{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261007-advance-20261009-r06-definite-integral-0046",
  "pilot_number": "R0006-03896",
  "number": 3896,
  "family_id": "qf-density-integral-parity-symmetric-pair",
  "title": {
    "zh-Hant": "第 3896 題：奇函數資格與對稱積分成對",
    "en": "Question 3896: Oddness Eligibility and Symmetric Integral Pair"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261007-advance-20261009-r06-definite-integral-0046",
    "batch_id": "batch-20261007-advance-20261009-r06-definite-integral-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-definite-integral"
    ],
    "title_zh": "奇函數資格與對稱積分成對",
    "stem_zh": "全實數論域；升冪係數的受限多項式模型。升冪係數 f=[8, 1, 21, -1]，r=1≥0。列 (f是否為奇函數, I(−r,r))，第二項採最簡分數。",
    "answer": {
      "kind": "expression",
      "value": "(否, 30/1)",
      "explanation_zh": "奇函數資格與對稱多項式矩=(否, 30/1)。多項式的全部偶次係數為零才是奇函數；對稱積分0不推出奇函數，亦可能來自零半徑或偶次項抵消。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 3,
      "skills": [
        "奇函數資格與對稱積分成對"
      ]
    },
    "variant": {
      "family_id": "qf-density-integral-parity-symmetric-pair",
      "seed": 202610096046,
      "parameters": {
        "f": [
          8,
          1,
          21,
          -1
        ],
        "r": 1,
        "operation": "density_integral_parity_symmetric_pair",
        "domain": "reals",
        "representation": "ascending_powers",
        "semantic_template": "qf-density-integral-parity-symmetric-pair/[[8,0,21],1]"
      }
    },
    "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-r06-definite-integral-0046",
    "batch_id": "batch-20261007-advance-20261009-r06-definite-integral-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "8aaabc45b52c2af1680127cc46d514338b69392fd310264e4e7b0aee62a7bdfe",
      "family_id": "qf-density-integral-parity-symmetric-pair",
      "seed": 202610096046,
      "semantic_template": "qf-density-integral-parity-symmetric-pair/[[8,0,21],1]"
    },
    "fields": {
      "title": "Oddness Eligibility and Symmetric Integral Pair",
      "stem": "On all R, f has ascending coefficients [8, 1, 21, -1], constant term first. For r=1≥0, give (is f odd?,I(f,−r,r)) as Yes/No followed by its exact signed polynomial moment. Use reduced positive-denominator fractions, retaining /1 for integers.",
      "answer": {
        "kind": "expression",
        "value": "(No, 30/1)",
        "explanation": "Oddness requires all even-power coefficients to be zero; the scoped symmetric integral is computed separately. The pair is (No, 30/1). A zero symmetric integral does not imply oddness, including the r=0 case."
      },
      "skills": [
        "Oddness Eligibility and Symmetric Integral Pair"
      ],
      "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；升冪係數的受限多項式模型。升冪係數 f=[8, 1, 21, -1]，r=1≥0。列 (f是否為奇函數, I(−r,r))，第二項採最簡分數。",
        "en": "On all R, f has ascending coefficients [8, 1, 21, -1], constant term first. For r=1≥0, give (is f odd?,I(f,−r,r)) as Yes/No followed by its exact signed polynomial moment. Use reduced positive-denominator fractions, retaining /1 for integers."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "explicit real polynomial moment or constant integrand model",
      "after": "oriented I(a,b)=H(b)−H(a);do not sort endpoints",
      "relation": "implies",
      "rule_id": "polynomial_moment_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "受限多項式在實數上連續可積；用逐項原函數端點差求精確有向積分，不把絕對面積或有限和當積分。",
        "en": "Real polynomials are continuous and integrable;use termwise primitive endpoint differences for exact oriented integrals,not absolute area or a finite sum."
      },
      "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": "even-power coefficient degree0=8",
      "after": "required zero=否",
      "relation": "implies",
      "rule_id": "odd_polynomial_even_coefficient",
      "conditions": [],
      "explanation": {
        "zh-Hant": "f(−x)+f(x)只保留偶次項的兩倍；多項式恆等為0當且僅當所有偶次係數0，含常數項。",
        "en": "f(−x)+f(x) keeps twice the even-power terms;the polynomial is identically0 exactly when every even coefficient,including the constant,is0."
      },
      "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": "even-power coefficient degree2=21",
      "after": "required zero=否",
      "relation": "implies",
      "rule_id": "odd_polynomial_even_coefficient",
      "conditions": [],
      "explanation": {
        "zh-Hant": "f(−x)+f(x)只保留偶次項的兩倍；多項式恆等為0當且僅當所有偶次係數0，含常數項。",
        "en": "f(−x)+f(x) keeps twice the even-power terms;the polynomial is identically0 exactly when every even coefficient,including the constant,is0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "symmetric interval primitive coefficient1 := div(8, 1)",
      "after": "symmetric interval primitive coefficient1 = 8",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "x^k的原函數係數是c/(k+1)，正整數k+1非零。",
        "en": "The primitive coefficient for c·x^k is c/(k+1),with positive nonzero k+1."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 symmetric interval primitive coefficient1，不以小數近似替代。",
          "en": "Recompute symmetric interval primitive coefficient1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "8",
          "1"
        ],
        "output": "8",
        "label": "symmetric interval primitive coefficient1"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "symmetric interval primitive coefficient2 := div(1, 2)",
      "after": "symmetric interval primitive coefficient2 = 1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "x^k的原函數係數是c/(k+1)，正整數k+1非零。",
        "en": "The primitive coefficient for c·x^k is c/(k+1),with positive nonzero k+1."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 symmetric interval primitive coefficient2，不以小數近似替代。",
          "en": "Recompute symmetric interval primitive coefficient2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "1",
          "2"
        ],
        "output": "1/2",
        "label": "symmetric interval primitive coefficient2"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "symmetric interval primitive coefficient3 := div(21, 3)",
      "after": "symmetric interval primitive coefficient3 = 7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "x^k的原函數係數是c/(k+1)，正整數k+1非零。",
        "en": "The primitive coefficient for c·x^k is c/(k+1),with positive nonzero k+1."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 symmetric interval primitive coefficient3，不以小數近似替代。",
          "en": "Recompute symmetric interval primitive coefficient3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "21",
          "3"
        ],
        "output": "7",
        "label": "symmetric interval primitive coefficient3"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "symmetric interval primitive coefficient4 := div(-1, 4)",
      "after": "symmetric interval primitive coefficient4 = -1/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "x^k的原函數係數是c/(k+1)，正整數k+1非零。",
        "en": "The primitive coefficient for c·x^k is c/(k+1),with positive nonzero k+1."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 symmetric interval primitive coefficient4，不以小數近似替代。",
          "en": "Recompute symmetric interval primitive coefficient4 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "-1",
          "4"
        ],
        "output": "-1/4",
        "label": "symmetric interval primitive coefficient4"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "[0, 8, 1/2, 7, -1/4]",
      "after": "H′=the supplied polynomial;integrate from -1 to 1",
      "relation": "implies",
      "rule_id": "polynomial_primitive_identity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐項微分回原係數確定原函數；積分常數在端點差相消。",
        "en": "Termwise differentiation recovers the original coefficients;the integration constant cancels in the endpoint difference."
      },
      "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": "symmetric interval H(b) ascending coefficients=[0, 8, 1/2, 7, -1/4]",
      "after": "evaluate at x=1",
      "relation": "implies",
      "rule_id": "polynomial_evaluation_order",
      "conditions": [],
      "explanation": {
        "zh-Hant": "升冪係數用Horner由最高非零項向常數項求值；略去零最高項不改函數，也不改原始係數清單。",
        "en": "Evaluate ascending coefficients by Horner from the highest nonzero term down;ignoring redundant top zero terms changes neither the function nor the retained original list."
      },
      "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": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "symmetric interval H(b) Horner multiply before degree3 := mul(-1/4, 1)",
      "after": "symmetric interval H(b) Horner multiply before degree3 = -1/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先乘同一代入點。",
        "en": "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 symmetric interval H(b) Horner multiply before degree3，不以小數近似替代。",
          "en": "Recompute symmetric interval H(b) Horner multiply before degree3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1/4",
          "1"
        ],
        "output": "-1/4",
        "label": "symmetric interval H(b) Horner multiply before degree3"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "symmetric interval H(b) Horner add degree3 := sum(-1/4, 7)",
      "after": "symmetric interval H(b) Horner add degree3 = 27/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加此位置的完整係數。",
        "en": "Then add the full coefficient at this position."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 symmetric interval H(b) Horner add degree3，不以小數近似替代。",
          "en": "Recompute symmetric interval H(b) Horner add degree3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-1/4",
          "7"
        ],
        "output": "27/4",
        "label": "symmetric interval H(b) Horner add degree3"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "symmetric interval H(b) Horner multiply before degree2 := mul(27/4, 1)",
      "after": "symmetric interval H(b) Horner multiply before degree2 = 27/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先乘同一代入點。",
        "en": "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 symmetric interval H(b) Horner multiply before degree2，不以小數近似替代。",
          "en": "Recompute symmetric interval H(b) Horner multiply before degree2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "27/4",
          "1"
        ],
        "output": "27/4",
        "label": "symmetric interval H(b) Horner multiply before degree2"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "symmetric interval H(b) Horner add degree2 := sum(27/4, 1/2)",
      "after": "symmetric interval H(b) Horner add degree2 = 29/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加此位置的完整係數。",
        "en": "Then add the full coefficient at this position."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 symmetric interval H(b) Horner add degree2，不以小數近似替代。",
          "en": "Recompute symmetric interval H(b) Horner add degree2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "27/4",
          "1/2"
        ],
        "output": "29/4",
        "label": "symmetric interval H(b) Horner add degree2"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "symmetric interval H(b) Horner multiply before degree1 := mul(29/4, 1)",
      "after": "symmetric interval H(b) Horner multiply before degree1 = 29/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先乘同一代入點。",
        "en": "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 symmetric interval H(b) Horner multiply before degree1，不以小數近似替代。",
          "en": "Recompute symmetric interval H(b) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "29/4",
          "1"
        ],
        "output": "29/4",
        "label": "symmetric interval H(b) Horner multiply before degree1"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "symmetric interval H(b) Horner add degree1 := sum(29/4, 8)",
      "after": "symmetric interval H(b) Horner add degree1 = 61/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加此位置的完整係數。",
        "en": "Then add the full coefficient at this position."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 symmetric interval H(b) Horner add degree1，不以小數近似替代。",
          "en": "Recompute symmetric interval H(b) Horner add degree1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "29/4",
          "8"
        ],
        "output": "61/4",
        "label": "symmetric interval H(b) Horner add degree1"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "symmetric interval H(b) Horner multiply before degree0 := mul(61/4, 1)",
      "after": "symmetric interval H(b) Horner multiply before degree0 = 61/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先乘同一代入點。",
        "en": "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 symmetric interval H(b) Horner multiply before degree0，不以小數近似替代。",
          "en": "Recompute symmetric interval H(b) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "61/4",
          "1"
        ],
        "output": "61/4",
        "label": "symmetric interval H(b) Horner multiply before degree0"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "symmetric interval H(b) Horner add degree0 := sum(61/4, 0)",
      "after": "symmetric interval H(b) Horner add degree0 = 61/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加此位置的完整係數。",
        "en": "Then add the full coefficient at this position."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 symmetric interval H(b) Horner add degree0，不以小數近似替代。",
          "en": "Recompute symmetric interval H(b) Horner add degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "61/4",
          "0"
        ],
        "output": "61/4",
        "label": "symmetric interval H(b) Horner add degree0"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "symmetric interval H(a) ascending coefficients=[0, 8, 1/2, 7, -1/4]",
      "after": "evaluate at x=-1",
      "relation": "implies",
      "rule_id": "polynomial_evaluation_order",
      "conditions": [],
      "explanation": {
        "zh-Hant": "升冪係數用Horner由最高非零項向常數項求值；略去零最高項不改函數，也不改原始係數清單。",
        "en": "Evaluate ascending coefficients by Horner from the highest nonzero term down;ignoring redundant top zero terms changes neither the function nor the retained original list."
      },
      "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": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "symmetric interval H(a) Horner multiply before degree3 := mul(-1/4, -1)",
      "after": "symmetric interval H(a) Horner multiply before degree3 = 1/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先乘同一代入點。",
        "en": "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 symmetric interval H(a) Horner multiply before degree3，不以小數近似替代。",
          "en": "Recompute symmetric interval H(a) Horner multiply before degree3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1/4",
          "-1"
        ],
        "output": "1/4",
        "label": "symmetric interval H(a) Horner multiply before degree3"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "symmetric interval H(a) Horner add degree3 := sum(1/4, 7)",
      "after": "symmetric interval H(a) Horner add degree3 = 29/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加此位置的完整係數。",
        "en": "Then add the full coefficient at this position."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 symmetric interval H(a) Horner add degree3，不以小數近似替代。",
          "en": "Recompute symmetric interval H(a) Horner add degree3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1/4",
          "7"
        ],
        "output": "29/4",
        "label": "symmetric interval H(a) Horner add degree3"
      }
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "symmetric interval H(a) Horner multiply before degree2 := mul(29/4, -1)",
      "after": "symmetric interval H(a) Horner multiply before degree2 = -29/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先乘同一代入點。",
        "en": "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 symmetric interval H(a) Horner multiply before degree2，不以小數近似替代。",
          "en": "Recompute symmetric interval H(a) Horner multiply before degree2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "29/4",
          "-1"
        ],
        "output": "-29/4",
        "label": "symmetric interval H(a) Horner multiply before degree2"
      }
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "symmetric interval H(a) Horner add degree2 := sum(-29/4, 1/2)",
      "after": "symmetric interval H(a) Horner add degree2 = -27/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加此位置的完整係數。",
        "en": "Then add the full coefficient at this position."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 symmetric interval H(a) Horner add degree2，不以小數近似替代。",
          "en": "Recompute symmetric interval H(a) Horner add degree2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-29/4",
          "1/2"
        ],
        "output": "-27/4",
        "label": "symmetric interval H(a) Horner add degree2"
      }
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "symmetric interval H(a) Horner multiply before degree1 := mul(-27/4, -1)",
      "after": "symmetric interval H(a) Horner multiply before degree1 = 27/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先乘同一代入點。",
        "en": "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 symmetric interval H(a) Horner multiply before degree1，不以小數近似替代。",
          "en": "Recompute symmetric interval H(a) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-27/4",
          "-1"
        ],
        "output": "27/4",
        "label": "symmetric interval H(a) Horner multiply before degree1"
      }
    },
    {
      "id": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "symmetric interval H(a) Horner add degree1 := sum(27/4, 8)",
      "after": "symmetric interval H(a) Horner add degree1 = 59/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加此位置的完整係數。",
        "en": "Then add the full coefficient at this position."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 symmetric interval H(a) Horner add degree1，不以小數近似替代。",
          "en": "Recompute symmetric interval H(a) Horner add degree1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "27/4",
          "8"
        ],
        "output": "59/4",
        "label": "symmetric interval H(a) Horner add degree1"
      }
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "symmetric interval H(a) Horner multiply before degree0 := mul(59/4, -1)",
      "after": "symmetric interval H(a) Horner multiply before degree0 = -59/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先乘同一代入點。",
        "en": "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 symmetric interval H(a) Horner multiply before degree0，不以小數近似替代。",
          "en": "Recompute symmetric interval H(a) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "59/4",
          "-1"
        ],
        "output": "-59/4",
        "label": "symmetric interval H(a) Horner multiply before degree0"
      }
    },
    {
      "id": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "symmetric interval H(a) Horner add degree0 := sum(-59/4, 0)",
      "after": "symmetric interval H(a) Horner add degree0 = -59/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加此位置的完整係數。",
        "en": "Then add the full coefficient at this position."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 symmetric interval H(a) Horner add degree0，不以小數近似替代。",
          "en": "Recompute symmetric interval H(a) Horner add degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-59/4",
          "0"
        ],
        "output": "-59/4",
        "label": "symmetric interval H(a) Horner add degree0"
      }
    },
    {
      "id": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "symmetric interval oriented moment := sub(61/4, -59/4)",
      "after": "symmetric interval oriented moment = 30",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "終點值減起點值，端點逆序時自然變號。",
        "en": "Subtract the starting primitive value from the ending value;reversing endpoints naturally reverses the sign."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 symmetric interval oriented moment，不以小數近似替代。",
          "en": "Recompute symmetric interval oriented moment with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "61/4",
          "-59/4"
        ],
        "output": "30",
        "label": "symmetric interval oriented moment"
      }
    },
    {
      "id": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "odd polynomial=否;r=1;integral=30",
      "after": "zero symmetric integral alone does not imply oddness",
      "relation": "implies",
      "rule_id": "odd_integral_converse_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "奇函數的對稱積分0，但反向不成立：3x²−1在[−1,1]積分0卻不奇；r=0時任何多項式積分都0。",
        "en": "An odd function has symmetric integral0,but not conversely:3x²−1 integrates to0 on[−1,1] and is not odd.For r=0 every polynomial has integral0."
      },
      "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,
      "30"
    ]
  },
  "conclusion": {
    "canonical_value": "(否, 30/1)",
    "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-20261009-r06-definite-integral-foundations",
      "batch_path": "public/data/questions/batches/batch-20261007-advance-20261009-r06-definite-integral-foundations/questions.jsonl",
      "batch_sha256": "00d0f47b298d3ba323a3ab2d1343f4bc4c7badcb593d0c1c108a6245ce665cfa",
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      "original_line_sha256": "9a0e3feced02b8ee28519087fafee30ebc60570286344455373c08c9bf79bde3",
      "question_compact_sha256": "8aaabc45b52c2af1680127cc46d514338b69392fd310264e4e7b0aee62a7bdfe",
      "english_row_compact_sha256": "1a5cfb6ad51bda999888ba27f86ea7275a6a2c6fd209d868650e4a4c7799baf5",
      "english_manifest_path": "public/data/i18n/editions/batch-20261007-advance-20261009-r06-definite-integral-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261007-advance-20261009-r06-definite-integral-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-definite-integral": {
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        "original_path": "public/data/mko/definite-integral.json",
        "english_path": "public/data/i18n/editions/mko-definite-integral.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "6bac5f83f28cdd9c992a469306bfdf5c5415de17d11847cb9b17b4d13934c7e6",
    "original_candidate_sha256": "c43c963f4353c5a72be4c30adc945eed290bbdc5740d92df74679aa348ce6e0b",
    "original_candidate_line_sha256": "c43c963f4353c5a72be4c30adc945eed290bbdc5740d92df74679aa348ce6e0b"
  },
  "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-p7",
    "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": "568eefd71709ac3dc16a0d5b3b4ad3e83376a8ef300c9e5493eee0721500b29d",
      "question_first_witness_sha256": "d964b91bedb9d4918bbc055f567762a09f5a58af6c876268e719d5100332d855"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 28,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "daf3e0de7e180b12511b39ffcf389bcb589a0556f393c013702bcadd73ef315e",
      "arithmetic_steps": 21
    },
    "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."
  }
}