesis",
      "conditions": [],
      "explanation": {
        "zh-Hant": "本題明示全R連續多項式，積分採有向端點。FTC關係只在這些充分前提下使用，不以有限核對宣稱新增一般FTC形式證明。",
        "en": "The declared integrand is a continuous real polynomial on all R,and endpoints are oriented.Use FTC only under these supplied sufficient hypotheses;finite checks are not a new general formal FTC proof."
      },
      "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": "raw candidate F=[[10, 1], [10, 1], [23, 2], [0, 1], [7, 4]]",
      "after": "normalized candidate coefficients=[10/1, 10/1, 23/2, 0/1, 7/4]",
      "relation": "implies",
      "rule_id": "candidate_primitive_original_denominators",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個原分母須正，先約成精確有理係數；保留原候選，不用另建的原函數覆寫它。",
        "en": "Every original denominator must be positive.Normalize to exact rationals while retaining the supplied candidate,not replacing it by a constructed primitive."
      },
      "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 F derivative coefficient0 := mul(1, 10)",
      "after": "candidate F derivative coefficient0 = 10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。",
        "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 candidate F derivative coefficient0，不以小數近似替代。",
          "en": "Recompute candidate F derivative coefficient0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "10"
        ],
        "output": "10",
        "label": "candidate F derivative coefficient0"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "candidate F:source power=1;coefficient=10/1",
      "after": "derivative power=0;coefficient=10/1",
      "relation": "implies",
      "rule_id": "polynomial_derivative_coefficient_rule",
      "conditions": [],
      "explanation": {
        "zh-Hant": "次數k乘原係數並把次數降1，常數項不參與；完整保留所有非零高次項。",
        "en": "Multiply the coefficient by k and lower its power by1.The constant contributes nothing;retain every higher nonzero term."
      },
      "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": "candidate F derivative coefficient1 := mul(2, 23/2)",
      "after": "candidate F derivative coefficient1 = 23",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。",
        "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 candidate F derivative coefficient1，不以小數近似替代。",
          "en": "Recompute candidate F derivative coefficient1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "23/2"
        ],
        "output": "23",
        "label": "candidate F derivative coefficient1"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "candidate F:source power=2;coefficient=23/2",
      "after": "derivative power=1;coefficient=23/1",
      "relation": "implies",
      "rule_id": "polynomial_derivative_coefficient_rule",
      "conditions": [],
      "explanation": {
        "zh-Hant": "次數k乘原係數並把次數降1，常數項不參與；完整保留所有非零高次項。",
        "en": "Multiply the coefficient by k and lower its power by1.The constant contributes nothing;retain every higher nonzero term."
      },
      "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": "candidate F derivative coefficient2 := mul(3, 0)",
      "after": "candidate F derivative coefficient2 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。",
        "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 candidate F derivative coefficient2，不以小數近似替代。",
          "en": "Recompute candidate F derivative coefficient2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3",
          "0"
        ],
        "output": "0",
        "label": "candidate F derivative coefficient2"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "candidate F:source power=3;coefficient=0/1",
      "after": "derivative power=2;coefficient=0/1",
      "relation": "implies",
      "rule_id": "polynomial_derivative_coefficient_rule",
      "conditions": [],
      "explanation": {
        "zh-Hant": "次數k乘原係數並把次數降1，常數項不參與；完整保留所有非零高次項。",
        "en": "Multiply the coefficient by k and lower its power by1.The constant contributes nothing;retain every higher nonzero term."
      },
      "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": "candidate F derivative coefficient3 := mul(4, 7/4)",
      "after": "candidate F derivative coefficient3 = 7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。",
        "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 candidate F derivative coefficient3，不以小數近似替代。",
          "en": "Recompute candidate F derivative coefficient3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "4",
          "7/4"
        ],
        "output": "7",
        "label": "candidate F derivative coefficient3"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "candidate F:source power=4;coefficient=7/4",
      "after": "derivative power=3;coefficient=7/1",
      "relation": "implies",
      "rule_id": "polynomial_derivative_coefficient_rule",
      "conditions": [],
      "explanation": {
        "zh-Hant": "次數k乘原係數並把次數降1，常數項不參與；完整保留所有非零高次項。",
        "en": "Multiply the coefficient by k and lower its power by1.The constant contributes nothing;retain every higher nonzero term."
      },
      "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": "power=0;candidate derivative coefficient=10/1;target f coefficient=10/1",
      "after": "coefficient equality=是",
      "relation": "implies",
      "rule_id": "FTC_candidate_full_derivative_identity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐係數比較F′與f到完整最大次數，不以一個相等點通過原函數前提。",
        "en": "Compare F′ and f coefficient by coefficient through the full larger degree;one matching point does not establish the primitive premise."
      },
      "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": "power=1;candidate derivative coefficient=23/1;target f coefficient=23/1",
      "after": "coefficient equality=是",
      "relation": "implies",
      "rule_id": "FTC_candidate_full_derivative_identity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐係數比較F′與f到完整最大次數，不以一個相等點通過原函數前提。",
        "en": "Compare F′ and f coefficient by coefficient through the full larger degree;one matching point does not establish the primitive premise."
      },
      "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": "power=2;candidate derivative coefficient=0/1;target f coefficient=0/1",
      "after": "coefficient equality=是",
      "relation": "implies",
      "rule_id": "FTC_candidate_full_derivative_identity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐係數比較F′與f到完整最大次數，不以一個相等點通過原函數前提。",
        "en": "Compare F′ and f coefficient by coefficient through the full larger degree;one matching point does not establish the primitive premise."
      },
      "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": "power=3;candidate derivative coefficient=7/1;target f coefficient=3/1",
      "after": "coefficient equality=否",
      "relation": "implies",
      "rule_id": "FTC_candidate_full_derivative_identity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐係數比較F′與f到完整最大次數，不以一個相等點通過原函數前提。",
        "en": "Compare F′ and f coefficient by coefficient through the full larger degree;one matching point does not establish the primitive premise."
      },
      "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": "candidate derivative=[10/1, 23/1, 0/1, 7/1];integrand=[10/1, 23/1, 0/1, 3/1]",
      "after": "this candidate satisfies F'=f=否",
      "relation": "implies",
      "rule_id": "failed_primitive_candidate_not_all_primitives",
      "conditions": [],
      "explanation": {
        "zh-Hant": "候選失敗只否定此F，不否定其他原函數；候選常數自由，係數身份才是本題端點公式的資格。",
        "en": "Failure rejects this F only,not other primitives.The candidate constant is free;coefficient identity is the requested endpoint-formula qualification."
      },
      "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
  },
  "conclusion": {
    "canonical_value": "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-20261007-advance-20261009-r01-fundamental-theorem-calculus-foundations",
      "batch_path": "public/data/questions/batches/batch-20261007-advance-20261009-r01-fundamental-theorem-calculus-foundations/questions.jsonl",
      "batch_sha256": "d2d6fe2173f2779b3f2778eb8354bae1c05ca3ef96e61fac19f950257f89c643",
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