{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261007-advance-20261009-r06-definite-integral-0006",
  "pilot_number": "R0006-03806",
  "number": 3806,
  "family_id": "qf-density-integral-additivity-triple",
  "title": {
    "zh-Hant": "第 3806 題：三段有向多項式矩積分",
    "en": "Question 3806: Three Oriented Polynomial-Moment Integrals"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261007-advance-20261009-r06-definite-integral-0006",
    "batch_id": "batch-20261007-advance-20261009-r06-definite-integral-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-definite-integral"
    ],
    "title_zh": "三段有向多項式矩積分",
    "stem_zh": "全實數論域；升冪係數的受限多項式模型。升冪係數 f=[-2, 6, 7, 1]，a=0、b=-6、c=-4。按明示多項式矩模型求 (I(a,b), I(b,c), I(a,c))，端點不預設排序。",
    "answer": {
      "kind": "expression",
      "value": "(-60/1, 92/3, -88/3)",
      "explanation_zh": "有向三段多項式矩=(-60/1, 92/3, -88/3)。產生端逐段以canonical primitive端點差另算，保留方向及符號；只核對此多項式模型。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "三段有向多項式矩積分"
      ]
    },
    "variant": {
      "family_id": "qf-density-integral-additivity-triple",
      "seed": 202610096006,
      "parameters": {
        "f": [
          -2,
          6,
          7,
          1
        ],
        "a": 0,
        "b": -6,
        "c": -4,
        "operation": "density_integral_additivity_triple",
        "domain": "reals",
        "representation": "ascending_powers",
        "semantic_template": "qf-density-integral-additivity-triple/[[-2,6,7,1],0,-6,-4]"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_integer_rational_arithmetic",
        "producer_finite_enumeration",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-07",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261007-advance-20261009-r06-definite-integral-0006",
    "batch_id": "batch-20261007-advance-20261009-r06-definite-integral-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "4cd0f20254a74b20254eef89c790e2ea95eb277094c07d2b1fd9ade19211c057",
      "family_id": "qf-density-integral-additivity-triple",
      "seed": 202610096006,
      "semantic_template": "qf-density-integral-additivity-triple/[[-2,6,7,1],0,-6,-4]"
    },
    "fields": {
      "title": "Three Oriented Polynomial-Moment Integrals",
      "stem": "On all R, f has ascending coefficients [-2, 6, 7, 1], constant term first. For arbitrary oriented endpoints a=0,b=-6,c=-4, give (I(f,a,b),I(f,b,c),I(f,a,c)) using the scoped polynomial moments. Use reduced positive-denominator fractions, retaining /1 for integers.",
      "answer": {
        "kind": "expression",
        "value": "(-60/1, 92/3, -88/3)",
        "explanation": "Exact signed moments give (-60/1, 92/3, -88/3); the first two sum to the third, retaining orientation even when the middle endpoint lies outside the other interval. This is not a new general integral proof."
      },
      "skills": [
        "Three Oriented Polynomial-Moment Integrals"
      ],
      "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=[-2, 6, 7, 1]，a=0、b=-6、c=-4。按明示多項式矩模型求 (I(a,b), I(b,c), I(a,c))，端點不預設排序。",
        "en": "On all R, f has ascending coefficients [-2, 6, 7, 1], constant term first. For arbitrary oriented endpoints a=0,b=-6,c=-4, give (I(f,a,b),I(f,b,c),I(f,a,c)) using the scoped polynomial moments. 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": "I(a,b) primitive coefficient1 := div(-2, 1)",
      "after": "I(a,b) primitive coefficient1 = -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 I(a,b) primitive coefficient1，不以小數近似替代。",
          "en": "Recompute I(a,b) primitive coefficient1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "-2",
          "1"
        ],
        "output": "-2",
        "label": "I(a,b) primitive coefficient1"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "I(a,b) primitive coefficient2 := div(6, 2)",
      "after": "I(a,b) primitive coefficient2 = 3",
      "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 I(a,b) primitive coefficient2，不以小數近似替代。",
          "en": "Recompute I(a,b) primitive coefficient2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "6",
          "2"
        ],
        "output": "3",
        "label": "I(a,b) primitive coefficient2"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "I(a,b) primitive coefficient3 := div(7, 3)",
      "after": "I(a,b) primitive coefficient3 = 7/3",
      "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 I(a,b) primitive coefficient3，不以小數近似替代。",
          "en": "Recompute I(a,b) primitive coefficient3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "7",
          "3"
        ],
        "output": "7/3",
        "label": "I(a,b) primitive coefficient3"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "I(a,b) primitive coefficient4 := div(1, 4)",
      "after": "I(a,b) 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 I(a,b) primitive coefficient4，不以小數近似替代。",
          "en": "Recompute I(a,b) primitive coefficient4 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "1",
          "4"
        ],
        "output": "1/4",
        "label": "I(a,b) primitive coefficient4"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "[0, -2, 3, 7/3, 1/4]",
      "after": "H′=the supplied polynomial;integrate from 0 to -6",
      "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": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "I(a,b) H(b) ascending coefficients=[0, -2, 3, 7/3, 1/4]",
      "after": "evaluate at x=-6",
      "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": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "I(a,b) H(b) Horner multiply before degree3 := mul(1/4, -6)",
      "after": "I(a,b) H(b) Horner multiply before degree3 = -3/2",
      "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 I(a,b) H(b) Horner multiply before degree3，不以小數近似替代。",
          "en": "Recompute I(a,b) H(b) Horner multiply before degree3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/4",
          "-6"
        ],
        "output": "-3/2",
        "label": "I(a,b) H(b) Horner multiply before degree3"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "I(a,b) H(b) Horner add degree3 := sum(-3/2, 7/3)",
      "after": "I(a,b) H(b) Horner add degree3 = 5/6",
      "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 I(a,b) H(b) Horner add degree3，不以小數近似替代。",
          "en": "Recompute I(a,b) H(b) Horner add degree3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-3/2",
          "7/3"
        ],
        "output": "5/6",
        "label": "I(a,b) H(b) Horner add degree3"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "I(a,b) H(b) Horner multiply before degree2 := mul(5/6, -6)",
      "after": "I(a,b) H(b) Horner multiply before degree2 = -5",
      "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 I(a,b) H(b) Horner multiply before degree2，不以小數近似替代。",
          "en": "Recompute I(a,b) H(b) Horner multiply before degree2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "5/6",
          "-6"
        ],
        "output": "-5",
        "label": "I(a,b) H(b) Horner multiply before degree2"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "I(a,b) H(b) Horner add degree2 := sum(-5, 3)",
      "after": "I(a,b) H(b) Horner add degree2 = -2",
      "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 I(a,b) H(b) Horner add degree2，不以小數近似替代。",
          "en": "Recompute I(a,b) H(b) Horner add degree2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-5",
          "3"
        ],
        "output": "-2",
        "label": "I(a,b) H(b) Horner add degree2"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "I(a,b) H(b) Horner multiply before degree1 := mul(-2, -6)",
      "after": "I(a,b) H(b) Horner multiply before degree1 = 12",
      "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 I(a,b) H(b) Horner multiply before degree1，不以小數近似替代。",
          "en": "Recompute I(a,b) H(b) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-2",
          "-6"
        ],
        "output": "12",
        "label": "I(a,b) H(b) Horner multiply before degree1"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "I(a,b) H(b) Horner add degree1 := sum(12, -2)",
      "after": "I(a,b) H(b) Horner add degree1 = 10",
      "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 I(a,b) H(b) Horner add degree1，不以小數近似替代。",
          "en": "Recompute I(a,b) H(b) Horner add degree1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "12",
          "-2"
        ],
        "output": "10",
        "label": "I(a,b) H(b) Horner add degree1"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "I(a,b) H(b) Horner multiply before degree0 := mul(10, -6)",
      "after": "I(a,b) H(b) Horner multiply before degree0 = -60",
      "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 I(a,b) H(b) Horner multiply before degree0，不以小數近似替代。",
          "en": "Recompute I(a,b) H(b) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "10",
          "-6"
        ],
        "output": "-60",
        "label": "I(a,b) H(b) Horner multiply before degree0"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "I(a,b) H(b) Horner add degree0 := sum(-60, 0)",
      "after": "I(a,b) H(b) Horner add degree0 = -60",
      "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 I(a,b) H(b) Horner add degree0，不以小數近似替代。",
          "en": "Recompute I(a,b) H(b) Horner add degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-60",
          "0"
        ],
        "output": "-60",
        "label": "I(a,b) H(b) Horner add degree0"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "I(a,b) H(a) ascending coefficients=[0, -2, 3, 7/3, 1/4]",
      "after": "evaluate at x=0",
      "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": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "I(a,b) H(a) Horner multiply before degree3 := mul(1/4, 0)",
      "after": "I(a,b) H(a) Horner multiply before degree3 = 0",
      "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 I(a,b) H(a) Horner multiply before degree3，不以小數近似替代。",
          "en": "Recompute I(a,b) H(a) Horner multiply before degree3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/4",
          "0"
        ],
        "output": "0",
        "label": "I(a,b) H(a) Horner multiply before degree3"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "I(a,b) H(a) Horner add degree3 := sum(0, 7/3)",
      "after": "I(a,b) H(a) Horner add degree3 = 7/3",
      "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 I(a,b) H(a) Horner add degree3，不以小數近似替代。",
          "en": "Recompute I(a,b) H(a) Horner add degree3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "7/3"
        ],
        "output": "7/3",
        "label": "I(a,b) H(a) Horner add degree3"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "I(a,b) H(a) Horner multiply before degree2 := mul(7/3, 0)",
      "after": "I(a,b) H(a) Horner multiply before degree2 = 0",
      "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 I(a,b) H(a) Horner multiply before degree2，不以小數近似替代。",
          "en": "Recompute I(a,b) H(a) Horner multiply before degree2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "7/3",
          "0"
        ],
        "output": "0",
        "label": "I(a,b) H(a) Horner multiply before degree2"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "I(a,b) H(a) Horner add degree2 := sum(0, 3)",
      "after": "I(a,b) H(a) Horner add degree2 = 3",
      "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 I(a,b) H(a) Horner add degree2，不以小數近似替代。",
          "en": "Recompute I(a,b) H(a) Horner add degree2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "3"
        ],
        "output": "3",
        "label": "I(a,b) H(a) Horner add degree2"
      }
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "I(a,b) H(a) Horner multiply before degree1 := mul(3, 0)",
      "after": "I(a,b) H(a) Horner multiply before degree1 = 0",
      "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 I(a,b) H(a) Horner multiply before degree1，不以小數近似替代。",
          "en": "Recompute I(a,b) H(a) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3",
          "0"
        ],
        "output": "0",
        "label": "I(a,b) H(a) Horner multiply before degree1"
      }
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "I(a,b) H(a) Horner add degree1 := sum(0, -2)",
      "after": "I(a,b) H(a) Horner add degree1 = -2",
      "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 I(a,b) H(a) Horner add degree1，不以小數近似替代。",
          "en": "Recompute I(a,b) H(a) Horner add degree1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "-2"
        ],
        "output": "-2",
        "label": "I(a,b) H(a) Horner add degree1"
      }
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "I(a,b) H(a) Horner multiply before degree0 := mul(-2, 0)",
      "after": "I(a,b) H(a) Horner multiply before degree0 = 0",
      "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 I(a,b) H(a) Horner multiply before degree0，不以小數近似替代。",
          "en": "Recompute I(a,b) H(a) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-2",
          "0"
        ],
        "output": "0",
        "label": "I(a,b) H(a) Horner multiply before degree0"
      }
    },
    {
      "id": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "I(a,b) H(a) Horner add degree0 := sum(0, 0)",
      "after": "I(a,b) H(a) Horner add degree0 = 0",
      "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 I(a,b) H(a) Horner add degree0，不以小數近似替代。",
          "en": "Recompute I(a,b) H(a) Horner add degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "I(a,b) H(a) Horner add degree0"
      }
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "I(a,b) oriented moment := sub(-60, 0)",
      "after": "I(a,b) oriented moment = -60",
      "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 I(a,b) oriented moment，不以小數近似替代。",
          "en": "Recompute I(a,b) oriented moment with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-60",
          "0"
        ],
        "output": "-60",
        "label": "I(a,b) oriented moment"
      }
    },
    {
      "id": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "I(b,c) primitive coefficient1 := div(-2, 1)",
      "after": "I(b,c) primitive coefficient1 = -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 I(b,c) primitive coefficient1，不以小數近似替代。",
          "en": "Recompute I(b,c) primitive coefficient1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "-2",
          "1"
        ],
        "output": "-2",
        "label": "I(b,c) primitive coefficient1"
      }
    },
    {
      "id": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "I(b,c) primitive coefficient2 := div(6, 2)",
      "after": "I(b,c) primitive coefficient2 = 3",
      "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 I(b,c) primitive coefficient2，不以小數近似替代。",
          "en": "Recompute I(b,c) primitive coefficient2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "6",
          "2"
        ],
        "output": "3",
        "label": "I(b,c) primitive coefficient2"
      }
    },
    {
      "id": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "I(b,c) primitive coefficient3 := div(7, 3)",
      "after": "I(b,c) primitive coefficient3 = 7/3",
      "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 I(b,c) primitive coefficient3，不以小數近似替代。",
          "en": "Recompute I(b,c) primitive coefficient3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "7",
          "3"
        ],
        "output": "7/3",
        "label": "I(b,c) primitive coefficient3"
      }
    },
    {
      "id": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "I(b,c) primitive coefficient4 := div(1, 4)",
      "after": "I(b,c) 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 I(b,c) primitive coefficient4，不以小數近似替代。",
          "en": "Recompute I(b,c) primitive coefficient4 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "1",
          "4"
        ],
        "output": "1/4",
        "label": "I(b,c) primitive coefficient4"
      }
    },
    {
      "id": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "[0, -2, 3, 7/3, 1/4]",
      "after": "H′=the supplied polynomial;integrate from -6 to -4",
      "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": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "I(b,c) H(b) ascending coefficients=[0, -2, 3, 7/3, 1/4]",
      "after": "evaluate at x=-4",
      "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": "S032",
      "uses": [
        "G001",
        "S031"
      ],
      "before": "I(b,c) H(b) Horner multiply before degree3 := mul(1/4, -4)",
      "after": "I(b,c) H(b) Horner multiply before degree3 = -1",
      "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 I(b,c) H(b) Horner multiply before degree3，不以小數近似替代。",
          "en": "Recompute I(b,c) H(b) Horner multiply before degree3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/4",
          "-4"
        ],
        "output": "-1",
        "label": "I(b,c) H(b) Horner multiply before degree3"
      }
    },
    {
      "id": "S033",
      "uses": [
        "G001",
        "S032"
      ],
      "before": "I(b,c) H(b) Horner add degree3 := sum(-1, 7/3)",
      "after": "I(b,c) H(b) Horner add degree3 = 4/3",
      "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 I(b,c) H(b) Horner add degree3，不以小數近似替代。",
          "en": "Recompute I(b,c) H(b) Horner add degree3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-1",
          "7/3"
        ],
        "output": "4/3",
        "label": "I(b,c) H(b) Horner add degree3"
      }
    },
    {
      "id": "S034",
      "uses": [
        "G001",
        "S033"
      ],
      "before": "I(b,c) H(b) Horner multiply before degree2 := mul(4/3, -4)",
      "after": "I(b,c) H(b) Horner multiply before degree2 = -16/3",
      "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 I(b,c) H(b) Horner multiply before degree2，不以小數近似替代。",
          "en": "Recompute I(b,c) H(b) Horner multiply before degree2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "4/3",
          "-4"
        ],
        "output": "-16/3",
        "label": "I(b,c) H(b) Horner multiply before degree2"
      }
    },
    {
      "id": "S035",
      "uses": [
        "G001",
        "S034"
      ],
      "before": "I(b,c) H(b) Horner add degree2 := sum(-16/3, 3)",
      "after": "I(b,c) H(b) Horner add degree2 = -7/3",
      "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 I(b,c) H(b) Horner add degree2，不以小數近似替代。",
          "en": "Recompute I(b,c) H(b) Horner add degree2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-16/3",
          "3"
        ],
        "output": "-7/3",
        "label": "I(b,c) H(b) Horner add degree2"
      }
    },
    {
      "id": "S036",
      "uses": [
        "G001",
        "S035"
      ],
      "before": "I(b,c) H(b) Horner multiply before degree1 := mul(-7/3, -4)",
      "after": "I(b,c) H(b) Horner multiply before degree1 = 28/3",
      "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 I(b,c) H(b) Horner multiply before degree1，不以小數近似替代。",
          "en": "Recompute I(b,c) H(b) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-7/3",
          "-4"
        ],
        "output": "28/3",
        "label": "I(b,c) H(b) Horner multiply before degree1"
      }
    },
    {
      "id": "S037",
      "uses": [
        "G001",
        "S036"
      ],
      "before": "I(b,c) H(b) Horner add degree1 := sum(28/3, -2)",
      "after": "I(b,c) H(b) Horner add degree1 = 22/3",
      "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 I(b,c) H(b) Horner add degree1，不以小數近似替代。",
          "en": "Recompute I(b,c) H(b) Horner add degree1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "28/3",
          "-2"
        ],
        "output": "22/3",
        "label": "I(b,c) H(b) Horner add degree1"
      }
    },
    {
      "id": "S038",
      "uses": [
        "G001",
        "S037"
      ],
      "before": "I(b,c) H(b) Horner multiply before degree0 := mul(22/3, -4)",
      "after": "I(b,c) H(b) Horner multiply before degree0 = -88/3",
      "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 I(b,c) H(b) Horner multiply before degree0，不以小數近似替代。",
          "en": "Recompute I(b,c) H(b) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "22/3",
          "-4"
        ],
        "output": "-88/3",
        "label": "I(b,c) H(b) Horner multiply before degree0"
      }
    },
    {
      "id": "S039",
      "uses": [
        "G001",
        "S038"
      ],
      "before": "I(b,c) H(b) Horner add degree0 := sum(-88/3, 0)",
      "after": "I(b,c) H(b) Horner add degree0 = -88/3",
      "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 I(b,c) H(b) Horner add degree0，不以小數近似替代。",
          "en": "Recompute I(b,c) H(b) Horner add degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-88/3",
          "0"
        ],
        "output": "-88/3",
        "label": "I(b,c) H(b) Horner add degree0"
      }
    },
    {
      "id": "S040",
      "uses": [
        "G001",
        "S039"
      ],
      "before": "I(b,c) H(a) ascending coefficients=[0, -2, 3, 7/3, 1/4]",
      "after": "evaluate at x=-6",
      "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": "S041",
      "uses": [
        "G001",
        "S040"
      ],
      "before": "I(b,c) H(a) Horner multiply before degree3 := mul(1/4, -6)",
      "after": "I(b,c) H(a) Horner multiply before degree3 = -3/2",
      "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 I(b,c) H(a) Horner multiply before degree3，不以小數近似替代。",
          "en": "Recompute I(b,c) H(a) Horner multiply before degree3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/4",
          "-6"
        ],
        "output": "-3/2",
        "label": "I(b,c) H(a) Horner multiply before degree3"
      }
    },
    {
      "id": "S042",
      "uses": [
        "G001",
        "S041"
      ],
      "before": "I(b,c) H(a) Horner add degree3 := sum(-3/2, 7/3)",
      "after": "I(b,c) H(a) Horner add degree3 = 5/6",
      "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 I(b,c) H(a) Horner add degree3，不以小數近似替代。",
          "en": "Recompute I(b,c) H(a) Horner add degree3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-3/2",
          "7/3"
        ],
        "output": "5/6",
        "label": "I(b,c) H(a) Horner add degree3"
      }
    },
    {
      "id": "S043",
      "uses": [
        "G001",
        "S042"
      ],
      "before": "I(b,c) H(a) Horner multiply before degree2 := mul(5/6, -6)",
      "after": "I(b,c) H(a) Horner multiply before degree2 = -5",
      "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 I(b,c) H(a) Horner multiply before degree2，不以小數近似替代。",
          "en": "Recompute I(b,c) H(a) Horner multiply before degree2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "5/6",
          "-6"
        ],
        "output": "-5",
        "label": "I(b,c) H(a) Horner multiply before degree2"
      }
    },
    {
      "id": "S044",
      "uses": [
        "G001",
        "S043"
      ],
      "before": "I(b,c) H(a) Horner add degree2 := sum(-5, 3)",
      "after": "I(b,c) H(a) Horner add degree2 = -2",
      "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 I(b,c) H(a) Horner add degree2，不以小數近似替代。",
          "en": "Recompute I(b,c) H(a) Horner add degree2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-5",
          "3"
        ],
        "output": "-2",
        "label": "I(b,c) H(a) Horner add degree2"
      }
    },
    {
      "id": "S045",
      "uses": [
        "G001",
        "S044"
      ],
      "before": "I(b,c) H(a) Horner multiply before degree1 := mul(-2, -6)",
      "after": "I(b,c) H(a) Horner multiply before degree1 = 12",
      "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 I(b,c) H(a) Horner multiply before degree1，不以小數近似替代。",
          "en": "Recompute I(b,c) H(a) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-2",
          "-6"
        ],
        "output": "12",
        "label": "I(b,c) H(a) Horner multiply before degree1"
      }
    },
    {
      "id": "S046",
      "uses": [
        "G001",
        "S045"
      ],
      "before": "I(b,c) H(a) Horner add degree1 := sum(12, -2)",
      "after": "I(b,c) H(a) Horner add degree1 = 10",
      "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 I(b,c) H(a) Horner add degree1，不以小數近似替代。",
          "en": "Recompute I(b,c) H(a) Horner add degree1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "12",
          "-2"
        ],
        "output": "10",
        "label": "I(b,c) H(a) Horner add degree1"
      }
    },
    {
      "id": "S047",
      "uses": [
        "G001",
        "S046"
      ],
      "before": "I(b,c) H(a) Horner multiply before degree0 := mul(10, -6)",
      "after": "I(b,c) H(a) Horner multiply before degree0 = -60",
      "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 I(b,c) H(a) Horner multiply before degree0，不以小數近似替代。",
          "en": "Recompute I(b,c) H(a) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "10",
          "-6"
        ],
        "output": "-60",
        "label": "I(b,c) H(a) Horner multiply before degree0"
      }
    },
    {
      "id": "S048",
      "uses": [
        "G001",
        "S047"
      ],
      "before": "I(b,c) H(a) Horner add degree0 := sum(-60, 0)",
      "after": "I(b,c) H(a) Horner add degree0 = -60",
      "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 I(b,c) H(a) Horner add degree0，不以小數近似替代。",
          "en": "Recompute I(b,c) H(a) Horner add degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-60",
          "0"
        ],
        "output": "-60",
        "label": "I(b,c) H(a) Horner add degree0"
      }
    },
    {
      "id": "S049",
      "uses": [
        "G001",
        "S048"
      ],
      "before": "I(b,c) oriented moment := sub(-88/3, -60)",
      "after": "I(b,c) oriented moment = 92/3",
      "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 I(b,c) oriented moment，不以小數近似替代。",
          "en": "Recompute I(b,c) oriented moment with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-88/3",
          "-60"
        ],
        "output": "92/3",
        "label": "I(b,c) oriented moment"
      }
    },
    {
      "id": "S050",
      "uses": [
        "G001",
        "S049"
      ],
      "before": "I(a,c) primitive coefficient1 := div(-2, 1)",
      "after": "I(a,c) primitive coefficient1 = -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 I(a,c) primitive coefficient1，不以小數近似替代。",
          "en": "Recompute I(a,c) primitive coefficient1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "-2",
          "1"
        ],
        "output": "-2",
        "label": "I(a,c) primitive coefficient1"
      }
    },
    {
      "id": "S051",
      "uses": [
        "G001",
        "S050"
      ],
      "before": "I(a,c) primitive coefficient2 := div(6, 2)",
      "after": "I(a,c) primitive coefficient2 = 3",
      "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 I(a,c) primitive coefficient2，不以小數近似替代。",
          "en": "Recompute I(a,c) primitive coefficient2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "6",
          "2"
        ],
        "output": "3",
        "label": "I(a,c) primitive coefficient2"
      }
    },
    {
      "id": "S052",
      "uses": [
        "G001",
        "S051"
      ],
      "before": "I(a,c) primitive coefficient3 := div(7, 3)",
      "after": "I(a,c) primitive coefficient3 = 7/3",
      "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 I(a,c) primitive coefficient3，不以小數近似替代。",
          "en": "Recompute I(a,c) primitive coefficient3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "7",
          "3"
        ],
        "output": "7/3",
        "label": "I(a,c) primitive coefficient3"
      }
    },
    {
      "id": "S053",
      "uses": [
        "G001",
        "S052"
      ],
      "before": "I(a,c) primitive coefficient4 := div(1, 4)",
      "after": "I(a,c) 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 I(a,c) primitive coefficient4，不以小數近似替代。",
          "en": "Recompute I(a,c) primitive coefficient4 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "1",
          "4"
        ],
        "output": "1/4",
        "label": "I(a,c) primitive coefficient4"
      }
    },
    {
      "id": "S054",
      "uses": [
        "G001",
        "S053"
      ],
      "before": "[0, -2, 3, 7/3, 1/4]",
      "after": "H′=the supplied polynomial;integrate from 0 to -4",
      "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": "S055",
      "uses": [
        "G001",
        "S054"
      ],
      "before": "I(a,c) H(b) ascending coefficients=[0, -2, 3, 7/3, 1/4]",
      "after": "evaluate at x=-4",
      "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": "S056",
      "uses": [
        "G001",
        "S055"
      ],
      "before": "I(a,c) H(b) Horner multiply before degree3 := mul(1/4, -4)",
      "after": "I(a,c) H(b) Horner multiply before degree3 = -1",
      "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 I(a,c) H(b) Horner multiply before degree3，不以小數近似替代。",
          "en": "Recompute I(a,c) H(b) Horner multiply before degree3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/4",
          "-4"
        ],
        "output": "-1",
        "label": "I(a,c) H(b) Horner multiply before degree3"
      }
    },
    {
      "id": "S057",
      "uses": [
        "G001",
        "S056"
      ],
      "before": "I(a,c) H(b) Horner add degree3 := sum(-1, 7/3)",
      "after": "I(a,c) H(b) Horner add degree3 = 4/3",
      "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 I(a,c) H(b) Horner add degree3，不以小數近似替代。",
          "en": "Recompute I(a,c) H(b) Horner add degree3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-1",
          "7/3"
        ],
        "output": "4/3",
        "label": "I(a,c) H(b) Horner add degree3"
      }
    },
    {
      "id": "S058",
      "uses": [
        "G001",
        "S057"
      ],
      "before": "I(a,c) H(b) Horner multiply before degree2 := mul(4/3, -4)",
      "after": "I(a,c) H(b) Horner multiply before degree2 = -16/3",
      "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 I(a,c) H(b) Horner multiply before degree2，不以小數近似替代。",
          "en": "Recompute I(a,c) H(b) Horner multiply before degree2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "4/3",
          "-4"
        ],
        "output": "-16/3",
        "label": "I(a,c) H(b) Horner multiply before degree2"
      }
    },
    {
      "id": "S059",
      "uses": [
        "G001",
        "S058"
      ],
      "before": "I(a,c) H(b) Horner add degree2 := sum(-16/3, 3)",
      "after": "I(a,c) H(b) Horner add degree2 = -7/3",
      "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 I(a,c) H(b) Horner add degree2，不以小數近似替代。",
          "en": "Recompute I(a,c) H(b) Horner add degree2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-16/3",
          "3"
        ],
        "output": "-7/3",
        "label": "I(a,c) H(b) Horner add degree2"
      }
    },
    {
      "id": "S060",
      "uses": [
        "G001",
        "S059"
      ],
      "before": "I(a,c) H(b) Horner multiply before degree1 := mul(-7/3, -4)",
      "after": "I(a,c) H(b) Horner multiply before degree1 = 28/3",
      "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 I(a,c) H(b) Horner multiply before degree1，不以小數近似替代。",
          "en": "Recompute I(a,c) H(b) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-7/3",
          "-4"
        ],
        "output": "28/3",
        "label": "I(a,c) H(b) Horner multiply before degree1"
      }
    },
    {
      "id": "S061",
      "uses": [
        "G001",
        "S060"
      ],
      "before": "I(a,c) H(b) Horner add degree1 := sum(28/3, -2)",
      "after": "I(a,c) H(b) Horner add degree1 = 22/3",
      "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 I(a,c) H(b) Horner add degree1，不以小數近似替代。",
          "en": "Recompute I(a,c) H(b) Horner add degree1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "28/3",
          "-2"
        ],
        "output": "22/3",
        "label": "I(a,c) H(b) Horner add degree1"
      }
    },
    {
      "id": "S062",
      "uses": [
        "G001",
        "S061"
      ],
      "before": "I(a,c) H(b) Horner multiply before degree0 := mul(22/3, -4)",
      "after": "I(a,c) H(b) Horner multiply before degree0 = -88/3",
      "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 I(a,c) H(b) Horner multiply before degree0，不以小數近似替代。",
          "en": "Recompute I(a,c) H(b) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "22/3",
          "-4"
        ],
        "output": "-88/3",
        "label": "I(a,c) H(b) Horner multiply before degree0"
      }
    },
    {
      "id": "S063",
      "uses": [
        "G001",
        "S062"
      ],
      "before": "I(a,c) H(b) Horner add degree0 := sum(-88/3, 0)",
      "after": "I(a,c) H(b) Horner add degree0 = -88/3",
      "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 I(a,c) H(b) Horner add degree0，不以小數近似替代。",
          "en": "Recompute I(a,c) H(b) Horner add degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-88/3",
          "0"
        ],
        "output": "-88/3",
        "label": "I(a,c) H(b) Horner add degree0"
      }
    },
    {
      "id": "S064",
      "uses": [
        "G001",
        "S063"
      ],
      "before": "I(a,c) H(a) ascending coefficients=[0, -2, 3, 7/3, 1/4]",
      "after": "evaluate at x=0",
      "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": "S065",
      "uses": [
        "G001",
        "S064"
      ],
      "before": "I(a,c) H(a) Horner multiply before degree3 := mul(1/4, 0)",
      "after": "I(a,c) H(a) Horner multiply before degree3 = 0",
      "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 I(a,c) H(a) Horner multiply before degree3，不以小數近似替代。",
          "en": "Recompute I(a,c) H(a) Horner multiply before degree3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/4",
          "0"
        ],
        "output": "0",
        "label": "I(a,c) H(a) Horner multiply before degree3"
      }
    },
    {
      "id": "S066",
      "uses": [
        "G001",
        "S065"
      ],
      "before": "I(a,c) H(a) Horner add degree3 := sum(0, 7/3)",
      "after": "I(a,c) H(a) Horner add degree3 = 7/3",
      "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 I(a,c) H(a) Horner add degree3，不以小數近似替代。",
          "en": "Recompute I(a,c) H(a) Horner add degree3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "7/3"
        ],
        "output": "7/3",
        "label": "I(a,c) H(a) Horner add degree3"
      }
    },
    {
      "id": "S067",
      "uses": [
        "G001",
        "S066"
      ],
      "before": "I(a,c) H(a) Horner multiply before degree2 := mul(7/3, 0)",
      "after": "I(a,c) H(a) Horner multiply before degree2 = 0",
      "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 I(a,c) H(a) Horner multiply before degree2，不以小數近似替代。",
          "en": "Recompute I(a,c) H(a) Horner multiply before degree2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "7/3",
          "0"
        ],
        "output": "0",
        "label": "I(a,c) H(a) Horner multiply before degree2"
      }
    },
    {
      "id": "S068",
      "uses": [
        "G001",
        "S067"
      ],
      "before": "I(a,c) H(a) Horner add degree2 := sum(0, 3)",
      "after": "I(a,c) H(a) Horner add degree2 = 3",
      "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 I(a,c) H(a) Horner add degree2，不以小數近似替代。",
          "en": "Recompute I(a,c) H(a) Horner add degree2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "3"
        ],
        "output": "3",
        "label": "I(a,c) H(a) Horner add degree2"
      }
    },
    {
      "id": "S069",
      "uses": [
        "G001",
        "S068"
      ],
      "before": "I(a,c) H(a) Horner multiply before degree1 := mul(3, 0)",
      "after": "I(a,c) H(a) Horner multiply before degree1 = 0",
      "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 I(a,c) H(a) Horner multiply before degree1，不以小數近似替代。",
          "en": "Recompute I(a,c) H(a) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3",
          "0"
        ],
        "output": "0",
        "label": "I(a,c) H(a) Horner multiply before degree1"
      }
    },
    {
      "id": "S070",
      "uses": [
        "G001",
        "S069"
      ],
      "before": "I(a,c) H(a) Horner add degree1 := sum(0, -2)",
      "after": "I(a,c) H(a) Horner add degree1 = -2",
      "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 I(a,c) H(a) Horner add degree1，不以小數近似替代。",
          "en": "Recompute I(a,c) H(a) Horner add degree1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "-2"
        ],
        "output": "-2",
        "label": "I(a,c) H(a) Horner add degree1"
      }
    },
    {
      "id": "S071",
      "uses": [
        "G001",
        "S070"
      ],
      "before": "I(a,c) H(a) Horner multiply before degree0 := mul(-2, 0)",
      "after": "I(a,c) H(a) Horner multiply before degree0 = 0",
      "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 I(a,c) H(a) Horner multiply before degree0，不以小數近似替代。",
          "en": "Recompute I(a,c) H(a) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-2",
          "0"
        ],
        "output": "0",
        "label": "I(a,c) H(a) Horner multiply before degree0"
      }
    },
    {
      "id": "S072",
      "uses": [
        "G001",
        "S071"
      ],
      "before": "I(a,c) H(a) Horner add degree0 := sum(0, 0)",
      "after": "I(a,c) H(a) Horner add degree0 = 0",
      "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 I(a,c) H(a) Horner add degree0，不以小數近似替代。",
          "en": "Recompute I(a,c) H(a) Horner add degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "I(a,c) H(a) Horner add degree0"
      }
    },
    {
      "id": "S073",
      "uses": [
        "G001",
        "S072"
      ],
      "before": "I(a,c) oriented moment := sub(-88/3, 0)",
      "after": "I(a,c) oriented moment = -88/3",
      "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 I(a,c) oriented moment，不以小數近似替代。",
          "en": "Recompute I(a,c) oriented moment with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-88/3",
          "0"
        ],
        "output": "-88/3",
        "label": "I(a,c) oriented moment"
      }
    },
    {
      "id": "S074",
      "uses": [
        "G001",
        "S073"
      ],
      "before": "adjacent oriented moment sum := sum(-60, 92/3)",
      "after": "adjacent oriented moment sum = -88/3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "H(b)在兩段端點差中相消，因此不必要求a≤b≤c。",
        "en": "H(b) cancels between endpoint differences,so a≤b≤c is not required."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 adjacent oriented moment sum，不以小數近似替代。",
          "en": "Recompute adjacent oriented moment sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-60",
          "92/3"
        ],
        "output": "-88/3",
        "label": "adjacent oriented moment sum"
      }
    },
    {
      "id": "S075",
      "uses": [
        "G001",
        "S074"
      ],
      "before": "[H(b)−H(a)]+[H(c)−H(b)]",
      "after": "H(c)−H(a)=-88/3",
      "relation": "implies",
      "rule_id": "oriented_integral_additivity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "此代數消去對任意實端點次序成立；未把負積分改成正面積。",
        "en": "This algebraic cancellation works for any endpoint order;negative integrals are not replaced by positive area."
      },
      "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": [
      "-60",
      "92/3",
      "-88/3"
    ]
  },
  "conclusion": {
    "canonical_value": "(-60/1, 92/3, -88/3)",
    "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",
      "original_line": 6,
      "original_line_sha256": "8b331c0196dbea7fc5dca75033514f849884ae85cb2d467ba06f20e9868a6f41",
      "question_compact_sha256": "4cd0f20254a74b20254eef89c790e2ea95eb277094c07d2b1fd9ade19211c057",
      "english_row_compact_sha256": "9a1a946b497b8a15fe608749327524fd68374968dbe1f1ec5fd58f96b7ead5b3",
      "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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        "english_sha256": "cd470242748859a6d2bf765f69c47fa6a116eca1f2e64fb1a5edea6fe686d6a4",
        "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": "b095a4fff86516153a6134e39fc6b0c340caa45ea5afbf71a8d22ddd09455b7c",
    "original_candidate_line_sha256": "b095a4fff86516153a6134e39fc6b0c340caa45ea5afbf71a8d22ddd09455b7c"
  },
  "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": {
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}