{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261007-advance-20261008-r05-derivative-rules-0002",
  "pilot_number": "R0061-31202",
  "number": 31202,
  "family_id": "qf-derivative-rule-sum",
  "title": {
    "zh-Hant": "第 31202 題：多項式和的導數",
    "en": "Question 31202: Derivative of a Polynomial Sum"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261007-advance-20261008-r05-derivative-rules-0002",
    "batch_id": "batch-20261007-advance-20261008-r05-derivative-rules-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-derivative-rules"
    ],
    "title_zh": "多項式和的導數",
    "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=[-11, 2, 0, 2]、g=[1, -1, 1, 1]，兩輸入多項式各至多三次；求 (f+g) 在 x=-1 的導數值。",
    "answer": {
      "kind": "short_text",
      "value": "8",
      "explanation_zh": "規則導數值=8。只對已明示多項式使用和／純量法則，不處理任意未宣告可微函數。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "多項式和的導數"
      ]
    },
    "variant": {
      "family_id": "qf-derivative-rule-sum",
      "seed": 202610085002,
      "parameters": {
        "f": [
          -11,
          2,
          0,
          2
        ],
        "g": [
          1,
          -1,
          1,
          1
        ],
        "x": -1,
        "operation": "derivative_sum",
        "domain": "reals",
        "representation": "ascending_powers",
        "semantic_template": "qf-derivative-rule-sum/{\"inputs\":[[\"1\",\"1\",\"3\"],\"-1\"],\"operation\":\"derivative_sum\"}"
      }
    },
    "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-r05-derivative-rules-0002",
    "batch_id": "batch-20261007-advance-20261008-r05-derivative-rules-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "d7398e9d38b31e806ad821c8fcfc4f04ff1af650c00c9d0865810c0816c5bac8",
      "family_id": "qf-derivative-rule-sum",
      "seed": 202610085002,
      "semantic_template": "qf-derivative-rule-sum/{\"inputs\":[[\"1\",\"1\",\"3\"],\"-1\"],\"operation\":\"derivative_sum\"}"
    },
    "fields": {
      "title": "Derivative of a Polynomial Sum",
      "stem": "On all R, f has ascending coefficients [-11, 2, 0, 2], starting with its constant term; g has ascending coefficients [1, -1, 1, 1], starting with its constant term. Find the derivative of f+g at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.",
      "answer": {
        "kind": "short_text",
        "value": "8",
        "explanation": "Use f′+g′. The exact result is 8; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."
      },
      "skills": [
        "Derivative of a Polynomial Sum"
      ],
      "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=[-11, 2, 0, 2]、g=[1, -1, 1, 1]，兩輸入多項式各至多三次；求 (f+g) 在 x=-1 的導數值。",
        "en": "On all R, f has ascending coefficients [-11, 2, 0, 2], starting with its constant term; g has ascending coefficients [1, -1, 1, 1], starting with its constant term. Find the derivative of f+g at x=-1. Input degrees are at most three; retain all higher terms produced by the operation."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [qf-derivative-rule-sum, derivative_sum, domain, reals]",
      "after": "derived = [[integer_derivative_value]]",
      "relation": "implies",
      "rule_id": "R61_SOURCE_exact_keys_operation_domain_OR_frame_and_output_roles",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先核本題欄位與來源協議：distance只用完整座標frame，其他用domain；離散運算local-hyphen，舊bank用underscore。輸出欄序及型別依題族，保留原始中英資料與狀態。",
        "en": "Check THIS source interface:distance uses ONLY its complete coordinate frame;other families use a domain.Discrete operations use local hyphens,legacy banks underscores.Retain family-specific ordered output types and original bilingual data/status."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始輸入、型別及本步條件反向代回核對，不聲稱結論能唯一還原所有前提。這是來源綁定候選，不是獨立審查或形式證明。",
          "en": "Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "source roles = [f OUTER or first, [-11, 2, 0, 2]]",
      "after": "derived = [[-11, 2, 0, 2]]",
      "relation": "implies",
      "rule_id": "Derivative_RAW_ascending_arity_BEFORE_trim_whole_polynomial_R",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先驗raw係數向量1..4及每個SAFE整數，再修去尾零但保留零多項式[0]。已給的是整個R上多項式法則，不能把少數樣本當可微性證明。",
        "en": "Check RAW ascending coefficient arity1..4 and EVERY SAFE integer BEFORE trimming trailing zeros,retaining[0].The GIVEN whole polynomial law holds on R;sample values alone do not establish differentiability."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始輸入、型別及本步條件反向代回核對，不聲稱結論能唯一還原所有前提。這是來源綁定候選，不是獨立審查或形式證明。",
          "en": "Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "source roles = [g INNER or second, [1, -1, 1, 1]]",
      "after": "derived = [[1, -1, 1, 1]]",
      "relation": "implies",
      "rule_id": "Derivative_RAW_ascending_arity_BEFORE_trim_whole_polynomial_R",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先驗raw係數向量1..4及每個SAFE整數，再修去尾零但保留零多項式[0]。已給的是整個R上多項式法則，不能把少數樣本當可微性證明。",
        "en": "Check RAW ascending coefficient arity1..4 and EVERY SAFE integer BEFORE trimming trailing zeros,retaining[0].The GIVEN whole polynomial law holds on R;sample values alone do not establish differentiability."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始輸入、型別及本步條件反向代回核對，不聲稱結論能唯一還原所有前提。這是來源綁定候選，不是獨立審查或形式證明。",
          "en": "Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "f derivative coefficient power1 := mul(1, 2)",
      "after": "f derivative coefficient power1 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序，作精確整數計算；只對原始輸入檢查SAFE，衍生整數不再套上輸入界，且不用浮點近似。",
        "en": "Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 f derivative coefficient power1，不以小數近似替代。",
          "en": "Recompute f derivative coefficient power1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "2"
        ],
        "output": "2",
        "label": "f derivative coefficient power1"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "f derivative coefficient power2 := mul(2, 0)",
      "after": "f derivative coefficient power2 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序，作精確整數計算；只對原始輸入檢查SAFE，衍生整數不再套上輸入界，且不用浮點近似。",
        "en": "Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 f derivative coefficient power2，不以小數近似替代。",
          "en": "Recompute f derivative coefficient power2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "0"
        ],
        "output": "0",
        "label": "f derivative coefficient power2"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "f derivative coefficient power3 := mul(3, 2)",
      "after": "f derivative coefficient power3 = 6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序，作精確整數計算；只對原始輸入檢查SAFE，衍生整數不再套上輸入界，且不用浮點近似。",
        "en": "Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 f derivative coefficient power3，不以小數近似替代。",
          "en": "Recompute f derivative coefficient power3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3",
          "2"
        ],
        "output": "6",
        "label": "f derivative coefficient power3"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "source roles = [f, [-11, 2, 0, 2]]",
      "after": "derived = [[2, 0, 6]]",
      "relation": "implies",
      "rule_id": "Derivative_POWER_rule_EVERY_nonconstant_term_constant_derivative_zero",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐項用(a_k x^k)′=k a_k x^(k−1)，常數導數為0。全部非零高次項都保留，不截成raw向量界。",
        "en": "For EVERY term use(a_k x^k)′=k a_k x^(k−1);a constant differentiates to0.Keep ALL higher terms without reapplying raw-vector bounds."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始輸入、型別及本步條件反向代回核對，不聲稱結論能唯一還原所有前提。這是來源綁定候選，不是獨立審查或形式證明。",
          "en": "Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "g derivative coefficient power1 := mul(1, -1)",
      "after": "g derivative coefficient power1 = -1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序，作精確整數計算；只對原始輸入檢查SAFE，衍生整數不再套上輸入界，且不用浮點近似。",
        "en": "Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 g derivative coefficient power1，不以小數近似替代。",
          "en": "Recompute g derivative coefficient power1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "-1"
        ],
        "output": "-1",
        "label": "g derivative coefficient power1"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "g derivative coefficient power2 := mul(2, 1)",
      "after": "g derivative coefficient power2 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序，作精確整數計算；只對原始輸入檢查SAFE，衍生整數不再套上輸入界，且不用浮點近似。",
        "en": "Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 g derivative coefficient power2，不以小數近似替代。",
          "en": "Recompute g derivative coefficient power2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "1"
        ],
        "output": "2",
        "label": "g derivative coefficient power2"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "g derivative coefficient power3 := mul(3, 1)",
      "after": "g derivative coefficient power3 = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序，作精確整數計算；只對原始輸入檢查SAFE，衍生整數不再套上輸入界，且不用浮點近似。",
        "en": "Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 g derivative coefficient power3，不以小數近似替代。",
          "en": "Recompute g derivative coefficient power3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3",
          "1"
        ],
        "output": "3",
        "label": "g derivative coefficient power3"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "source roles = [g, [1, -1, 1, 1]]",
      "after": "derived = [[-1, 2, 3]]",
      "relation": "implies",
      "rule_id": "Derivative_POWER_rule_EVERY_nonconstant_term_constant_derivative_zero",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐項用(a_k x^k)′=k a_k x^(k−1)，常數導數為0。全部非零高次項都保留，不截成raw向量界。",
        "en": "For EVERY term use(a_k x^k)′=k a_k x^(k−1);a constant differentiates to0.Keep ALL higher terms without reapplying raw-vector bounds."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始輸入、型別及本步條件反向代回核對，不聲稱結論能唯一還原所有前提。這是來源綁定候選，不是獨立審查或形式證明。",
          "en": "Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "g derivative at ORIGINAL x Horner power2 times argument := mul(0, -1)",
      "after": "g derivative at ORIGINAL x Horner power2 times argument = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序，作精確整數計算；只對原始輸入檢查SAFE，衍生整數不再套上輸入界，且不用浮點近似。",
        "en": "Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 g derivative at ORIGINAL x Horner power2 times argument，不以小數近似替代。",
          "en": "Recompute g derivative at ORIGINAL x Horner power2 times argument with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "-1"
        ],
        "output": "0",
        "label": "g derivative at ORIGINAL x Horner power2 times argument"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "g derivative at ORIGINAL x Horner power2 plus coefficient := sum(0, 3)",
      "after": "g derivative at ORIGINAL x Horner power2 plus coefficient = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序，作精確整數計算；只對原始輸入檢查SAFE，衍生整數不再套上輸入界，且不用浮點近似。",
        "en": "Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 g derivative at ORIGINAL x Horner power2 plus coefficient，不以小數近似替代。",
          "en": "Recompute g derivative at ORIGINAL x Horner power2 plus coefficient with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "3"
        ],
        "output": "3",
        "label": "g derivative at ORIGINAL x Horner power2 plus coefficient"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "g derivative at ORIGINAL x Horner power1 times argument := mul(3, -1)",
      "after": "g derivative at ORIGINAL x Horner power1 times argument = -3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序，作精確整數計算；只對原始輸入檢查SAFE，衍生整數不再套上輸入界，且不用浮點近似。",
        "en": "Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 g derivative at ORIGINAL x Horner power1 times argument，不以小數近似替代。",
          "en": "Recompute g derivative at ORIGINAL x Horner power1 times argument with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3",
          "-1"
        ],
        "output": "-3",
        "label": "g derivative at ORIGINAL x Horner power1 times argument"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "g derivative at ORIGINAL x Horner power1 plus coefficient := sum(-3, 2)",
      "after": "g derivative at ORIGINAL x Horner power1 plus coefficient = -1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序，作精確整數計算；只對原始輸入檢查SAFE，衍生整數不再套上輸入界，且不用浮點近似。",
        "en": "Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 g derivative at ORIGINAL x Horner power1 plus coefficient，不以小數近似替代。",
          "en": "Recompute g derivative at ORIGINAL x Horner power1 plus coefficient with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-3",
          "2"
        ],
        "output": "-1",
        "label": "g derivative at ORIGINAL x Horner power1 plus coefficient"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "g derivative at ORIGINAL x Horner power0 times argument := mul(-1, -1)",
      "after": "g derivative at ORIGINAL x Horner power0 times argument = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序，作精確整數計算；只對原始輸入檢查SAFE，衍生整數不再套上輸入界，且不用浮點近似。",
        "en": "Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 g derivative at ORIGINAL x Horner power0 times argument，不以小數近似替代。",
          "en": "Recompute g derivative at ORIGINAL x Horner power0 times argument with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "-1"
        ],
        "output": "1",
        "label": "g derivative at ORIGINAL x Horner power0 times argument"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "g derivative at ORIGINAL x Horner power0 plus coefficient := sum(1, -1)",
      "after": "g derivative at ORIGINAL x Horner power0 plus coefficient = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序，作精確整數計算；只對原始輸入檢查SAFE，衍生整數不再套上輸入界，且不用浮點近似。",
        "en": "Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 g derivative at ORIGINAL x Horner power0 plus coefficient，不以小數近似替代。",
          "en": "Recompute g derivative at ORIGINAL x Horner power0 plus coefficient with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "-1"
        ],
        "output": "0",
        "label": "g derivative at ORIGINAL x Horner power0 plus coefficient"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "source roles = [g derivative at ORIGINAL x, [-1, 2, 3], -1]",
      "after": "derived = [0]",
      "relation": "implies",
      "rule_id": "Derivative_HORNER_ALL_coefficients_AT_declared_argument",
      "conditions": [],
      "explanation": {
        "zh-Hant": "升冪資料由最高次開始Horner，逐次乘明示argument再加原係數，包含所有零項；不將inner值誤換為原x。",
        "en": "Evaluate ascending coefficients by Horner from the highest power,multiplying the DECLARED argument then adding the ORIGINAL coefficient at every step,including zeros.Do not replace an inner value with original x."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始輸入、型別及本步條件反向代回核對，不聲稱結論能唯一還原所有前提。這是來源綁定候選，不是獨立審查或形式證明。",
          "en": "Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "f derivative at ORIGINAL x Horner power2 times argument := mul(0, -1)",
      "after": "f derivative at ORIGINAL x Horner power2 times argument = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序，作精確整數計算；只對原始輸入檢查SAFE，衍生整數不再套上輸入界，且不用浮點近似。",
        "en": "Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 f derivative at ORIGINAL x Horner power2 times argument，不以小數近似替代。",
          "en": "Recompute f derivative at ORIGINAL x Horner power2 times argument with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "-1"
        ],
        "output": "0",
        "label": "f derivative at ORIGINAL x Horner power2 times argument"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "f derivative at ORIGINAL x Horner power2 plus coefficient := sum(0, 6)",
      "after": "f derivative at ORIGINAL x Horner power2 plus coefficient = 6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序，作精確整數計算；只對原始輸入檢查SAFE，衍生整數不再套上輸入界，且不用浮點近似。",
        "en": "Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 f derivative at ORIGINAL x Horner power2 plus coefficient，不以小數近似替代。",
          "en": "Recompute f derivative at ORIGINAL x Horner power2 plus coefficient with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "6"
        ],
        "output": "6",
        "label": "f derivative at ORIGINAL x Horner power2 plus coefficient"
      }
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "f derivative at ORIGINAL x Horner power1 times argument := mul(6, -1)",
      "after": "f derivative at ORIGINAL x Horner power1 times argument = -6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序，作精確整數計算；只對原始輸入檢查SAFE，衍生整數不再套上輸入界，且不用浮點近似。",
        "en": "Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 f derivative at ORIGINAL x Horner power1 times argument，不以小數近似替代。",
          "en": "Recompute f derivative at ORIGINAL x Horner power1 times argument with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "6",
          "-1"
        ],
        "output": "-6",
        "label": "f derivative at ORIGINAL x Horner power1 times argument"
      }
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "f derivative at ORIGINAL x Horner power1 plus coefficient := sum(-6, 0)",
      "after": "f derivative at ORIGINAL x Horner power1 plus coefficient = -6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序，作精確整數計算；只對原始輸入檢查SAFE，衍生整數不再套上輸入界，且不用浮點近似。",
        "en": "Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 f derivative at ORIGINAL x Horner power1 plus coefficient，不以小數近似替代。",
          "en": "Recompute f derivative at ORIGINAL x Horner power1 plus coefficient with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-6",
          "0"
        ],
        "output": "-6",
        "label": "f derivative at ORIGINAL x Horner power1 plus coefficient"
      }
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "f derivative at ORIGINAL x Horner power0 times argument := mul(-6, -1)",
      "after": "f derivative at ORIGINAL x Horner power0 times argument = 6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序，作精確整數計算；只對原始輸入檢查SAFE，衍生整數不再套上輸入界，且不用浮點近似。",
        "en": "Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 f derivative at ORIGINAL x Horner power0 times argument，不以小數近似替代。",
          "en": "Recompute f derivative at ORIGINAL x Horner power0 times argument with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-6",
          "-1"
        ],
        "output": "6",
        "label": "f derivative at ORIGINAL x Horner power0 times argument"
      }
    },
    {
      "id": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "f derivative at ORIGINAL x Horner power0 plus coefficient := sum(6, 2)",
      "after": "f derivative at ORIGINAL x Horner power0 plus coefficient = 8",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序，作精確整數計算；只對原始輸入檢查SAFE，衍生整數不再套上輸入界，且不用浮點近似。",
        "en": "Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 f derivative at ORIGINAL x Horner power0 plus coefficient，不以小數近似替代。",
          "en": "Recompute f derivative at ORIGINAL x Horner power0 plus coefficient with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "6",
          "2"
        ],
        "output": "8",
        "label": "f derivative at ORIGINAL x Horner power0 plus coefficient"
      }
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "source roles = [f derivative at ORIGINAL x, [2, 0, 6], -1]",
      "after": "derived = [8]",
      "relation": "implies",
      "rule_id": "Derivative_HORNER_ALL_coefficients_AT_declared_argument",
      "conditions": [],
      "explanation": {
        "zh-Hant": "升冪資料由最高次開始Horner，逐次乘明示argument再加原係數，包含所有零項；不將inner值誤換為原x。",
        "en": "Evaluate ascending coefficients by Horner from the highest power,multiplying the DECLARED argument then adding the ORIGINAL coefficient at every step,including zeros.Do not replace an inner value with original x."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始輸入、型別及本步條件反向代回核對，不聲稱結論能唯一還原所有前提。這是來源綁定候選，不是獨立審查或形式證明。",
          "en": "Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "Derivative sum ORIGINAL fprime plus gprime := sum(8, 0)",
      "after": "Derivative sum ORIGINAL fprime plus gprime = 8",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序，作精確整數計算；只對原始輸入檢查SAFE，衍生整數不再套上輸入界，且不用浮點近似。",
        "en": "Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Derivative sum ORIGINAL fprime plus gprime，不以小數近似替代。",
          "en": "Recompute Derivative sum ORIGINAL fprime plus gprime with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "8",
          "0"
        ],
        "output": "8",
        "label": "Derivative sum ORIGINAL fprime plus gprime"
      }
    },
    {
      "id": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "source roles = [[-11, 2, 0, 2], [1, -1, 1, 1], -1, [2, 0, 6], [-1, 2, 3]]",
      "after": "derived = [8, 0, 8]",
      "relation": "implies",
      "rule_id": "Derivative_SUM_rule_same_original_argument",
      "conditions": [],
      "explanation": {
        "zh-Hant": "和的導數逐項相加，在同一個原x代值；答案是此點的整數導數值，不是係數表或帶/1字串。",
        "en": "Differentiate the sum termwise and evaluate BOTH derivatives at the SAME original x.The answer is the integer derivative VALUE,not a coefficient list or a/1 string."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始輸入、型別及本步條件反向代回核對，不聲稱結論能唯一還原所有前提。這是來源綁定候選，不是獨立審查或形式證明。",
          "en": "Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": "8"
  },
  "conclusion": {
    "canonical_value": "8",
    "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-r05-derivative-rules-foundations",
      "batch_path": "public/data/questions/batches/batch-20261007-advance-20261008-r05-derivative-rules-foundations/questions.jsonl",
      "batch_sha256": "43a070f36cb018a249764a06a259cf826f90c4d01b3e12150de2148cf65be618",
      "original_line": 2,
      "original_line_sha256": "bf0f1815af9accaa6ab6e5efe44a815495e0b32b7d7b87ff6b1f254469710b3f",
      "question_compact_sha256": "d7398e9d38b31e806ad821c8fcfc4f04ff1af650c00c9d0865810c0816c5bac8",
      "english_row_compact_sha256": "26c714fb966636f469e900e0a73a8a0d01ed10bf6d44b9f0ca416ad66f4b9d2e",
      "english_manifest_path": "public/data/i18n/editions/batch-20261007-advance-20261008-r05-derivative-rules-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261007-advance-20261008-r05-derivative-rules-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-derivative-rules": {
        "sha256": "4756c1f6226c8cecc330975377ab5d42ed0e461b0fe039c05959fc60ab7c628f",
        "english_sha256": "ed443b25aa7217b775b464078d3230bca85b98df615f3517b6284c5f828ded44",
        "original_path": "public/data/mko/derivative-rules.json",
        "english_path": "public/data/i18n/editions/mko-derivative-rules.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "912db26eb6c90aa6e256f4356cb5ee94cd0a352f04f667876ae6739f8181a17c",
    "original_candidate_sha256": "06d5d5707bb3e7bc0bfa474927cbf5ce9ed420e41b3dffd86ad383d090bda6fb",
    "original_candidate_line_sha256": "06d5d5707bb3e7bc0bfa474927cbf5ce9ed420e41b3dffd86ad383d090bda6fb"
  },
  "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-20261026-011-p62",
    "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": "5403317467ff7c6b74c18ae7211530d0156ebc67d63fc00a3e07b81a201d3d69",
      "question_first_witness_sha256": "c2b5022aa77f803cd3e1ba9ce162a707da0281aa5bb5289111a97de51fe8c495"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 27,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "b0a52f73c21efc03f0539f6610a8ed0bc0b6caaf447430e9d9271e1a28e3f4ef",
      "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."
  }
}