{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261015-pulse-20261119-r02-0092",
  "pilot_number": "R0055-28167",
  "number": 28167,
  "family_id": "qf-mod50-trap-error-boundary",
  "title": {
    "zh-Hant": "第 28167 題：複合梯形法",
    "en": "Question 28167: Quadratic Reference Error Is Not a General Quadrature Error Theorem"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261015-pulse-20261119-r02-0092",
    "batch_id": "batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-trapezoidal-quadrature"
    ],
    "title_zh": "複合梯形法",
    "stem_zh": "嚴格遞增partition、n+1 nodes與逐subinterval寬度。操作=trap-error-boundary；明示資料={\"mode\":\"quadratic_reference\",\"model\":{\"coefficients\":[17,1,1],\"nodes\":[0,1,19]}}；請依 exact_ordered_tuple_declared_by_interface 回答。",
    "answer": {
      "kind": "expression",
      "value": "(quadratic_reference, [17, 1, 1], [0, 1, 19], 3762, 16739/6, -5833/6, 5833/6, 否, exact_polynomial_reference_error_is_not_a_general_second_derivative_error_bound)",
      "explanation_zh": "Wave6第十三層核對=(quadratic_reference, [17, 1, 1], [0, 1, 19], 3762, 16739/6, -5833/6, 5833/6, 否, exact_polynomial_reference_error_is_not_a_general_second_derivative_error_bound)；依完整明示資料精確重算；複合梯形逐段使用endpoint average與自己的width；signed error固定為I-T；affine exact或指定quadratic reference不推出一般誤差界、收斂階或唯一integrand。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "複合梯形法"
      ]
    },
    "variant": {
      "family_id": "qf-mod50-trap-error-boundary",
      "seed": 202611192092,
      "parameters": {
        "mode": "quadratic_reference",
        "model": {
          "coefficients": [
            17,
            1,
            1
          ],
          "nodes": [
            0,
            1,
            19
          ]
        },
        "domain": "Exact_composite_trapezoidal_quadrature",
        "operation": "trap-error-boundary",
        "semantic_template": "trap-error-boundary|{\"mode\":\"quadratic_reference\",\"model\":{\"coefficients\":[[17,1],[1,1],[1,1]],\"nodes\":[[0,1],[1,1],[19,1]]}}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_contract_recomputation",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "GPT-5.6 Terra",
      "generated_at": "2026-10-15",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261015-pulse-20261119-r02-0092",
    "batch_id": "batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "c43b15304d369781a5878698ac7e7d57c58b98a49ae7c7026bcbed75e0d03876",
      "family_id": "qf-mod50-trap-error-boundary",
      "seed": 202611192092,
      "semantic_template": "trap-error-boundary|{\"mode\":\"quadratic_reference\",\"model\":{\"coefficients\":[[17,1],[1,1],[1,1]],\"nodes\":[[0,1],[1,1],[19,1]]}}"
    },
    "fields": {
      "title": "Quadratic Reference Error Is Not a General Quadrature Error Theorem",
      "stem": "Use mode=quadratic_reference, complete model={\"coefficients\":[17,1,1],\"nodes\":[0,1,19]}. Give (mode, coefficients_low_to_high, nodes, trapezoid_value, exact_integral, signed_error_I_minus_T, absolute_error, general_error_bound_claim, scope).",
      "answer": {
        "kind": "expression",
        "value": "(quadratic_reference, [17, 1, 1], [0, 1, 19], 3762, 16739/6, -5833/6, 5833/6, No, exact_polynomial_reference_error_is_not_a_general_second_derivative_error_bound)",
        "explanation": "The full quadratic polynomial permits an exact reference integral and signed error I−T, with absolute error separate. A nonincreasing node list is invalid and not repaired by sorting. Endpoint samples alone do not specify the integrand or exact integral. A discriminating nonuniform example refutes using the first interval width globally. The exact result, retaining the requested order, is (quadratic_reference, [17, 1, 1], [0, 1, 19], 3762, 16739/6, -5833/6, 5833/6, No, exact_polynomial_reference_error_is_not_a_general_second_derivative_error_bound)."
      },
      "skills": [
        "Quadratic Reference Error Is Not a General Quadrature Error Theorem"
      ],
      "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": "嚴格遞增partition、n+1 nodes與逐subinterval寬度。操作=trap-error-boundary；明示資料={\"mode\":\"quadratic_reference\",\"model\":{\"coefficients\":[17,1,1],\"nodes\":[0,1,19]}}；請依 exact_ordered_tuple_declared_by_interface 回答。",
        "en": "Use mode=quadratic_reference, complete model={\"coefficients\":[17,1,1],\"nodes\":[0,1,19]}. Give (mode, coefficients_low_to_high, nodes, trapezoid_value, exact_integral, signed_error_I_minus_T, absolute_error, general_error_bound_claim, scope)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [qf-mod50-trap-error-boundary, trap-error-boundary]",
      "after": "derived = [[mode, coefficients_low_to_high, nodes, trapezoid_value, exact_integral, signed_error_I_minus_T, absolute_error, general_error_bound_claim, scope]]",
      "relation": "implies",
      "rule_id": "source_answer_roles_or_ordered_fields_without_JS",
      "conditions": [],
      "explanation": {
        "zh-Hant": "此頁明列本題scalar／label／set／point或mode欄序，不強套現代tuple。舊題domain與operation底線格式按原資料，分數整數亦保/1；原中英快照不改。",
        "en": "List this scalar/label/set/point answer role or mode-specific ordered fields,not an imposed universal tuple.Retain legacy domain/underscore operation and integer fractions with/1;original bilingual snapshots stay unchanged."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查完整來源資料、原操作數及本步條件；有限核對、條件式理由、形式證明與發布資格分開。",
          "en": "Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "Trapezoid quadratic reference polynomial ORIGINAL derivative coefficient1 := mul(1, 1)",
      "after": "Trapezoid quadratic reference polynomial ORIGINAL derivative coefficient1 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Trapezoid quadratic reference polynomial ORIGINAL derivative coefficient1，不以小數近似替代。",
          "en": "Recompute Trapezoid quadratic reference polynomial ORIGINAL derivative coefficient1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "Trapezoid quadratic reference polynomial ORIGINAL derivative coefficient1"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "Trapezoid quadratic reference polynomial ORIGINAL derivative coefficient2 := mul(2, 1)",
      "after": "Trapezoid quadratic reference polynomial ORIGINAL derivative coefficient2 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Trapezoid quadratic reference polynomial ORIGINAL derivative coefficient2，不以小數近似替代。",
          "en": "Recompute Trapezoid quadratic reference polynomial ORIGINAL derivative coefficient2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "1"
        ],
        "output": "2",
        "label": "Trapezoid quadratic reference polynomial ORIGINAL derivative coefficient2"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "source roles = [Trapezoid quadratic reference polynomial, [17, 1, 1]]",
      "after": "derived = [[17, 1, 1], [1, 2]]",
      "relation": "implies",
      "rule_id": "Newton_polynomial_raw_arity_trim_nonconstant_AND_ALL_derivative_coefficients",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原vector先要求2..6，再只刪trailing zeros且仍須非constant。逐x^i微分得i*c_i*x^(i-1)，來源derivative係數亦先過1e8 profile，即使稍後root停止或steps0也不跳過。",
        "en": "Require raw length2..6 before trimming trailing zeros,and remain nonconstant.Differentiate every monomial to i*c_i*x^(i-1).Source derivative coefficients must pass their1e8 profile even if a root later stops or steps=0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查完整來源資料、原操作數及本步條件；有限核對、條件式理由、形式證明與發布資格分開。",
          "en": "Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "Trapezoid quadratic ORIGINAL node0 ORIGINAL Horner multiply degree2 := mul(0, 0)",
      "after": "Trapezoid quadratic ORIGINAL node0 ORIGINAL Horner multiply degree2 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Trapezoid quadratic ORIGINAL node0 ORIGINAL Horner multiply degree2，不以小數近似替代。",
          "en": "Recompute Trapezoid quadratic ORIGINAL node0 ORIGINAL Horner multiply degree2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "Trapezoid quadratic ORIGINAL node0 ORIGINAL Horner multiply degree2"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "Trapezoid quadratic ORIGINAL node0 ORIGINAL Horner add coefficient2 := sum(0, 1)",
      "after": "Trapezoid quadratic ORIGINAL node0 ORIGINAL Horner add coefficient2 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Trapezoid quadratic ORIGINAL node0 ORIGINAL Horner add coefficient2，不以小數近似替代。",
          "en": "Recompute Trapezoid quadratic ORIGINAL node0 ORIGINAL Horner add coefficient2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "1"
        ],
        "output": "1",
        "label": "Trapezoid quadratic ORIGINAL node0 ORIGINAL Horner add coefficient2"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "Trapezoid quadratic ORIGINAL node0 ORIGINAL Horner multiply degree1 := mul(1, 0)",
      "after": "Trapezoid quadratic ORIGINAL node0 ORIGINAL Horner multiply degree1 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Trapezoid quadratic ORIGINAL node0 ORIGINAL Horner multiply degree1，不以小數近似替代。",
          "en": "Recompute Trapezoid quadratic ORIGINAL node0 ORIGINAL Horner multiply degree1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "0"
        ],
        "output": "0",
        "label": "Trapezoid quadratic ORIGINAL node0 ORIGINAL Horner multiply degree1"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "Trapezoid quadratic ORIGINAL node0 ORIGINAL Horner add coefficient1 := sum(0, 1)",
      "after": "Trapezoid quadratic ORIGINAL node0 ORIGINAL Horner add coefficient1 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Trapezoid quadratic ORIGINAL node0 ORIGINAL Horner add coefficient1，不以小數近似替代。",
          "en": "Recompute Trapezoid quadratic ORIGINAL node0 ORIGINAL Horner add coefficient1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "1"
        ],
        "output": "1",
        "label": "Trapezoid quadratic ORIGINAL node0 ORIGINAL Horner add coefficient1"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "Trapezoid quadratic ORIGINAL node0 ORIGINAL Horner multiply degree0 := mul(1, 0)",
      "after": "Trapezoid quadratic ORIGINAL node0 ORIGINAL Horner multiply degree0 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Trapezoid quadratic ORIGINAL node0 ORIGINAL Horner multiply degree0，不以小數近似替代。",
          "en": "Recompute Trapezoid quadratic ORIGINAL node0 ORIGINAL Horner multiply degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "0"
        ],
        "output": "0",
        "label": "Trapezoid quadratic ORIGINAL node0 ORIGINAL Horner multiply degree0"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "Trapezoid quadratic ORIGINAL node0 ORIGINAL Horner add coefficient0 := sum(0, 17)",
      "after": "Trapezoid quadratic ORIGINAL node0 ORIGINAL Horner add coefficient0 = 17",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Trapezoid quadratic ORIGINAL node0 ORIGINAL Horner add coefficient0，不以小數近似替代。",
          "en": "Recompute Trapezoid quadratic ORIGINAL node0 ORIGINAL Horner add coefficient0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "17"
        ],
        "output": "17",
        "label": "Trapezoid quadratic ORIGINAL node0 ORIGINAL Horner add coefficient0"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "Trapezoid quadratic ORIGINAL node1 ORIGINAL Horner multiply degree2 := mul(0, 1)",
      "after": "Trapezoid quadratic ORIGINAL node1 ORIGINAL Horner multiply degree2 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Trapezoid quadratic ORIGINAL node1 ORIGINAL Horner multiply degree2，不以小數近似替代。",
          "en": "Recompute Trapezoid quadratic ORIGINAL node1 ORIGINAL Horner multiply degree2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "1"
        ],
        "output": "0",
        "label": "Trapezoid quadratic ORIGINAL node1 ORIGINAL Horner multiply degree2"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "Trapezoid quadratic ORIGINAL node1 ORIGINAL Horner add coefficient2 := sum(0, 1)",
      "after": "Trapezoid quadratic ORIGINAL node1 ORIGINAL Horner add coefficient2 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Trapezoid quadratic ORIGINAL node1 ORIGINAL Horner add coefficient2，不以小數近似替代。",
          "en": "Recompute Trapezoid quadratic ORIGINAL node1 ORIGINAL Horner add coefficient2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "1"
        ],
        "output": "1",
        "label": "Trapezoid quadratic ORIGINAL node1 ORIGINAL Horner add coefficient2"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "Trapezoid quadratic ORIGINAL node1 ORIGINAL Horner multiply degree1 := mul(1, 1)",
      "after": "Trapezoid quadratic ORIGINAL node1 ORIGINAL Horner multiply degree1 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Trapezoid quadratic ORIGINAL node1 ORIGINAL Horner multiply degree1，不以小數近似替代。",
          "en": "Recompute Trapezoid quadratic ORIGINAL node1 ORIGINAL Horner multiply degree1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "Trapezoid quadratic ORIGINAL node1 ORIGINAL Horner multiply degree1"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "Trapezoid quadratic ORIGINAL node1 ORIGINAL Horner add coefficient1 := sum(1, 1)",
      "after": "Trapezoid quadratic ORIGINAL node1 ORIGINAL Horner add coefficient1 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Trapezoid quadratic ORIGINAL node1 ORIGINAL Horner add coefficient1，不以小數近似替代。",
          "en": "Recompute Trapezoid quadratic ORIGINAL node1 ORIGINAL Horner add coefficient1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "1"
        ],
        "output": "2",
        "label": "Trapezoid quadratic ORIGINAL node1 ORIGINAL Horner add coefficient1"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "Trapezoid quadratic ORIGINAL node1 ORIGINAL Horner multiply degree0 := mul(2, 1)",
      "after": "Trapezoid quadratic ORIGINAL node1 ORIGINAL Horner multiply degree0 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Trapezoid quadratic ORIGINAL node1 ORIGINAL Horner multiply degree0，不以小數近似替代。",
          "en": "Recompute Trapezoid quadratic ORIGINAL node1 ORIGINAL Horner multiply degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "1"
        ],
        "output": "2",
        "label": "Trapezoid quadratic ORIGINAL node1 ORIGINAL Horner multiply degree0"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "Trapezoid quadratic ORIGINAL node1 ORIGINAL Horner add coefficient0 := sum(2, 17)",
      "after": "Trapezoid quadratic ORIGINAL node1 ORIGINAL Horner add coefficient0 = 19",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Trapezoid quadratic ORIGINAL node1 ORIGINAL Horner add coefficient0，不以小數近似替代。",
          "en": "Recompute Trapezoid quadratic ORIGINAL node1 ORIGINAL Horner add coefficient0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "2",
          "17"
        ],
        "output": "19",
        "label": "Trapezoid quadratic ORIGINAL node1 ORIGINAL Horner add coefficient0"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner multiply degree2 := mul(0, 19)",
      "after": "Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner multiply degree2 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner multiply degree2，不以小數近似替代。",
          "en": "Recompute Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner multiply degree2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "19"
        ],
        "output": "0",
        "label": "Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner multiply degree2"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner add coefficient2 := sum(0, 1)",
      "after": "Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner add coefficient2 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner add coefficient2，不以小數近似替代。",
          "en": "Recompute Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner add coefficient2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "1"
        ],
        "output": "1",
        "label": "Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner add coefficient2"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner multiply degree1 := mul(1, 19)",
      "after": "Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner multiply degree1 = 19",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner multiply degree1，不以小數近似替代。",
          "en": "Recompute Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner multiply degree1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "19"
        ],
        "output": "19",
        "label": "Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner multiply degree1"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner add coefficient1 := sum(19, 1)",
      "after": "Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner add coefficient1 = 20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner add coefficient1，不以小數近似替代。",
          "en": "Recompute Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner add coefficient1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "19",
          "1"
        ],
        "output": "20",
        "label": "Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner add coefficient1"
      }
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner multiply degree0 := mul(20, 19)",
      "after": "Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner multiply degree0 = 380",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner multiply degree0，不以小數近似替代。",
          "en": "Recompute Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner multiply degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "19"
        ],
        "output": "380",
        "label": "Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner multiply degree0"
      }
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner add coefficient0 := sum(380, 17)",
      "after": "Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner add coefficient0 = 397",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner add coefficient0，不以小數近似替代。",
          "en": "Recompute Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner add coefficient0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "380",
          "17"
        ],
        "output": "397",
        "label": "Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner add coefficient0"
      }
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "Trapezoid quadratic partition ORIGINAL width0 := sub(1, 0)",
      "after": "Trapezoid quadratic partition ORIGINAL width0 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Trapezoid quadratic partition ORIGINAL width0，不以小數近似替代。",
          "en": "Recompute Trapezoid quadratic partition ORIGINAL width0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "1",
          "0"
        ],
        "output": "1",
        "label": "Trapezoid quadratic partition ORIGINAL width0"
      }
    },
    {
      "id": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "Trapezoid quadratic partition ORIGINAL width1 := sub(19, 1)",
      "after": "Trapezoid quadratic partition ORIGINAL width1 = 18",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Trapezoid quadratic partition ORIGINAL width1，不以小數近似替代。",
          "en": "Recompute Trapezoid quadratic partition ORIGINAL width1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "19",
          "1"
        ],
        "output": "18",
        "label": "Trapezoid quadratic partition ORIGINAL width1"
      }
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "Trapezoid quadratic partition ORIGINAL width over2 interval0 := div(1, 2)",
      "after": "Trapezoid quadratic partition ORIGINAL width over2 interval0 = 1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Trapezoid quadratic partition ORIGINAL width over2 interval0，不以小數近似替代。",
          "en": "Recompute Trapezoid quadratic partition ORIGINAL width over2 interval0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "1",
          "2"
        ],
        "output": "1/2",
        "label": "Trapezoid quadratic partition ORIGINAL width over2 interval0"
      }
    },
    {
      "id": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "Trapezoid quadratic partition ORIGINAL endpoint sum interval0 := sum(17, 19)",
      "after": "Trapezoid quadratic partition ORIGINAL endpoint sum interval0 = 36",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Trapezoid quadratic partition ORIGINAL endpoint sum interval0，不以小數近似替代。",
          "en": "Recompute Trapezoid quadratic partition ORIGINAL endpoint sum interval0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "17",
          "19"
        ],
        "output": "36",
        "label": "Trapezoid quadratic partition ORIGINAL endpoint sum interval0"
      }
    },
    {
      "id": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "Trapezoid quadratic partition ORIGINAL individual trapezoid interval0 := mul(1/2, 36)",
      "after": "Trapezoid quadratic partition ORIGINAL individual trapezoid interval0 = 18",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Trapezoid quadratic partition ORIGINAL individual trapezoid interval0，不以小數近似替代。",
          "en": "Recompute Trapezoid quadratic partition ORIGINAL individual trapezoid interval0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/2",
          "36"
        ],
        "output": "18",
        "label": "Trapezoid quadratic partition ORIGINAL individual trapezoid interval0"
      }
    },
    {
      "id": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "source roles = [Trapezoid quadratic partition, 0, 0, 1, 17, 19]",
      "after": "derived = [1, 1/2, 36, 18]",
      "relation": "implies",
      "rule_id": "Trapezoid_EACH_original_interval_own_width_and_aligned_endpoints",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用這一段的原始兩端與自己的width，乘endpoint average；width可不同、values可負，不排序或把所有段換第一段width。",
        "en": "Use the own width and aligned endpoints of this ORIGINAL interval times the endpoint average.Widths may differ and values may be negative;do not sort or replace all widths with the first."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查完整來源資料、原操作數及本步條件；有限核對、條件式理由、形式證明與發布資格分開。",
          "en": "Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "Trapezoid quadratic partition ORIGINAL width over2 interval1 := div(18, 2)",
      "after": "Trapezoid quadratic partition ORIGINAL width over2 interval1 = 9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Trapezoid quadratic partition ORIGINAL width over2 interval1，不以小數近似替代。",
          "en": "Recompute Trapezoid quadratic partition ORIGINAL width over2 interval1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "18",
          "2"
        ],
        "output": "9",
        "label": "Trapezoid quadratic partition ORIGINAL width over2 interval1"
      }
    },
    {
      "id": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "Trapezoid quadratic partition ORIGINAL endpoint sum interval1 := sum(19, 397)",
      "after": "Trapezoid quadratic partition ORIGINAL endpoint sum interval1 = 416",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Trapezoid quadratic partition ORIGINAL endpoint sum interval1，不以小數近似替代。",
          "en": "Recompute Trapezoid quadratic partition ORIGINAL endpoint sum interval1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "19",
          "397"
        ],
        "output": "416",
        "label": "Trapezoid quadratic partition ORIGINAL endpoint sum interval1"
      }
    },
    {
      "id": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "Trapezoid quadratic partition ORIGINAL individual trapezoid interval1 := mul(9, 416)",
      "after": "Trapezoid quadratic partition ORIGINAL individual trapezoid interval1 = 3744",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Trapezoid quadratic partition ORIGINAL individual trapezoid interval1，不以小數近似替代。",
          "en": "Recompute Trapezoid quadratic partition ORIGINAL individual trapezoid interval1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "9",
          "416"
        ],
        "output": "3744",
        "label": "Trapezoid quadratic partition ORIGINAL individual trapezoid interval1"
      }
    },
    {
      "id": "S032",
      "uses": [
        "G001",
        "S031"
      ],
      "before": "source roles = [Trapezoid quadratic partition, 1, 1, 19, 19, 397]",
      "after": "derived = [18, 9, 416, 3744]",
      "relation": "implies",
      "rule_id": "Trapezoid_EACH_original_interval_own_width_and_aligned_endpoints",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用這一段的原始兩端與自己的width，乘endpoint average；width可不同、values可負，不排序或把所有段換第一段width。",
        "en": "Use the own width and aligned endpoints of this ORIGINAL interval times the endpoint average.Widths may differ and values may be negative;do not sort or replace all widths with the first."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查完整來源資料、原操作數及本步條件；有限核對、條件式理由、形式證明與發布資格分開。",
          "en": "Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S033",
      "uses": [
        "G001",
        "S032"
      ],
      "before": "Trapezoid quadratic partition ORIGINAL trapezoid running total0 := sum(0, 18)",
      "after": "Trapezoid quadratic partition ORIGINAL trapezoid running total0 = 18",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Trapezoid quadratic partition ORIGINAL trapezoid running total0，不以小數近似替代。",
          "en": "Recompute Trapezoid quadratic partition ORIGINAL trapezoid running total0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "18"
        ],
        "output": "18",
        "label": "Trapezoid quadratic partition ORIGINAL trapezoid running total0"
      }
    },
    {
      "id": "S034",
      "uses": [
        "G001",
        "S033"
      ],
      "before": "Trapezoid quadratic partition ORIGINAL trapezoid running total1 := sum(18, 3744)",
      "after": "Trapezoid quadratic partition ORIGINAL trapezoid running total1 = 3762",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Trapezoid quadratic partition ORIGINAL trapezoid running total1，不以小數近似替代。",
          "en": "Recompute Trapezoid quadratic partition ORIGINAL trapezoid running total1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "18",
          "3744"
        ],
        "output": "3762",
        "label": "Trapezoid quadratic partition ORIGINAL trapezoid running total1"
      }
    },
    {
      "id": "S035",
      "uses": [
        "G001",
        "S034"
      ],
      "before": "source roles = [Trapezoid quadratic partition, [[0, 1], [1, 1], [19, 1]], [[17, 1], [19, 1], [397, 1]]]",
      "after": "derived = [[0, 1, 19], [17, 19, 397], [1, 18], [18, 3744], 3762]",
      "relation": "implies",
      "rule_id": "Trapezoid_COMPLETE_partition_n_intervals_nplus1_nodes_no_unique_integrand",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整partition須原次序strictly increasing，n段用n+1個aligned nodes/values（上界24不是vector8）。表可決定這個sum，未給full integrand時不決定唯一積分。",
        "en": "A complete partition is STRICTLY increasing in original order,with n+1 aligned nodes/values for n intervals;the24-node cap is not the vector8 cap.The table determines this sum,not a unique integral without a full integrand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查完整來源資料、原操作數及本步條件；有限核對、條件式理由、形式證明與發布資格分開。",
          "en": "Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S036",
      "uses": [
        "G001",
        "S035"
      ],
      "before": "Trapezoid quadratic reference ORIGINAL coefficient over1 := div(17, 1)",
      "after": "Trapezoid quadratic reference ORIGINAL coefficient over1 = 17",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Trapezoid quadratic reference ORIGINAL coefficient over1，不以小數近似替代。",
          "en": "Recompute Trapezoid quadratic reference ORIGINAL coefficient over1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "17",
          "1"
        ],
        "output": "17",
        "label": "Trapezoid quadratic reference ORIGINAL coefficient over1"
      }
    },
    {
      "id": "S037",
      "uses": [
        "G001",
        "S036"
      ],
      "before": "Trapezoid quadratic reference right term0 ORIGINAL power multiply1 := mul(1, 19)",
      "after": "Trapezoid quadratic reference right term0 ORIGINAL power multiply1 = 19",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Trapezoid quadratic reference right term0 ORIGINAL power multiply1，不以小數近似替代。",
          "en": "Recompute Trapezoid quadratic reference right term0 ORIGINAL power multiply1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "19"
        ],
        "output": "19",
        "label": "Trapezoid quadratic reference right term0 ORIGINAL power multiply1"
      }
    },
    {
      "id": "S038",
      "uses": [
        "G001",
        "S037"
      ],
      "before": "Trapezoid quadratic reference left term0 ORIGINAL power multiply1 := mul(1, 0)",
      "after": "Trapezoid quadratic reference left term0 ORIGINAL power multiply1 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Trapezoid quadratic reference left term0 ORIGINAL power multiply1，不以小數近似替代。",
          "en": "Recompute Trapezoid quadratic reference left term0 ORIGINAL power multiply1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "0"
        ],
        "output": "0",
        "label": "Trapezoid quadratic reference left term0 ORIGINAL power multiply1"
      }
    },
    {
      "id": "S039",
      "uses": [
        "G001",
        "S038"
      ],
      "before": "Trapezoid quadratic reference ORIGINAL power difference0 := sub(19, 0)",
      "after": "Trapezoid quadratic reference ORIGINAL power difference0 = 19",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Trapezoid quadratic reference ORIGINAL power difference0，不以小數近似替代。",
          "en": "Recompute Trapezoid quadratic reference ORIGINAL power difference0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "19",
          "0"
        ],
        "output": "19",
        "label": "Trapezoid quadratic reference ORIGINAL power difference0"
      }
    },
    {
      "id": "S040",
      "uses": [
        "G001",
        "S039"
      ],
      "before": "Trapezoid quadratic reference ORIGINAL integrated term0 := mul(17, 19)",
      "after": "Trapezoid quadratic reference ORIGINAL integrated term0 = 323",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Trapezoid quadratic reference ORIGINAL integrated term0，不以小數近似替代。",
          "en": "Recompute Trapezoid quadratic reference ORIGINAL integrated term0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "17",
          "19"
        ],
        "output": "323",
        "label": "Trapezoid quadratic reference ORIGINAL integrated term0"
      }
    },
    {
      "id": "S041",
      "uses": [
        "G001",
        "S040"
      ],
      "before": "Trapezoid quadratic reference ORIGINAL integral running sum0 := sum(0, 323)",
      "after": "Trapezoid quadratic reference ORIGINAL integral running sum0 = 323",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Trapezoid quadratic reference ORIGINAL integral running sum0，不以小數近似替代。",
          "en": "Recompute Trapezoid quadratic reference ORIGINAL integral running sum0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "323"
        ],
        "output": "323",
        "label": "Trapezoid quadratic reference ORIGINAL integral running sum0"
      }
    },
    {
      "id": "S042",
      "uses": [
        "G001",
        "S041"
      ],
      "before": "Trapezoid quadratic reference ORIGINAL coefficient over2 := div(1, 2)",
      "after": "Trapezoid quadratic reference ORIGINAL coefficient over2 = 1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Trapezoid quadratic reference ORIGINAL coefficient over2，不以小數近似替代。",
          "en": "Recompute Trapezoid quadratic reference ORIGINAL coefficient over2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "1",
          "2"
        ],
        "output": "1/2",
        "label": "Trapezoid quadratic reference ORIGINAL coefficient over2"
      }
    },
    {
      "id": "S043",
      "uses": [
        "G001",
        "S042"
      ],
      "before": "Trapezoid quadratic reference right term1 ORIGINAL power multiply1 := mul(1, 19)",
      "after": "Trapezoid quadratic reference right term1 ORIGINAL power multiply1 = 19",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Trapezoid quadratic reference right term1 ORIGINAL power multiply1，不以小數近似替代。",
          "en": "Recompute Trapezoid quadratic reference right term1 ORIGINAL power multiply1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "19"
        ],
        "output": "19",
        "label": "Trapezoid quadratic reference right term1 ORIGINAL power multiply1"
      }
    },
    {
      "id": "S044",
      "uses": [
        "G001",
        "S043"
      ],
      "before": "Trapezoid quadratic reference right term1 ORIGINAL power multiply2 := mul(19, 19)",
      "after": "Trapezoid quadratic reference right term1 ORIGINAL power multiply2 = 361",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Trapezoid quadratic reference right term1 ORIGINAL power multiply2，不以小數近似替代。",
          "en": "Recompute Trapezoid quadratic reference right term1 ORIGINAL power multiply2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "19",
          "19"
        ],
        "output": "361",
        "label": "Trapezoid quadratic reference right term1 ORIGINAL power multiply2"
      }
    },
    {
      "id": "S045",
      "uses": [
        "G001",
        "S044"
      ],
      "before": "Trapezoid quadratic reference left term1 ORIGINAL power multiply1 := mul(1, 0)",
      "after": "Trapezoid quadratic reference left term1 ORIGINAL power multiply1 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Trapezoid quadratic reference left term1 ORIGINAL power multiply1，不以小數近似替代。",
          "en": "Recompute Trapezoid quadratic reference left term1 ORIGINAL power multiply1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "0"
        ],
        "output": "0",
        "label": "Trapezoid quadratic reference left term1 ORIGINAL power multiply1"
      }
    },
    {
      "id": "S046",
      "uses": [
        "G001",
        "S045"
      ],
      "before": "Trapezoid quadratic reference left term1 ORIGINAL power multiply2 := mul(0, 0)",
      "after": "Trapezoid quadratic reference left term1 ORIGINAL power multiply2 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Trapezoid quadratic reference left term1 ORIGINAL power multiply2，不以小數近似替代。",
          "en": "Recompute Trapezoid quadratic reference left term1 ORIGINAL power multiply2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "Trapezoid quadratic reference left term1 ORIGINAL power multiply2"
      }
    },
    {
      "id": "S047",
      "uses": [
        "G001",
        "S046"
      ],
      "before": "Trapezoid quadratic reference ORIGINAL power difference1 := sub(361, 0)",
      "after": "Trapezoid quadratic reference ORIGINAL power difference1 = 361",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Trapezoid quadratic reference ORIGINAL power difference1，不以小數近似替代。",
          "en": "Recompute Trapezoid quadratic reference ORIGINAL power difference1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "361",
          "0"
        ],
        "output": "361",
        "label": "Trapezoid quadratic reference ORIGINAL power difference1"
      }
    },
    {
      "id": "S048",
      "uses": [
        "G001",
        "S047"
      ],
      "before": "Trapezoid quadratic reference ORIGINAL integrated term1 := mul(1/2, 361)",
      "after": "Trapezoid quadratic reference ORIGINAL integrated term1 = 361/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Trapezoid quadratic reference ORIGINAL integrated term1，不以小數近似替代。",
          "en": "Recompute Trapezoid quadratic reference ORIGINAL integrated term1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/2",
          "361"
        ],
        "output": "361/2",
        "label": "Trapezoid quadratic reference ORIGINAL integrated term1"
      }
    },
    {
      "id": "S049",
      "uses": [
        "G001",
        "S048"
      ],
      "before": "Trapezoid quadratic reference ORIGINAL integral running sum1 := sum(323, 361/2)",
      "after": "Trapezoid quadratic reference ORIGINAL integral running sum1 = 1007/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Trapezoid quadratic reference ORIGINAL integral running sum1，不以小數近似替代。",
          "en": "Recompute Trapezoid quadratic reference ORIGINAL integral running sum1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "323",
          "361/2"
        ],
        "output": "1007/2",
        "label": "Trapezoid quadratic reference ORIGINAL integral running sum1"
      }
    },
    {
      "id": "S050",
      "uses": [
        "G001",
        "S049"
      ],
      "before": "Trapezoid quadratic reference ORIGINAL coefficient over3 := div(1, 3)",
      "after": "Trapezoid quadratic reference ORIGINAL coefficient over3 = 1/3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Trapezoid quadratic reference ORIGINAL coefficient over3，不以小數近似替代。",
          "en": "Recompute Trapezoid quadratic reference ORIGINAL coefficient over3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "1",
          "3"
        ],
        "output": "1/3",
        "label": "Trapezoid quadratic reference ORIGINAL coefficient over3"
      }
    },
    {
      "id": "S051",
      "uses": [
        "G001",
        "S050"
      ],
      "before": "Trapezoid quadratic reference right term2 ORIGINAL power multiply1 := mul(1, 19)",
      "after": "Trapezoid quadratic reference right term2 ORIGINAL power multiply1 = 19",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Trapezoid quadratic reference right term2 ORIGINAL power multiply1，不以小數近似替代。",
          "en": "Recompute Trapezoid quadratic reference right term2 ORIGINAL power multiply1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "19"
        ],
        "output": "19",
        "label": "Trapezoid quadratic reference right term2 ORIGINAL power multiply1"
      }
    },
    {
      "id": "S052",
      "uses": [
        "G001",
        "S051"
      ],
      "before": "Trapezoid quadratic reference right term2 ORIGINAL power multiply2 := mul(19, 19)",
      "after": "Trapezoid quadratic reference right term2 ORIGINAL power multiply2 = 361",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Trapezoid quadratic reference right term2 ORIGINAL power multiply2，不以小數近似替代。",
          "en": "Recompute Trapezoid quadratic reference right term2 ORIGINAL power multiply2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "19",
          "19"
        ],
        "output": "361",
        "label": "Trapezoid quadratic reference right term2 ORIGINAL power multiply2"
      }
    },
    {
      "id": "S053",
      "uses": [
        "G001",
        "S052"
      ],
      "before": "Trapezoid quadratic reference right term2 ORIGINAL power multiply3 := mul(361, 19)",
      "after": "Trapezoid quadratic reference right term2 ORIGINAL power multiply3 = 6859",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Trapezoid quadratic reference right term2 ORIGINAL power multiply3，不以小數近似替代。",
          "en": "Recompute Trapezoid quadratic reference right term2 ORIGINAL power multiply3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "361",
          "19"
        ],
        "output": "6859",
        "label": "Trapezoid quadratic reference right term2 ORIGINAL power multiply3"
      }
    },
    {
      "id": "S054",
      "uses": [
        "G001",
        "S053"
      ],
      "before": "Trapezoid quadratic reference left term2 ORIGINAL power multiply1 := mul(1, 0)",
      "after": "Trapezoid quadratic reference left term2 ORIGINAL power multiply1 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Trapezoid quadratic reference left term2 ORIGINAL power multiply1，不以小數近似替代。",
          "en": "Recompute Trapezoid quadratic reference left term2 ORIGINAL power multiply1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "0"
        ],
        "output": "0",
        "label": "Trapezoid quadratic reference left term2 ORIGINAL power multiply1"
      }
    },
    {
      "id": "S055",
      "uses": [
        "G001",
        "S054"
      ],
      "before": "Trapezoid quadratic reference left term2 ORIGINAL power multiply2 := mul(0, 0)",
      "after": "Trapezoid quadratic reference left term2 ORIGINAL power multiply2 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Trapezoid quadratic reference left term2 ORIGINAL power multiply2，不以小數近似替代。",
          "en": "Recompute Trapezoid quadratic reference left term2 ORIGINAL power multiply2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "Trapezoid quadratic reference left term2 ORIGINAL power multiply2"
      }
    },
    {
      "id": "S056",
      "uses": [
        "G001",
        "S055"
      ],
      "before": "Trapezoid quadratic reference left term2 ORIGINAL power multiply3 := mul(0, 0)",
      "after": "Trapezoid quadratic reference left term2 ORIGINAL power multiply3 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Trapezoid quadratic reference left term2 ORIGINAL power multiply3，不以小數近似替代。",
          "en": "Recompute Trapezoid quadratic reference left term2 ORIGINAL power multiply3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "Trapezoid quadratic reference left term2 ORIGINAL power multiply3"
      }
    },
    {
      "id": "S057",
      "uses": [
        "G001",
        "S056"
      ],
      "before": "Trapezoid quadratic reference ORIGINAL power difference2 := sub(6859, 0)",
      "after": "Trapezoid quadratic reference ORIGINAL power difference2 = 6859",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Trapezoid quadratic reference ORIGINAL power difference2，不以小數近似替代。",
          "en": "Recompute Trapezoid quadratic reference ORIGINAL power difference2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "6859",
          "0"
        ],
        "output": "6859",
        "label": "Trapezoid quadratic reference ORIGINAL power difference2"
      }
    },
    {
      "id": "S058",
      "uses": [
        "G001",
        "S057"
      ],
      "before": "Trapezoid quadratic reference ORIGINAL integrated term2 := mul(1/3, 6859)",
      "after": "Trapezoid quadratic reference ORIGINAL integrated term2 = 6859/3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Trapezoid quadratic reference ORIGINAL integrated term2，不以小數近似替代。",
          "en": "Recompute Trapezoid quadratic reference ORIGINAL integrated term2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/3",
          "6859"
        ],
        "output": "6859/3",
        "label": "Trapezoid quadratic reference ORIGINAL integrated term2"
      }
    },
    {
      "id": "S059",
      "uses": [
        "G001",
        "S058"
      ],
      "before": "Trapezoid quadratic reference ORIGINAL integral running sum2 := sum(1007/2, 6859/3)",
      "after": "Trapezoid quadratic reference ORIGINAL integral running sum2 = 16739/6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Trapezoid quadratic reference ORIGINAL integral running sum2，不以小數近似替代。",
          "en": "Recompute Trapezoid quadratic reference ORIGINAL integral running sum2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1007/2",
          "6859/3"
        ],
        "output": "16739/6",
        "label": "Trapezoid quadratic reference ORIGINAL integral running sum2"
      }
    },
    {
      "id": "S060",
      "uses": [
        "G001",
        "S059"
      ],
      "before": "Trapezoid ORIGINAL signed error I minus T := sub(16739/6, 3762)",
      "after": "Trapezoid ORIGINAL signed error I minus T = -5833/6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Trapezoid ORIGINAL signed error I minus T，不以小數近似替代。",
          "en": "Recompute Trapezoid ORIGINAL signed error I minus T with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "16739/6",
          "3762"
        ],
        "output": "-5833/6",
        "label": "Trapezoid ORIGINAL signed error I minus T"
      }
    },
    {
      "id": "S061",
      "uses": [
        "G001",
        "S060"
      ],
      "before": "source roles = [[17, 1, 1], [0, 1, 19]]",
      "after": "derived = [3762, 16739/6, -5833/6, 5833/6, 否]",
      "relation": "implies",
      "rule_id": "Trapezoid_ACTUAL_full_quadratic_reference_IminusT_NOT_general_error_theorem",
      "conditions": [],
      "explanation": {
        "zh-Hant": "提供完整quadratic才可精確積分作reference，signed error固定I-T、abs另列；這不是由endpoint table推唯一integrand，也不自動給一般二階導數誤差界或order。",
        "en": "The supplied WHOLE quadratic permits an exact reference integral.Signed error is I-T with absolute error separate;this neither infers a unique integrand from samples nor supplies a general second-derivative/error-order theorem."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查完整來源資料、原操作數及本步條件；有限核對、條件式理由、形式證明與發布資格分開。",
          "en": "Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      "quadratic_reference",
      [
        "17",
        "1",
        "1"
      ],
      [
        "0",
        "1",
        "19"
      ],
      "3762",
      "16739/6",
      "-5833/6",
      "5833/6",
      false,
      "exact_polynomial_reference_error_is_not_a_general_second_derivative_error_bound"
    ]
  },
  "conclusion": {
    "canonical_value": "(quadratic_reference, [17, 1, 1], [0, 1, 19], 3762, 16739/6, -5833/6, 5833/6, 否, exact_polynomial_reference_error_is_not_a_general_second_derivative_error_bound)",
    "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-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations",
      "batch_path": "public/data/questions/batches/batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations/questions.jsonl",
      "batch_sha256": "42c2a2253cda785cc89b1d9f6e2ca25dc3eedad80f592c69c8f2637ae2511f18",
      "original_line": 92,
      "original_line_sha256": "4af821a7811581d51d4f43ccd5b1f6d4ee7b18157d9c7f99df25855c86bebb71",
      "question_compact_sha256": "c43b15304d369781a5878698ac7e7d57c58b98a49ae7c7026bcbed75e0d03876",
      "english_row_compact_sha256": "a7df371fd591046b7996d1dffaea5810201da81227effe9058b04b688737b308",
      "english_manifest_path": "public/data/i18n/editions/batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-trapezoidal-quadrature": {
        "sha256": "bd70a77af9c5b1fc3dbca7b29f06de4dbb579be5a6c48bbb16df6141f4f90001",
        "english_sha256": "00cf21dfab1295bd666c4883317d8bf1a960f6d5c2020d9b82fa4147c1779e80",
        "original_path": "public/data/mko/trapezoidal-quadrature.json",
        "english_path": "public/data/i18n/editions/mko-trapezoidal-quadrature.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "b5763df142e86ef7f9787b6ea86050afc60bd3c5c53538d4839606827cc44686",
    "original_candidate_sha256": "1658580b06c3bdd6defd9c569720f6d6982433ec37270263352b75ef9b3e5d1c",
    "original_candidate_line_sha256": "1658580b06c3bdd6defd9c569720f6d6982433ec37270263352b75ef9b3e5d1c"
  },
  "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-20261026-010-p56",
    "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": "6af367dae24a584f3596525c4953bbc2115343efd68cc9083693eb5c1ef2e683",
      "question_first_witness_sha256": "611eb69f3716404b968296a8b0d7d8c61b5ca18129d406f1ff9c7c11e10ffb4e"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 61,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "4e25da2b53430fb8b2e5fa68696cfd02288ed4e3192989888255ab40e4423f8b",
      "arithmetic_steps": 55
    },
    "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."
  }
}