{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261007-advance-20261008-r06-antiderivative-0080",
  "pilot_number": "R0001-01380",
  "number": 1380,
  "family_id": "qf-antiderivative-shifted-value",
  "title": {
    "zh-Hant": "第 1380 題：含實際常數的原函數值",
    "en": "Question 1380: Antiderivative Value Retaining the Actual Constant"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261007-advance-20261008-r06-antiderivative-0080",
    "batch_id": "batch-20261007-advance-20261008-r06-antiderivative-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-antiderivative"
    ],
    "title_zh": "含實際常數的原函數值",
    "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=[-8, 5, 0, 1]，F0 為零常數 canonical primitive。令 F=F0+(-8)，求 F(2)，採最簡正分母分數。",
    "answer": {
      "kind": "expression",
      "value": "-10/1",
      "explanation_zh": "含常數原函數值=-10/1。此題保留實際 C 與求值點，不混同忽略 C 的原函數資格檢查。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "含實際常數的原函數值"
      ]
    },
    "variant": {
      "family_id": "qf-antiderivative-shifted-value",
      "seed": 202610086080,
      "parameters": {
        "f": [
          -8,
          5,
          0,
          1
        ],
        "C": -8,
        "x": 2,
        "operation": "shifted_primitive_value",
        "domain": "reals",
        "representation": "ascending_powers",
        "semantic_template": "qf-antiderivative-shifted-value/{\"inputs\":[[\"-8\",\"5\",\"0\",\"1\"],\"-8\",\"2\"],\"operation\":\"shifted_primitive_value\"}"
      }
    },
    "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-0080",
    "batch_id": "batch-20261007-advance-20261008-r06-antiderivative-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "d8d7d4df98f017df57bab18c503219e8d32567482179344c0a17f18f7a6061dc",
      "family_id": "qf-antiderivative-shifted-value",
      "seed": 202610086080,
      "semantic_template": "qf-antiderivative-shifted-value/{\"inputs\":[[\"-8\",\"5\",\"0\",\"1\"],\"-8\",\"2\"],\"operation\":\"shifted_primitive_value\"}"
    },
    "fields": {
      "title": "Antiderivative Value Retaining the Actual Constant",
      "stem": "On all R, f has ascending coefficients [-8, 5, 0, 1], starting with its constant term. Let F0 be its zero-constant primitive. Find F0(2)+(-8), retaining the actual additive constant. Give a reduced fraction with positive denominator, retaining /1 for integers.",
      "answer": {
        "kind": "expression",
        "value": "-10/1",
        "explanation": "Evaluate the rational primitive exactly at 2, then add C=-8, obtaining -10/1. A constant irrelevant to differentiation remains relevant to this value."
      },
      "skills": [
        "Antiderivative Value Retaining the Actual Constant"
      ],
      "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=[-8, 5, 0, 1]，F0 為零常數 canonical primitive。令 F=F0+(-8)，求 F(2)，採最簡正分母分數。",
        "en": "On all R, f has ascending coefficients [-8, 5, 0, 1], starting with its constant term. Let F0 be its zero-constant primitive. Find F0(2)+(-8), retaining the actual additive constant. Give a reduced fraction with positive denominator, retaining /1 for integers."
      }
    }
  ],
  "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": "f coefficients ascending=[-8, 5, 0, 1]",
      "after": "seek F₀ with constant coefficient0 and F₀′=f",
      "relation": "implies",
      "rule_id": "canonical_primitive_domain",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原函數並不一般唯一；本題額外固定常數項為0。",
        "en": "Antiderivatives are not generally unique; this task additionally fixes the constant coefficient at0."
      },
      "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": "primitive coefficient at degree1 := div(-8, 1)",
      "after": "primitive coefficient at degree1 = -8",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "由(x^(k+1))′=(k+1)x^k，係數須除以非零正整數k+1。",
        "en": "Since (x^(k+1))′=(k+1)x^k,divide the coefficient by the positive nonzero integer k+1."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 primitive coefficient at degree1，不以小數近似替代。",
          "en": "Recompute primitive coefficient at degree1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "-8",
          "1"
        ],
        "output": "-8",
        "label": "primitive coefficient at degree1"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "differentiate back coefficient0 := mul(-8, 1)",
      "after": "differentiate back coefficient0 = -8",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "微分乘回k+1，應恢復原多項式對應係數。",
        "en": "Differentiation multiplies back by k+1 and must recover the original coefficient."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 differentiate back coefficient0，不以小數近似替代。",
          "en": "Recompute differentiate back coefficient0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-8",
          "1"
        ],
        "output": "-8",
        "label": "differentiate back coefficient0"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "primitive coefficient at degree2 := div(5, 2)",
      "after": "primitive coefficient at degree2 = 5/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "由(x^(k+1))′=(k+1)x^k，係數須除以非零正整數k+1。",
        "en": "Since (x^(k+1))′=(k+1)x^k,divide the coefficient by the positive nonzero integer k+1."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 primitive coefficient at degree2，不以小數近似替代。",
          "en": "Recompute primitive coefficient at degree2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "5",
          "2"
        ],
        "output": "5/2",
        "label": "primitive coefficient at degree2"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "differentiate back coefficient1 := mul(5/2, 2)",
      "after": "differentiate back coefficient1 = 5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "微分乘回k+1，應恢復原多項式對應係數。",
        "en": "Differentiation multiplies back by k+1 and must recover the original coefficient."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 differentiate back coefficient1，不以小數近似替代。",
          "en": "Recompute differentiate back coefficient1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "5/2",
          "2"
        ],
        "output": "5",
        "label": "differentiate back coefficient1"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "primitive coefficient at degree3 := div(0, 3)",
      "after": "primitive coefficient at degree3 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "由(x^(k+1))′=(k+1)x^k，係數須除以非零正整數k+1。",
        "en": "Since (x^(k+1))′=(k+1)x^k,divide the coefficient by the positive nonzero integer k+1."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 primitive coefficient at degree3，不以小數近似替代。",
          "en": "Recompute primitive coefficient at degree3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "0",
          "3"
        ],
        "output": "0",
        "label": "primitive coefficient at degree3"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "differentiate back coefficient2 := mul(0, 3)",
      "after": "differentiate back coefficient2 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "微分乘回k+1，應恢復原多項式對應係數。",
        "en": "Differentiation multiplies back by k+1 and must recover the original coefficient."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 differentiate back coefficient2，不以小數近似替代。",
          "en": "Recompute differentiate back coefficient2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "3"
        ],
        "output": "0",
        "label": "differentiate back coefficient2"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "primitive coefficient at degree4 := div(1, 4)",
      "after": "primitive coefficient at degree4 = 1/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "由(x^(k+1))′=(k+1)x^k，係數須除以非零正整數k+1。",
        "en": "Since (x^(k+1))′=(k+1)x^k,divide the coefficient by the positive nonzero integer k+1."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 primitive coefficient at degree4，不以小數近似替代。",
          "en": "Recompute primitive coefficient at degree4 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "1",
          "4"
        ],
        "output": "1/4",
        "label": "primitive coefficient at degree4"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "differentiate back coefficient3 := mul(1/4, 4)",
      "after": "differentiate back coefficient3 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "微分乘回k+1，應恢復原多項式對應係數。",
        "en": "Differentiation multiplies back by k+1 and must recover the original coefficient."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 differentiate back coefficient3，不以小數近似替代。",
          "en": "Recompute differentiate back coefficient3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/4",
          "4"
        ],
        "output": "1",
        "label": "differentiate back coefficient3"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "[0, -8, 5/2, 0, 1/4]",
      "after": "all derivative coefficients agree with f",
      "relation": "implies",
      "rule_id": "global_polynomial_identity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "係數逐項相同證明全實數上的多項式恆等，不是只在幾個點相等。",
        "en": "Coefficient agreement proves the polynomial identity on all reals,not merely equality at a few points."
      },
      "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": "constant coefficient+C := sum(0, -8)",
      "after": "constant coefficient+C = -8",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只改常數項，導數仍是同一f。",
        "en": "Change only the constant coefficient;the derivative remains the same f."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 constant coefficient+C，不以小數近似替代。",
          "en": "Recompute constant coefficient+C with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "-8"
        ],
        "output": "-8",
        "label": "constant coefficient+C"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "F(x) multiply at degree4 := mul(0, 2)",
      "after": "F(x) multiply at degree4 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依Horner順序先乘同一代入點。",
        "en": "In Horner order, multiply by the same evaluation point first."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 F(x) multiply at degree4，不以小數近似替代。",
          "en": "Recompute F(x) multiply at degree4 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "2"
        ],
        "output": "0",
        "label": "F(x) multiply at degree4"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "F(x) add coefficient4 := sum(0, 1/4)",
      "after": "F(x) add coefficient4 = 1/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加該次數的完整有理係數。",
        "en": "Then add the exact rational coefficient for this degree."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 F(x) add coefficient4，不以小數近似替代。",
          "en": "Recompute F(x) add coefficient4 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "1/4"
        ],
        "output": "1/4",
        "label": "F(x) add coefficient4"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "F(x) multiply at degree3 := mul(1/4, 2)",
      "after": "F(x) multiply at degree3 = 1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依Horner順序先乘同一代入點。",
        "en": "In Horner order, multiply by the same evaluation point first."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 F(x) multiply at degree3，不以小數近似替代。",
          "en": "Recompute F(x) multiply at degree3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/4",
          "2"
        ],
        "output": "1/2",
        "label": "F(x) multiply at degree3"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "F(x) add coefficient3 := sum(1/2, 0)",
      "after": "F(x) add coefficient3 = 1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加該次數的完整有理係數。",
        "en": "Then add the exact rational coefficient for this degree."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 F(x) add coefficient3，不以小數近似替代。",
          "en": "Recompute F(x) add coefficient3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1/2",
          "0"
        ],
        "output": "1/2",
        "label": "F(x) add coefficient3"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "F(x) multiply at degree2 := mul(1/2, 2)",
      "after": "F(x) multiply at degree2 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依Horner順序先乘同一代入點。",
        "en": "In Horner order, multiply by the same evaluation point first."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 F(x) multiply at degree2，不以小數近似替代。",
          "en": "Recompute F(x) multiply at degree2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/2",
          "2"
        ],
        "output": "1",
        "label": "F(x) multiply at degree2"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "F(x) add coefficient2 := sum(1, 5/2)",
      "after": "F(x) add coefficient2 = 7/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加該次數的完整有理係數。",
        "en": "Then add the exact rational coefficient for this degree."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 F(x) add coefficient2，不以小數近似替代。",
          "en": "Recompute F(x) add coefficient2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "5/2"
        ],
        "output": "7/2",
        "label": "F(x) add coefficient2"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "F(x) multiply at degree1 := mul(7/2, 2)",
      "after": "F(x) multiply at degree1 = 7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依Horner順序先乘同一代入點。",
        "en": "In Horner order, multiply by the same evaluation point first."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 F(x) multiply at degree1，不以小數近似替代。",
          "en": "Recompute F(x) multiply at degree1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "7/2",
          "2"
        ],
        "output": "7",
        "label": "F(x) multiply at degree1"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "F(x) add coefficient1 := sum(7, -8)",
      "after": "F(x) add coefficient1 = -1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加該次數的完整有理係數。",
        "en": "Then add the exact rational coefficient for this degree."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 F(x) add coefficient1，不以小數近似替代。",
          "en": "Recompute F(x) add coefficient1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "7",
          "-8"
        ],
        "output": "-1",
        "label": "F(x) add coefficient1"
      }
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "F(x) multiply at degree0 := mul(-1, 2)",
      "after": "F(x) multiply at degree0 = -2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依Horner順序先乘同一代入點。",
        "en": "In Horner order, multiply by the same evaluation point first."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 F(x) multiply at degree0，不以小數近似替代。",
          "en": "Recompute F(x) multiply at degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "2"
        ],
        "output": "-2",
        "label": "F(x) multiply at degree0"
      }
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "F(x) add coefficient0 := sum(-2, -8)",
      "after": "F(x) add coefficient0 = -10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加該次數的完整有理係數。",
        "en": "Then add the exact rational coefficient for this degree."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 F(x) add coefficient0，不以小數近似替代。",
          "en": "Recompute F(x) add coefficient0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "-8"
        ],
        "output": "-10",
        "label": "F(x) add coefficient0"
      }
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "x=2; additive constant=-8",
      "after": "F(x)=-10/1",
      "relation": "implies",
      "rule_id": "primitive_value_format",
      "conditions": [],
      "explanation": {
        "zh-Hant": "答案採最簡正分母分數，整數也保留/1。",
        "en": "Use a reduced positive-denominator fraction,retaining /1 even for integers."
      },
      "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": {
    "primitive": [
      "0",
      "-8",
      "5/2",
      "0",
      "1/4"
    ],
    "constant": "-8",
    "value": "-10"
  },
  "conclusion": {
    "canonical_value": "-10/1",
    "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-20261008-r06-antiderivative-foundations",
      "batch_path": "public/data/questions/batches/batch-20261007-advance-20261008-r06-antiderivative-foundations/questions.jsonl",
      "batch_sha256": "322101eaff25d0ec9ae01a8564baf7cf7de5678f3caeba49643ad8968169b98e",
      "original_line": 80,
      "original_line_sha256": "e81c7c9b2237a6e4991998f6d455b18a35e272f62dcd2d7b374d389d62ffd00b",
      "question_compact_sha256": "d8d7d4df98f017df57bab18c503219e8d32567482179344c0a17f18f7a6061dc",
      "english_row_compact_sha256": "c659dbf33bbe1f338da0c39da2a99568f446165f6ae267ca6b546063a49b8e28",
      "english_manifest_path": "public/data/i18n/editions/batch-20261007-advance-20261008-r06-antiderivative-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261007-advance-20261008-r06-antiderivative-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-antiderivative": {
        "sha256": "5f2ecff6866e471d1f8da7fea1fa5e121364ae2d24b2fea405d2300fe9ff1cae",
        "english_sha256": "f983dd8b898bc9eac9d5237c7895e5f7dc182ac46efb5f9b58b925fc6c2ecb9d",
        "original_path": "public/data/mko/antiderivative.json",
        "english_path": "public/data/i18n/editions/mko-antiderivative.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "b9fe70a7a264c90c20349be0449b2bd1a1b4d0e0a97092bc9fbcb7ca36ce0694",
    "original_candidate_sha256": "76f1b06612a2770233e9c86a5bb0daa99c83ffd511befdfe0b14908be52c5bca",
    "original_candidate_line_sha256": "76f1b06612a2770233e9c86a5bb0daa99c83ffd511befdfe0b14908be52c5bca"
  },
  "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-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": {
      "semantic_review_sha256": "c3b44ce8ed60ada6902fb206627537d5f90ba2a867eaba43a36f903280c496cc",
      "question_first_witness_sha256": "3e91602adc7f81a2671a2a3ffaf6a447028b56abc5853e8271673d76f30cd7ad"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 23,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "9435f1307bd23b1c0186543e450070276aa3dfba7a4703eaea51c7d12c2b0756",
      "arithmetic_steps": 19
    },
    "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."
  }
}